| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fzfi 14014 | . . . . . 6
⊢
(0...𝐾) ∈
Fin | 
| 2 |  | fzfi 14014 | . . . . . 6
⊢
(1...𝑁) ∈
Fin | 
| 3 |  | mapfi 9389 | . . . . . 6
⊢
(((0...𝐾) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((0...𝐾)
↑m (1...𝑁))
∈ Fin) | 
| 4 | 1, 2, 3 | mp2an 692 | . . . . 5
⊢
((0...𝐾)
↑m (1...𝑁))
∈ Fin | 
| 5 |  | fzfi 14014 | . . . . 5
⊢
(0...(𝑁 − 1))
∈ Fin | 
| 6 |  | mapfi 9389 | . . . . 5
⊢
((((0...𝐾)
↑m (1...𝑁))
∈ Fin ∧ (0...(𝑁
− 1)) ∈ Fin) → (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) ∈
Fin) | 
| 7 | 4, 5, 6 | mp2an 692 | . . . 4
⊢
(((0...𝐾)
↑m (1...𝑁))
↑m (0...(𝑁
− 1))) ∈ Fin | 
| 8 | 7 | a1i 11 | . . 3
⊢ (𝜑 → (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) ∈
Fin) | 
| 9 |  | 2z 12651 | . . . 4
⊢ 2 ∈
ℤ | 
| 10 | 9 | a1i 11 | . . 3
⊢ (𝜑 → 2 ∈
ℤ) | 
| 11 |  | fzofi 14016 | . . . . . . . 8
⊢
(0..^𝐾) ∈
Fin | 
| 12 |  | mapfi 9389 | . . . . . . . 8
⊢
(((0..^𝐾) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((0..^𝐾)
↑m (1...𝑁))
∈ Fin) | 
| 13 | 11, 2, 12 | mp2an 692 | . . . . . . 7
⊢
((0..^𝐾)
↑m (1...𝑁))
∈ Fin | 
| 14 |  | mapfi 9389 | . . . . . . . . 9
⊢
(((1...𝑁) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((1...𝑁)
↑m (1...𝑁))
∈ Fin) | 
| 15 | 2, 2, 14 | mp2an 692 | . . . . . . . 8
⊢
((1...𝑁)
↑m (1...𝑁))
∈ Fin | 
| 16 |  | f1of 6847 | . . . . . . . . . 10
⊢ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁)) | 
| 17 | 16 | ss2abi 4066 | . . . . . . . . 9
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} | 
| 18 |  | ovex 7465 | . . . . . . . . . 10
⊢
(1...𝑁) ∈
V | 
| 19 | 18, 18 | mapval 8879 | . . . . . . . . 9
⊢
((1...𝑁)
↑m (1...𝑁))
= {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} | 
| 20 | 17, 19 | sseqtrri 4032 | . . . . . . . 8
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑m (1...𝑁)) | 
| 21 |  | ssfi 9214 | . . . . . . . 8
⊢
((((1...𝑁)
↑m (1...𝑁))
∈ Fin ∧ {𝑓 ∣
𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑m (1...𝑁))) → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) | 
| 22 | 15, 20, 21 | mp2an 692 | . . . . . . 7
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin | 
| 23 |  | xpfi 9359 | . . . . . . 7
⊢
((((0..^𝐾)
↑m (1...𝑁))
∈ Fin ∧ {𝑓 ∣
𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) → (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin) | 
| 24 | 13, 22, 23 | mp2an 692 | . . . . . 6
⊢
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin | 
| 25 |  | fzfi 14014 | . . . . . 6
⊢
(0...𝑁) ∈
Fin | 
| 26 |  | xpfi 9359 | . . . . . 6
⊢
(((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin ∧ (0...𝑁) ∈ Fin) → ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin) | 
| 27 | 24, 25, 26 | mp2an 692 | . . . . 5
⊢
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin | 
| 28 |  | rabfi 9304 | . . . . 5
⊢
(((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin) | 
| 29 | 27, 28 | ax-mp 5 | . . . 4
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin | 
| 30 |  | hashcl 14396 | . . . . 5
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈
ℕ0) | 
| 31 | 30 | nn0zd 12641 | . . . 4
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈ ℤ) | 
| 32 | 29, 31 | mp1i 13 | . . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) →
(♯‘{𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈ ℤ) | 
| 33 |  | dfrex2 3072 | . . . . 5
⊢
(∃𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ↔ ¬ ∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) | 
| 34 |  | nfv 1913 | . . . . . 6
⊢
Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) | 
| 35 |  | nfcv 2904 | . . . . . . 7
⊢
Ⅎ𝑡2 | 
| 36 |  | nfcv 2904 | . . . . . . 7
⊢
Ⅎ𝑡
∥ | 
| 37 |  | nfcv 2904 | . . . . . . . 8
⊢
Ⅎ𝑡♯ | 
| 38 |  | nfrab1 3456 | . . . . . . . 8
⊢
Ⅎ𝑡{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} | 
| 39 | 37, 38 | nffv 6915 | . . . . . . 7
⊢
Ⅎ𝑡(♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) | 
| 40 | 35, 36, 39 | nfbr 5189 | . . . . . 6
⊢
Ⅎ𝑡2 ∥
(♯‘{𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) | 
| 41 |  | neq0 4351 | . . . . . . . . . . . 12
⊢ (¬
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ ↔
∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) | 
| 42 |  | iddvds 16308 | . . . . . . . . . . . . . . . . 17
⊢ (2 ∈
ℤ → 2 ∥ 2) | 
| 43 | 9, 42 | ax-mp 5 | . . . . . . . . . . . . . . . 16
⊢ 2 ∥
2 | 
| 44 |  | vex 3483 | . . . . . . . . . . . . . . . . . . 19
⊢ 𝑠 ∈ V | 
| 45 |  | hashsng 14409 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ V →
(♯‘{𝑠}) =
1) | 
| 46 | 44, 45 | ax-mp 5 | . . . . . . . . . . . . . . . . . 18
⊢
(♯‘{𝑠})
= 1 | 
| 47 | 46 | oveq2i 7443 | . . . . . . . . . . . . . . . . 17
⊢ (1 +
(♯‘{𝑠})) = (1 +
1) | 
| 48 |  | df-2 12330 | . . . . . . . . . . . . . . . . 17
⊢ 2 = (1 +
1) | 
| 49 | 47, 48 | eqtr4i 2767 | . . . . . . . . . . . . . . . 16
⊢ (1 +
(♯‘{𝑠})) =
2 | 
| 50 | 43, 49 | breqtrri 5169 | . . . . . . . . . . . . . . 15
⊢ 2 ∥
(1 + (♯‘{𝑠})) | 
| 51 |  | rabfi 9304 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∈
Fin) | 
| 52 |  | diffi 9216 | . . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∈ Fin →
({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin) | 
| 53 | 27, 51, 52 | mp2b 10 | . . . . . . . . . . . . . . . . . . 19
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin | 
| 54 |  | snfi 9084 | . . . . . . . . . . . . . . . . . . 19
⊢ {𝑠} ∈ Fin | 
| 55 |  | disjdifr 4472 | . . . . . . . . . . . . . . . . . . 19
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅ | 
| 56 |  | hashun 14422 | . . . . . . . . . . . . . . . . . . 19
⊢ ((({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin ∧ {𝑠} ∈ Fin ∧ (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅) → (♯‘(({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠}))) | 
| 57 | 53, 54, 55, 56 | mp3an 1462 | . . . . . . . . . . . . . . . . . 18
⊢
(♯‘(({𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠})) | 
| 58 |  | difsnid 4809 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠}) = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) | 
| 59 | 58 | fveq2d 6909 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} →
(♯‘(({𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) | 
| 60 | 57, 59 | eqtr3id 2790 | . . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} →
((♯‘({𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠})) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) | 
| 61 | 60 | adantl 481 | . . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) →
((♯‘({𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠})) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) | 
| 62 |  | poimir.0 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| 63 | 62 | ad3antrrr 730 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑁 ∈
ℕ) | 
| 64 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (2nd ‘𝑡) = (2nd ‘𝑢)) | 
| 65 | 64 | breq2d 5154 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 = 𝑢 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑢))) | 
| 66 | 65 | ifbid 4548 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑡 = 𝑢 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1))) | 
| 67 | 66 | csbeq1d 3902 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) | 
| 68 |  | 2fveq3 6910 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 = 𝑢 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑢))) | 
| 69 |  | 2fveq3 6910 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑡 = 𝑢 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑢))) | 
| 70 | 69 | imaeq1d 6076 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑡 = 𝑢 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑢)) “
(1...𝑗))) | 
| 71 | 70 | xpeq1d 5713 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1})) | 
| 72 | 69 | imaeq1d 6076 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑡 = 𝑢 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑢)) “ ((𝑗 + 1)...𝑁))) | 
| 73 | 72 | xpeq1d 5713 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})) | 
| 74 | 71, 73 | uneq12d 4168 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 = 𝑢 → ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) | 
| 75 | 68, 74 | oveq12d 7450 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑡 = 𝑢 → ((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) | 
| 76 | 75 | csbeq2dv 3905 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) | 
| 77 | 67, 76 | eqtrd 2776 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) | 
| 78 | 77 | mpteq2dv 5243 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 = 𝑢 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))) | 
| 79 |  | breq1 5145 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑤 → (𝑦 < (2nd ‘𝑢) ↔ 𝑤 < (2nd ‘𝑢))) | 
| 80 |  | id 22 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤) | 
| 81 |  | oveq1 7439 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1)) | 
| 82 | 79, 80, 81 | ifbieq12d 4553 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 = 𝑤 → if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) = if(𝑤 < (2nd ‘𝑢), 𝑤, (𝑤 + 1))) | 
| 83 | 82 | csbeq1d 3902 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = 𝑤 → ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) | 
| 84 |  | oveq2 7440 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖)) | 
| 85 | 84 | imaeq2d 6077 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 = 𝑖 → ((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑢)) “
(1...𝑖))) | 
| 86 | 85 | xpeq1d 5713 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 = 𝑖 → (((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑢)) “ (1...𝑖)) × {1})) | 
| 87 |  | oveq1 7439 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 = 𝑖 → (𝑗 + 1) = (𝑖 + 1)) | 
| 88 | 87 | oveq1d 7447 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 = 𝑖 → ((𝑗 + 1)...𝑁) = ((𝑖 + 1)...𝑁)) | 
| 89 | 88 | imaeq2d 6077 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 = 𝑖 → ((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑢)) “ ((𝑖 + 1)...𝑁))) | 
| 90 | 89 | xpeq1d 5713 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 = 𝑖 → (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})) | 
| 91 | 86, 90 | uneq12d 4168 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 = 𝑖 → ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))) | 
| 92 | 91 | oveq2d 7448 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 = 𝑖 → ((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))) | 
| 93 | 92 | cbvcsbv 3910 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢
⦋if(𝑤
< (2nd ‘𝑢), 𝑤, (𝑤 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))) | 
| 94 | 83, 93 | eqtrdi 2792 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = 𝑤 → ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))) | 
| 95 | 94 | cbvmptv 5254 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))) | 
| 96 | 78, 95 | eqtrdi 2792 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑢 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))) | 
| 97 | 96 | eqeq2d 2747 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑢 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))))) | 
| 98 | 97 | cbvrabv 3446 | . . . . . . . . . . . . . . . . . . 19
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = {𝑢 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘f + ((((2nd
‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))} | 
| 99 |  | elmapi 8890 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) → 𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) | 
| 100 | 99 | ad3antlr 731 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) | 
| 101 |  | simpr 484 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) | 
| 102 |  | simpl 482 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
((∃𝑝 ∈
ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) | 
| 103 | 102 | ralimi 3082 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) | 
| 104 | 103 | ad2antlr 727 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) | 
| 105 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑚 → (𝑝‘𝑛) = (𝑝‘𝑚)) | 
| 106 | 105 | neeq1d 2999 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑚 → ((𝑝‘𝑛) ≠ 0 ↔ (𝑝‘𝑚) ≠ 0)) | 
| 107 | 106 | rexbidv 3178 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑚) ≠ 0)) | 
| 108 |  | fveq1 6904 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑝 = 𝑞 → (𝑝‘𝑚) = (𝑞‘𝑚)) | 
| 109 | 108 | neeq1d 2999 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = 𝑞 → ((𝑝‘𝑚) ≠ 0 ↔ (𝑞‘𝑚) ≠ 0)) | 
| 110 | 109 | cbvrexvw 3237 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑝 ∈ ran
𝑥(𝑝‘𝑚) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0) | 
| 111 | 107, 110 | bitrdi 287 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0)) | 
| 112 | 111 | rspccva 3620 | . . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑛 ∈
(1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0) | 
| 113 | 104, 112 | sylan 580 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0) | 
| 114 |  | simpr 484 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
((∃𝑝 ∈
ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) | 
| 115 | 114 | ralimi 3082 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) | 
| 116 | 115 | ad2antlr 727 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) | 
| 117 | 105 | neeq1d 2999 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑚 → ((𝑝‘𝑛) ≠ 𝐾 ↔ (𝑝‘𝑚) ≠ 𝐾)) | 
| 118 | 117 | rexbidv 3178 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑚) ≠ 𝐾)) | 
| 119 | 108 | neeq1d 2999 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = 𝑞 → ((𝑝‘𝑚) ≠ 𝐾 ↔ (𝑞‘𝑚) ≠ 𝐾)) | 
| 120 | 119 | cbvrexvw 3237 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑝 ∈ ran
𝑥(𝑝‘𝑚) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾) | 
| 121 | 118, 120 | bitrdi 287 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾)) | 
| 122 | 121 | rspccva 3620 | . . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑛 ∈
(1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾) | 
| 123 | 116, 122 | sylan 580 | . . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾) | 
| 124 | 63, 98, 100, 101, 113, 123 | poimirlem22 37650 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∃!𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠) | 
| 125 |  | eldifsn 4785 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ↔ (𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠)) | 
| 126 | 125 | eubii 2584 | . . . . . . . . . . . . . . . . . . 19
⊢
(∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠)) | 
| 127 | 53 | elexi 3502 | . . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ V | 
| 128 |  | euhash1 14460 | . . . . . . . . . . . . . . . . . . . 20
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ V →
((♯‘({𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}))) | 
| 129 | 127, 128 | ax-mp 5 | . . . . . . . . . . . . . . . . . . 19
⊢
((♯‘({𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) | 
| 130 |  | df-reu 3380 | . . . . . . . . . . . . . . . . . . 19
⊢
(∃!𝑧 ∈
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠 ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠)) | 
| 131 | 126, 129,
130 | 3bitr4ri 304 | . . . . . . . . . . . . . . . . . 18
⊢
(∃!𝑧 ∈
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠 ↔ (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1) | 
| 132 | 124, 131 | sylib 218 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) →
(♯‘({𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1) | 
| 133 | 132 | oveq1d 7447 | . . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) →
((♯‘({𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠})) = (1 + (♯‘{𝑠}))) | 
| 134 | 61, 133 | eqtr3d 2778 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) →
(♯‘{𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (1 +
(♯‘{𝑠}))) | 
| 135 | 50, 134 | breqtrrid 5180 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 2 ∥
(♯‘{𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) | 
| 136 | 135 | ex 412 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 2 ∥
(♯‘{𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) | 
| 137 | 136 | exlimdv 1932 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 2 ∥
(♯‘{𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) | 
| 138 | 41, 137 | biimtrid 242 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (¬ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → 2
∥ (♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) | 
| 139 |  | dvds0 16310 | . . . . . . . . . . . . . 14
⊢ (2 ∈
ℤ → 2 ∥ 0) | 
| 140 | 9, 139 | ax-mp 5 | . . . . . . . . . . . . 13
⊢ 2 ∥
0 | 
| 141 |  | hash0 14407 | . . . . . . . . . . . . 13
⊢
(♯‘∅) = 0 | 
| 142 | 140, 141 | breqtrri 5169 | . . . . . . . . . . . 12
⊢ 2 ∥
(♯‘∅) | 
| 143 |  | fveq2 6905 | . . . . . . . . . . . 12
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ →
(♯‘{𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) =
(♯‘∅)) | 
| 144 | 142, 143 | breqtrrid 5180 | . . . . . . . . . . 11
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → 2
∥ (♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) | 
| 145 | 138, 144 | pm2.61d2 181 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) | 
| 146 | 145 | ex 412 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) →
(∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) | 
| 147 | 146 | adantld 490 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) →
(((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) | 
| 148 |  | iba 527 | . . . . . . . . . . 11
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) | 
| 149 | 148 | rabbidv 3443 | . . . . . . . . . 10
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) | 
| 150 | 149 | fveq2d 6909 | . . . . . . . . 9
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) | 
| 151 | 150 | breq2d 5154 | . . . . . . . 8
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ↔ 2 ∥
(♯‘{𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) | 
| 152 | 147, 151 | mpbidi 241 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) →
(((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) | 
| 153 | 152 | a1d 25 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})))) | 
| 154 | 34, 40, 153 | rexlimd 3265 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) →
(∃𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) | 
| 155 | 33, 154 | biimtrrid 243 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (¬
∀𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) | 
| 156 |  | simpr 484 | . . . . . . . . 9
⊢ ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) | 
| 157 | 156 | con3i 154 | . . . . . . . 8
⊢ (¬
((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) | 
| 158 | 157 | ralimi 3082 | . . . . . . 7
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → ∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) | 
| 159 |  | rabeq0 4387 | . . . . . . 7
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ∅ ↔ ∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) | 
| 160 | 158, 159 | sylibr 234 | . . . . . 6
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ∅) | 
| 161 | 160 | fveq2d 6909 | . . . . 5
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) =
(♯‘∅)) | 
| 162 | 142, 161 | breqtrrid 5180 | . . . 4
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) | 
| 163 | 155, 162 | pm2.61d2 181 | . . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → 2 ∥
(♯‘{𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) | 
| 164 | 8, 10, 32, 163 | fsumdvds 16346 | . 2
⊢ (𝜑 → 2 ∥ Σ𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 −
1)))(♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) | 
| 165 |  | rabfi 9304 | . . . . 5
⊢
(((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∈ Fin) | 
| 166 | 27, 165 | ax-mp 5 | . . . 4
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∈ Fin | 
| 167 |  | simp1 1136 | . . . . . . 7
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) | 
| 168 |  | sneq 4635 | . . . . . . . . . . . . 13
⊢
((2nd ‘𝑡) = 𝑁 → {(2nd ‘𝑡)} = {𝑁}) | 
| 169 | 168 | difeq2d 4125 | . . . . . . . . . . . 12
⊢
((2nd ‘𝑡) = 𝑁 → ((0...𝑁) ∖ {(2nd ‘𝑡)}) = ((0...𝑁) ∖ {𝑁})) | 
| 170 |  | difun2 4480 | . . . . . . . . . . . . 13
⊢
(((0...(𝑁 −
1)) ∪ {𝑁}) ∖
{𝑁}) = ((0...(𝑁 − 1)) ∖ {𝑁}) | 
| 171 | 62 | nnnn0d 12589 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈
ℕ0) | 
| 172 |  | nn0uz 12921 | . . . . . . . . . . . . . . . . . 18
⊢
ℕ0 = (ℤ≥‘0) | 
| 173 | 171, 172 | eleqtrdi 2850 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘0)) | 
| 174 |  | fzm1 13648 | . . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘0) → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) | 
| 175 | 173, 174 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) | 
| 176 |  | elun 4152 | . . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁})) | 
| 177 |  | velsn 4641 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ {𝑁} ↔ 𝑛 = 𝑁) | 
| 178 | 177 | orbi2i 912 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) | 
| 179 | 176, 178 | bitri 275 | . . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) | 
| 180 | 175, 179 | bitr4di 289 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}))) | 
| 181 | 180 | eqrdv 2734 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁})) | 
| 182 | 181 | difeq1d 4124 | . . . . . . . . . . . . 13
⊢ (𝜑 → ((0...𝑁) ∖ {𝑁}) = (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁})) | 
| 183 | 62 | nnzd 12642 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℤ) | 
| 184 |  | uzid 12894 | . . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
(ℤ≥‘𝑁)) | 
| 185 |  | uznfz 13651 | . . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘𝑁) → ¬ 𝑁 ∈ (0...(𝑁 − 1))) | 
| 186 | 183, 184,
185 | 3syl 18 | . . . . . . . . . . . . . 14
⊢ (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1))) | 
| 187 |  | disjsn 4710 | . . . . . . . . . . . . . . 15
⊢
(((0...(𝑁 −
1)) ∩ {𝑁}) = ∅
↔ ¬ 𝑁 ∈
(0...(𝑁 −
1))) | 
| 188 |  | disj3 4453 | . . . . . . . . . . . . . . 15
⊢
(((0...(𝑁 −
1)) ∩ {𝑁}) = ∅
↔ (0...(𝑁 − 1))
= ((0...(𝑁 − 1))
∖ {𝑁})) | 
| 189 | 187, 188 | bitr3i 277 | . . . . . . . . . . . . . 14
⊢ (¬
𝑁 ∈ (0...(𝑁 − 1)) ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁})) | 
| 190 | 186, 189 | sylib 218 | . . . . . . . . . . . . 13
⊢ (𝜑 → (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁})) | 
| 191 | 170, 182,
190 | 3eqtr4a 2802 | . . . . . . . . . . . 12
⊢ (𝜑 → ((0...𝑁) ∖ {𝑁}) = (0...(𝑁 − 1))) | 
| 192 | 169, 191 | sylan9eqr 2798 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (2nd
‘𝑡) = 𝑁) → ((0...𝑁) ∖ {(2nd ‘𝑡)}) = (0...(𝑁 − 1))) | 
| 193 | 192 | rexeqdv 3326 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (2nd
‘𝑡) = 𝑁) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) | 
| 194 | 193 | biimprd 248 | . . . . . . . . 9
⊢ ((𝜑 ∧ (2nd
‘𝑡) = 𝑁) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) | 
| 195 | 194 | ralimdv 3168 | . . . . . . . 8
⊢ ((𝜑 ∧ (2nd
‘𝑡) = 𝑁) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) | 
| 196 | 195 | expimpd 453 | . . . . . . 7
⊢ (𝜑 → (((2nd
‘𝑡) = 𝑁 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) | 
| 197 | 167, 196 | sylan2i 606 | . . . . . 6
⊢ (𝜑 → (((2nd
‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) | 
| 198 | 197 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) | 
| 199 | 198 | ss2rabdv 4075 | . . . 4
⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) | 
| 200 |  | hashssdif 14452 | . . . 4
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∈ Fin ∧ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}))) | 
| 201 | 166, 199,
200 | sylancr 587 | . . 3
⊢ (𝜑 → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}))) | 
| 202 | 62 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → 𝑁 ∈ ℕ) | 
| 203 |  | poimirlem28.1 | . . . . . . . . . 10
⊢ (𝑝 = ((1st ‘𝑠) ∘f +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) | 
| 204 |  | poimirlem28.2 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) | 
| 205 | 204 | adantlr 715 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) | 
| 206 |  | xp1st 8047 | . . . . . . . . . . . 12
⊢ (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) | 
| 207 |  | xp1st 8047 | . . . . . . . . . . . 12
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st
‘(1st ‘𝑡)) ∈ ((0..^𝐾) ↑m (1...𝑁))) | 
| 208 |  | elmapi 8890 | . . . . . . . . . . . 12
⊢
((1st ‘(1st ‘𝑡)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st
‘(1st ‘𝑡)):(1...𝑁)⟶(0..^𝐾)) | 
| 209 | 206, 207,
208 | 3syl 18 | . . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st
‘(1st ‘𝑡)):(1...𝑁)⟶(0..^𝐾)) | 
| 210 | 209 | adantl 481 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (1st
‘(1st ‘𝑡)):(1...𝑁)⟶(0..^𝐾)) | 
| 211 |  | xp2nd 8048 | . . . . . . . . . . . . 13
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) | 
| 212 |  | fvex 6918 | . . . . . . . . . . . . . 14
⊢
(2nd ‘(1st ‘𝑡)) ∈ V | 
| 213 |  | f1oeq1 6835 | . . . . . . . . . . . . . 14
⊢ (𝑓 = (2nd
‘(1st ‘𝑡)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))) | 
| 214 | 212, 213 | elab 3678 | . . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)) | 
| 215 | 211, 214 | sylib 218 | . . . . . . . . . . . 12
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)) | 
| 216 | 206, 215 | syl 17 | . . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)) | 
| 217 | 216 | adantl 481 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)) | 
| 218 |  | xp2nd 8048 | . . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑡) ∈ (0...𝑁)) | 
| 219 | 218 | adantl 481 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (2nd ‘𝑡) ∈ (0...𝑁)) | 
| 220 | 202, 203,
205, 210, 217, 219 | poimirlem24 37652 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) | 
| 221 | 206 | adantl 481 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (1st ‘𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) | 
| 222 |  | 1st2nd2 8054 | . . . . . . . . . . . . . . 15
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘𝑡) = 〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉) | 
| 223 | 222 | csbeq1d 3902 | . . . . . . . . . . . . . 14
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶) | 
| 224 | 223 | eqeq2d 2747 | . . . . . . . . . . . . 13
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶)) | 
| 225 | 224 | rexbidv 3178 | . . . . . . . . . . . 12
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶)) | 
| 226 | 225 | ralbidv 3177 | . . . . . . . . . . 11
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶)) | 
| 227 | 226 | anbi1d 631 | . . . . . . . . . 10
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) | 
| 228 | 221, 227 | syl 17 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) | 
| 229 | 220, 228 | bitr4d 282 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) | 
| 230 | 99 | frnd 6743 | . . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) → ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) | 
| 231 | 230 | anim2i 617 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)))) | 
| 232 |  | dfss3 3971 | . . . . . . . . . . . . . 14
⊢
((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵)) | 
| 233 |  | vex 3483 | . . . . . . . . . . . . . . . 16
⊢ 𝑛 ∈ V | 
| 234 |  | eqid 2736 | . . . . . . . . . . . . . . . . 17
⊢ (𝑝 ∈ ran 𝑥 ↦ 𝐵) = (𝑝 ∈ ran 𝑥 ↦ 𝐵) | 
| 235 | 234 | elrnmpt 5968 | . . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ V → (𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) | 
| 236 | 233, 235 | ax-mp 5 | . . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) | 
| 237 | 236 | ralbii 3092 | . . . . . . . . . . . . . 14
⊢
(∀𝑛 ∈
(0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) | 
| 238 | 232, 237 | sylbb 219 | . . . . . . . . . . . . 13
⊢
((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) → ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) | 
| 239 |  | 1eluzge0 12935 | . . . . . . . . . . . . . . . . 17
⊢ 1 ∈
(ℤ≥‘0) | 
| 240 |  | fzss1 13604 | . . . . . . . . . . . . . . . . 17
⊢ (1 ∈
(ℤ≥‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))) | 
| 241 |  | ssralv 4051 | . . . . . . . . . . . . . . . . 17
⊢
((1...(𝑁 − 1))
⊆ (0...(𝑁 − 1))
→ (∀𝑛 ∈
(0...(𝑁 −
1))∃𝑝 ∈ ran
𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) | 
| 242 | 239, 240,
241 | mp2b 10 | . . . . . . . . . . . . . . . 16
⊢
(∀𝑛 ∈
(0...(𝑁 −
1))∃𝑝 ∈ ran
𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) | 
| 243 | 62 | nncnd 12283 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝑁 ∈ ℂ) | 
| 244 |  | npcan1 11689 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) | 
| 245 | 243, 244 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) | 
| 246 |  | peano2zm 12662 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈
ℤ) | 
| 247 |  | uzid 12894 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑁 − 1) ∈ ℤ
→ (𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1))) | 
| 248 |  | peano2uz 12944 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) | 
| 249 | 183, 246,
247, 248 | 4syl 19 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) | 
| 250 | 245, 249 | eqeltrrd 2841 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) | 
| 251 |  | fzss2 13605 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈
(ℤ≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁)) | 
| 252 | 250, 251 | syl 17 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁)) | 
| 253 | 252 | sselda 3982 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁)) | 
| 254 | 253 | adantlr 715 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁)) | 
| 255 |  | simplr 768 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) | 
| 256 |  | ssel2 3977 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ran
𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝 ∈ ((0...𝐾) ↑m (1...𝑁))) | 
| 257 |  | elmapi 8890 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 ∈ ((0...𝐾) ↑m (1...𝑁)) → 𝑝:(1...𝑁)⟶(0...𝐾)) | 
| 258 | 256, 257 | syl 17 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ran
𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾)) | 
| 259 | 255, 258 | sylan 580 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾)) | 
| 260 |  | poimirlem28.3 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) | 
| 261 |  | elfzelz 13565 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ) | 
| 262 | 261 | zred 12724 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ) | 
| 263 | 262 | ltnrd 11396 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 ∈ (1...𝑁) → ¬ 𝑛 < 𝑛) | 
| 264 |  | breq1 5145 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑛 = 𝐵 → (𝑛 < 𝑛 ↔ 𝐵 < 𝑛)) | 
| 265 | 264 | notbid 318 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 = 𝐵 → (¬ 𝑛 < 𝑛 ↔ ¬ 𝐵 < 𝑛)) | 
| 266 | 263, 265 | syl5ibcom 245 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑛 ∈ (1...𝑁) → (𝑛 = 𝐵 → ¬ 𝐵 < 𝑛)) | 
| 267 | 266 | necon2ad 2954 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ (1...𝑁) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵)) | 
| 268 | 267 | 3ad2ant1 1133 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵)) | 
| 269 | 268 | adantl 481 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵)) | 
| 270 | 260, 269 | mpd 15 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝑛 ≠ 𝐵) | 
| 271 | 270 | 3exp2 1354 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝‘𝑛) = 0 → 𝑛 ≠ 𝐵)))) | 
| 272 | 271 | imp31 417 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝‘𝑛) = 0 → 𝑛 ≠ 𝐵)) | 
| 273 | 272 | necon2d 2962 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0)) | 
| 274 | 273 | adantllr 719 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0)) | 
| 275 | 259, 274 | syldan 591 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0)) | 
| 276 | 275 | reximdva 3167 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) | 
| 277 | 254, 276 | syldan 591 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) | 
| 278 | 277 | ralimdva 3166 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) → (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) | 
| 279 | 278 | imp 406 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) | 
| 280 | 242, 279 | sylan2 593 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) | 
| 281 | 280 | biantrurd 532 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) | 
| 282 |  | nnuz 12922 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ℕ =
(ℤ≥‘1) | 
| 283 | 62, 282 | eleqtrdi 2850 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘1)) | 
| 284 |  | fzm1 13648 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ≥‘1) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) | 
| 285 | 283, 284 | syl 17 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) | 
| 286 |  | elun 4152 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁})) | 
| 287 | 177 | orbi2i 912 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) | 
| 288 | 286, 287 | bitri 275 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) | 
| 289 | 285, 288 | bitr4di 289 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↔ 𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}))) | 
| 290 | 289 | eqrdv 2734 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁})) | 
| 291 | 290 | raleqdv 3325 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∀𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) | 
| 292 |  | ralunb 4196 | . . . . . . . . . . . . . . . . 17
⊢
(∀𝑛 ∈
((1...(𝑁 − 1)) ∪
{𝑁})∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) | 
| 293 | 291, 292 | bitrdi 287 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0))) | 
| 294 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑁 → (𝑝‘𝑛) = (𝑝‘𝑁)) | 
| 295 | 294 | neeq1d 2999 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 𝑁 → ((𝑝‘𝑛) ≠ 0 ↔ (𝑝‘𝑁) ≠ 0)) | 
| 296 | 295 | rexbidv 3178 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 𝑁 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) | 
| 297 | 296 | ralsng 4674 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℕ →
(∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) | 
| 298 | 62, 297 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) | 
| 299 | 298 | anbi2d 630 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) | 
| 300 | 293, 299 | bitrd 279 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) | 
| 301 | 300 | ad2antrr 726 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) | 
| 302 |  | 0z 12626 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ 0 ∈
ℤ | 
| 303 |  | 1z 12649 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ 1 ∈
ℤ | 
| 304 |  | fzshftral 13656 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((0
∈ ℤ ∧ (𝑁
− 1) ∈ ℤ ∧ 1 ∈ ℤ) → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) | 
| 305 | 302, 303,
304 | mp3an13 1453 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑁 − 1) ∈ ℤ
→ (∀𝑛 ∈
(0...(𝑁 −
1))∃𝑝 ∈ ran
𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) | 
| 306 | 183, 246,
305 | 3syl 18 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) | 
| 307 |  | 0p1e1 12389 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (0 + 1) =
1 | 
| 308 | 307 | a1i 11 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (0 + 1) =
1) | 
| 309 | 308, 245 | oveq12d 7450 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((0 + 1)...((𝑁 − 1) + 1)) = (1...𝑁)) | 
| 310 | 309 | raleqdv 3325 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) | 
| 311 | 306, 310 | bitrd 279 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) | 
| 312 |  | ovex 7465 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 − 1) ∈
V | 
| 313 |  | eqeq1 2740 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = (𝑚 − 1) → (𝑛 = 𝐵 ↔ (𝑚 − 1) = 𝐵)) | 
| 314 | 313 | rexbidv 3178 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = (𝑚 − 1) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)) | 
| 315 | 312, 314 | sbcie 3829 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
([(𝑚 −
1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵) | 
| 316 | 315 | ralbii 3092 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑚 ∈
(1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵) | 
| 317 |  | oveq1 7439 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1)) | 
| 318 | 317 | eqeq1d 2738 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 = 𝑛 → ((𝑚 − 1) = 𝐵 ↔ (𝑛 − 1) = 𝐵)) | 
| 319 | 318 | rexbidv 3178 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 = 𝑛 → (∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)) | 
| 320 | 319 | cbvralvw 3236 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑚 ∈
(1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) | 
| 321 | 316, 320 | bitri 275 | . . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑚 ∈
(1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) | 
| 322 | 311, 321 | bitrdi 287 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)) | 
| 323 | 322 | biimpa 476 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) | 
| 324 | 323 | adantlr 715 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) | 
| 325 |  | poimirlem28.4 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1)) | 
| 326 | 325 | necomd 2995 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → (𝑛 − 1) ≠ 𝐵) | 
| 327 | 326 | 3exp2 1354 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝‘𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵)))) | 
| 328 | 327 | imp31 417 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝‘𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵)) | 
| 329 | 328 | necon2d 2962 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾)) | 
| 330 | 329 | adantllr 719 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾)) | 
| 331 | 259, 330 | syldan 591 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾)) | 
| 332 | 331 | reximdva 3167 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) | 
| 333 | 332 | ralimdva 3166 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) | 
| 334 | 333 | imp 406 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) | 
| 335 | 324, 334 | syldan 591 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) | 
| 336 | 335 | biantrud 531 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) | 
| 337 |  | r19.26 3110 | . . . . . . . . . . . . . . 15
⊢
(∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) | 
| 338 | 336, 337 | bitr4di 289 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) | 
| 339 | 281, 301,
338 | 3bitr2d 307 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) | 
| 340 | 231, 238,
339 | syl2an 596 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ (0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵)) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) | 
| 341 | 340 | pm5.32da 579 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) →
(((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0) ↔ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) | 
| 342 | 341 | anbi2d 630 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) | 
| 343 | 342 | rexbidva 3176 | . . . . . . . . 9
⊢ (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) | 
| 344 | 343 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) | 
| 345 | 191 | rexeqdv 3326 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) | 
| 346 | 345 | biimpd 229 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) | 
| 347 | 346 | ralimdv 3168 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) | 
| 348 | 169 | rexeqdv 3326 | . . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑡) = 𝑁 → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) | 
| 349 | 348 | ralbidv 3177 | . . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) | 
| 350 | 349 | imbi1d 341 | . . . . . . . . . . . . . . . . 17
⊢
((2nd ‘𝑡) = 𝑁 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) | 
| 351 | 347, 350 | syl5ibrcom 247 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) | 
| 352 | 351 | com23 86 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ((2nd ‘𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) | 
| 353 | 352 | imp 406 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → ((2nd ‘𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) | 
| 354 | 353 | adantrd 491 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) | 
| 355 | 354 | pm4.71rd 562 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) | 
| 356 |  | an12 645 | . . . . . . . . . . . . 13
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))) | 
| 357 |  | 3anass 1094 | . . . . . . . . . . . . . 14
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) | 
| 358 | 357 | anbi2i 623 | . . . . . . . . . . . . 13
⊢
(((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))) | 
| 359 | 356, 358 | bitr4i 278 | . . . . . . . . . . . 12
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) | 
| 360 | 355, 359 | bitrdi 287 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))) | 
| 361 | 360 | notbid 318 | . . . . . . . . . 10
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (¬ ((2nd
‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ¬ ((2nd
‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))) | 
| 362 | 361 | pm5.32da 579 | . . . . . . . . 9
⊢ (𝜑 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) | 
| 363 | 362 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) | 
| 364 | 229, 344,
363 | 3bitr3d 309 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) | 
| 365 | 364 | rabbidva 3442 | . . . . . 6
⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))}) | 
| 366 |  | iunrab 5051 | . . . . . 6
⊢ ∪ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} | 
| 367 |  | difrab 4317 | . . . . . 6
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))} | 
| 368 | 365, 366,
367 | 3eqtr4g 2801 | . . . . 5
⊢ (𝜑 → ∪ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) | 
| 369 | 368 | fveq2d 6909 | . . . 4
⊢ (𝜑 → (♯‘∪ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}))) | 
| 370 | 27, 28 | mp1i 13 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin) | 
| 371 |  | simpl 482 | . . . . . . . . . . . 12
⊢ ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))) | 
| 372 | 371 | a1i 11 | . . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))))) | 
| 373 | 372 | ss2rabi 4076 | . . . . . . . . . 10
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} | 
| 374 | 373 | sseli 3978 | . . . . . . . . 9
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) | 
| 375 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (2nd ‘𝑡) = (2nd ‘𝑠)) | 
| 376 | 375 | breq2d 5154 | . . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑠 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑠))) | 
| 377 | 376 | ifbid 4548 | . . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑠 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1))) | 
| 378 | 377 | csbeq1d 3902 | . . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) | 
| 379 |  | 2fveq3 6910 | . . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑠 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑠))) | 
| 380 |  | 2fveq3 6910 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑠 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑠))) | 
| 381 | 380 | imaeq1d 6076 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑠 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑠)) “
(1...𝑗))) | 
| 382 | 381 | xpeq1d 5713 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1})) | 
| 383 | 380 | imaeq1d 6076 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑠 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑠)) “ ((𝑗 + 1)...𝑁))) | 
| 384 | 383 | xpeq1d 5713 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})) | 
| 385 | 382, 384 | uneq12d 4168 | . . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑠 → ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))) | 
| 386 | 379, 385 | oveq12d 7450 | . . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑠 → ((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) | 
| 387 | 386 | csbeq2dv 3905 | . . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) | 
| 388 | 378, 387 | eqtrd 2776 | . . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) | 
| 389 | 388 | mpteq2dv 5243 | . . . . . . . . . . . . 13
⊢ (𝑡 = 𝑠 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))) | 
| 390 | 389 | eqeq2d 2747 | . . . . . . . . . . . 12
⊢ (𝑡 = 𝑠 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))))) | 
| 391 |  | eqcom 2743 | . . . . . . . . . . . 12
⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥) | 
| 392 | 390, 391 | bitrdi 287 | . . . . . . . . . . 11
⊢ (𝑡 = 𝑠 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)) | 
| 393 | 392 | elrab 3691 | . . . . . . . . . 10
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ↔ (𝑠 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)) | 
| 394 | 393 | simprbi 496 | . . . . . . . . 9
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥) | 
| 395 | 374, 394 | syl 17 | . . . . . . . 8
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥) | 
| 396 | 395 | rgen 3062 | . . . . . . 7
⊢
∀𝑠 ∈
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥 | 
| 397 | 396 | rgenw 3064 | . . . . . 6
⊢
∀𝑥 ∈
(((0...𝐾)
↑m (1...𝑁))
↑m (0...(𝑁
− 1)))∀𝑠
∈ {𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥 | 
| 398 |  | invdisj 5128 | . . . . . 6
⊢
(∀𝑥 ∈
(((0...𝐾)
↑m (1...𝑁))
↑m (0...(𝑁
− 1)))∀𝑠
∈ {𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘f + ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥 → Disj 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) | 
| 399 | 397, 398 | mp1i 13 | . . . . 5
⊢ (𝜑 → Disj 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) | 
| 400 | 8, 370, 399 | hashiun 15859 | . . . 4
⊢ (𝜑 → (♯‘∪ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = Σ𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 −
1)))(♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) | 
| 401 | 369, 400 | eqtr3d 2778 | . . 3
⊢ (𝜑 → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = Σ𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 −
1)))(♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) | 
| 402 |  | fo1st 8035 | . . . . . . . . . . . . 13
⊢
1st :V–onto→V | 
| 403 |  | fofun 6820 | . . . . . . . . . . . . 13
⊢
(1st :V–onto→V → Fun 1st ) | 
| 404 | 402, 403 | ax-mp 5 | . . . . . . . . . . . 12
⊢ Fun
1st | 
| 405 |  | ssv 4007 | . . . . . . . . . . . . 13
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ V | 
| 406 |  | fof 6819 | . . . . . . . . . . . . . . 15
⊢
(1st :V–onto→V → 1st
:V⟶V) | 
| 407 | 402, 406 | ax-mp 5 | . . . . . . . . . . . . . 14
⊢
1st :V⟶V | 
| 408 | 407 | fdmi 6746 | . . . . . . . . . . . . 13
⊢ dom
1st = V | 
| 409 | 405, 408 | sseqtrri 4032 | . . . . . . . . . . . 12
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom
1st | 
| 410 |  | fores 6829 | . . . . . . . . . . . 12
⊢ ((Fun
1st ∧ {𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st ) →
(1st ↾ {𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) | 
| 411 | 404, 409,
410 | mp2an 692 | . . . . . . . . . . 11
⊢
(1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) | 
| 412 |  | fveqeq2 6914 | . . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑥 → ((2nd ‘𝑡) = 𝑁 ↔ (2nd ‘𝑥) = 𝑁)) | 
| 413 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 = 𝑥 → (1st ‘𝑡) = (1st ‘𝑥)) | 
| 414 | 413 | csbeq1d 3902 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑥 → ⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) | 
| 415 | 414 | eqeq2d 2747 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) | 
| 416 | 415 | rexbidv 3178 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) | 
| 417 | 416 | ralbidv 3177 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) | 
| 418 |  | 2fveq3 6910 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑥))) | 
| 419 | 418 | fveq1d 6907 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → ((1st
‘(1st ‘𝑡))‘𝑁) = ((1st ‘(1st
‘𝑥))‘𝑁)) | 
| 420 | 419 | eqeq1d 2738 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ↔ ((1st
‘(1st ‘𝑥))‘𝑁) = 0)) | 
| 421 |  | 2fveq3 6910 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑥))) | 
| 422 | 421 | fveq1d 6907 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → ((2nd
‘(1st ‘𝑡))‘𝑁) = ((2nd ‘(1st
‘𝑥))‘𝑁)) | 
| 423 | 422 | eqeq1d 2738 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁 ↔ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) | 
| 424 | 417, 420,
423 | 3anbi123d 1437 | . . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑥 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁))) | 
| 425 | 412, 424 | anbi12d 632 | . . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑥 → (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)))) | 
| 426 | 425 | rexrab 3701 | . . . . . . . . . . . . . . 15
⊢
(∃𝑥 ∈
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠 ↔ ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠)) | 
| 427 |  | xp1st 8047 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) | 
| 428 | 427 | anim1i 615 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁))) | 
| 429 |  | eleq1 2828 | . . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘𝑥) = 𝑠 → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))) | 
| 430 |  | csbeq1a 3912 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑠 = (1st ‘𝑥) → 𝐶 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) | 
| 431 | 430 | eqcoms 2744 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((1st ‘𝑥) = 𝑠 → 𝐶 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) | 
| 432 | 431 | eqcomd 2742 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((1st ‘𝑥) = 𝑠 → ⦋(1st
‘𝑥) / 𝑠⦌𝐶 = 𝐶) | 
| 433 | 432 | eqeq2d 2747 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1st ‘𝑥) = 𝑠 → (𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶)) | 
| 434 | 433 | rexbidv 3178 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑥) = 𝑠 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) | 
| 435 | 434 | ralbidv 3177 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
((1st ‘𝑥) = 𝑠 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) | 
| 436 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1st ‘𝑥) = 𝑠 → (1st
‘(1st ‘𝑥)) = (1st ‘𝑠)) | 
| 437 | 436 | fveq1d 6907 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑥) = 𝑠 → ((1st
‘(1st ‘𝑥))‘𝑁) = ((1st ‘𝑠)‘𝑁)) | 
| 438 | 437 | eqeq1d 2738 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
((1st ‘𝑥) = 𝑠 → (((1st
‘(1st ‘𝑥))‘𝑁) = 0 ↔ ((1st ‘𝑠)‘𝑁) = 0)) | 
| 439 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1st ‘𝑥) = 𝑠 → (2nd
‘(1st ‘𝑥)) = (2nd ‘𝑠)) | 
| 440 | 439 | fveq1d 6907 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑥) = 𝑠 → ((2nd
‘(1st ‘𝑥))‘𝑁) = ((2nd ‘𝑠)‘𝑁)) | 
| 441 | 440 | eqeq1d 2738 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
((1st ‘𝑥) = 𝑠 → (((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁 ↔ ((2nd ‘𝑠)‘𝑁) = 𝑁)) | 
| 442 | 435, 438,
441 | 3anbi123d 1437 | . . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘𝑥) = 𝑠 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))) | 
| 443 | 429, 442 | anbi12d 632 | . . . . . . . . . . . . . . . . . . . 20
⊢
((1st ‘𝑥) = 𝑠 → (((1st ‘𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) | 
| 444 | 428, 443 | syl5ibcom 245 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) | 
| 445 | 444 | adantrl 716 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁))) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) | 
| 446 | 445 | expimpd 453 | . . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) | 
| 447 | 446 | rexlimiv 3147 | . . . . . . . . . . . . . . . 16
⊢
(∃𝑥 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))) | 
| 448 |  | nn0fz0 13666 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℕ0
↔ 𝑁 ∈ (0...𝑁)) | 
| 449 | 171, 448 | sylib 218 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) | 
| 450 |  | opelxpi 5721 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑁 ∈ (0...𝑁)) → 〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) | 
| 451 | 449, 450 | sylan2 593 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝜑) → 〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) | 
| 452 | 451 | ancoms 458 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → 〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) | 
| 453 |  | opelxp2 5727 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → 𝑁 ∈ (0...𝑁)) | 
| 454 |  | op2ndg 8028 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd ‘〈𝑠, 𝑁〉) = 𝑁) | 
| 455 | 454 | biantrurd 532 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁) ↔ ((2nd ‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)))) | 
| 456 |  | op1stg 8027 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘〈𝑠, 𝑁〉) = 𝑠) | 
| 457 |  | csbeq1a 3912 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑠 = (1st
‘〈𝑠, 𝑁〉) → 𝐶 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶) | 
| 458 | 457 | eqcoms 2744 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((1st ‘〈𝑠, 𝑁〉) = 𝑠 → 𝐶 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶) | 
| 459 | 458 | eqcomd 2742 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((1st ‘〈𝑠, 𝑁〉) = 𝑠 → ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 = 𝐶) | 
| 460 | 456, 459 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 = 𝐶) | 
| 461 | 460 | eqeq2d 2747 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶)) | 
| 462 | 461 | rexbidv 3178 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) | 
| 463 | 462 | ralbidv 3177 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) | 
| 464 | 456 | fveq2d 6909 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st
‘(1st ‘〈𝑠, 𝑁〉)) = (1st ‘𝑠)) | 
| 465 | 464 | fveq1d 6907 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = ((1st ‘𝑠)‘𝑁)) | 
| 466 | 465 | eqeq1d 2738 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ↔ ((1st ‘𝑠)‘𝑁) = 0)) | 
| 467 | 456 | fveq2d 6909 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd
‘(1st ‘〈𝑠, 𝑁〉)) = (2nd ‘𝑠)) | 
| 468 | 467 | fveq1d 6907 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = ((2nd ‘𝑠)‘𝑁)) | 
| 469 | 468 | eqeq1d 2738 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁 ↔ ((2nd ‘𝑠)‘𝑁) = 𝑁)) | 
| 470 | 463, 466,
469 | 3anbi123d 1437 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))) | 
| 471 | 456 | biantrud 531 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ↔ (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠))) | 
| 472 | 455, 470,
471 | 3bitr3d 309 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁) ↔ (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠))) | 
| 473 | 44, 453, 472 | sylancr 587 | . . . . . . . . . . . . . . . . . . . 20
⊢
(〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁) ↔ (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠))) | 
| 474 | 473 | biimpa 476 | . . . . . . . . . . . . . . . . . . 19
⊢
((〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠)) | 
| 475 |  | fveqeq2 6914 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((2nd ‘𝑥) = 𝑁 ↔ (2nd ‘〈𝑠, 𝑁〉) = 𝑁)) | 
| 476 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (1st ‘𝑥) = (1st
‘〈𝑠, 𝑁〉)) | 
| 477 | 476 | csbeq1d 3902 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ⦋(1st
‘𝑥) / 𝑠⦌𝐶 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶) | 
| 478 | 477 | eqeq2d 2747 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶)) | 
| 479 | 478 | rexbidv 3178 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶)) | 
| 480 | 479 | ralbidv 3177 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶)) | 
| 481 |  | 2fveq3 6910 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (1st
‘(1st ‘𝑥)) = (1st ‘(1st
‘〈𝑠, 𝑁〉))) | 
| 482 | 481 | fveq1d 6907 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((1st
‘(1st ‘𝑥))‘𝑁) = ((1st ‘(1st
‘〈𝑠, 𝑁〉))‘𝑁)) | 
| 483 | 482 | eqeq1d 2738 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (((1st
‘(1st ‘𝑥))‘𝑁) = 0 ↔ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0)) | 
| 484 |  | 2fveq3 6910 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (2nd
‘(1st ‘𝑥)) = (2nd ‘(1st
‘〈𝑠, 𝑁〉))) | 
| 485 | 484 | fveq1d 6907 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((2nd
‘(1st ‘𝑥))‘𝑁) = ((2nd ‘(1st
‘〈𝑠, 𝑁〉))‘𝑁)) | 
| 486 | 485 | eqeq1d 2738 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁 ↔ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) | 
| 487 | 480, 483,
486 | 3anbi123d 1437 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁))) | 
| 488 | 475, 487 | anbi12d 632 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ↔ ((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)))) | 
| 489 |  | fveqeq2 6914 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((1st ‘𝑥) = 𝑠 ↔ (1st ‘〈𝑠, 𝑁〉) = 𝑠)) | 
| 490 | 488, 489 | anbi12d 632 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((((2nd
‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) ↔ (((2nd ‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠))) | 
| 491 | 490 | rspcev 3621 | . . . . . . . . . . . . . . . . . . 19
⊢
((〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠)) | 
| 492 | 474, 491 | syldan 591 | . . . . . . . . . . . . . . . . . 18
⊢
((〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠)) | 
| 493 | 452, 492 | sylan 580 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠)) | 
| 494 | 493 | expl 457 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠))) | 
| 495 | 447, 494 | impbid2 226 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) | 
| 496 | 426, 495 | bitrid 283 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠 ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) | 
| 497 | 496 | abbidv 2807 | . . . . . . . . . . . . 13
⊢ (𝜑 → {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))}) | 
| 498 |  | dfimafn 6970 | . . . . . . . . . . . . . . 15
⊢ ((Fun
1st ∧ {𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st ) →
(1st “ {𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦}) | 
| 499 | 404, 409,
498 | mp2an 692 | . . . . . . . . . . . . . 14
⊢
(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦} | 
| 500 |  | nfv 1913 | . . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑠(2nd ‘𝑡) = 𝑁 | 
| 501 |  | nfcv 2904 | . . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑠(0...(𝑁 − 1)) | 
| 502 |  | nfcsb1v 3922 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑠⦋(1st ‘𝑡) / 𝑠⦌𝐶 | 
| 503 | 502 | nfeq2 2922 | . . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑠 𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 | 
| 504 | 501, 503 | nfrexw 3312 | . . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑠∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 | 
| 505 | 501, 504 | nfralw 3310 | . . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 | 
| 506 |  | nfv 1913 | . . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠((1st ‘(1st
‘𝑡))‘𝑁) = 0 | 
| 507 |  | nfv 1913 | . . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠((2nd ‘(1st
‘𝑡))‘𝑁) = 𝑁 | 
| 508 | 505, 506,
507 | nf3an 1900 | . . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑠(∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁) | 
| 509 | 500, 508 | nfan 1898 | . . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑠((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) | 
| 510 |  | nfcv 2904 | . . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑠((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) | 
| 511 | 509, 510 | nfrabw 3474 | . . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑠{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} | 
| 512 |  | nfv 1913 | . . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑠(1st ‘𝑥) = 𝑦 | 
| 513 | 511, 512 | nfrexw 3312 | . . . . . . . . . . . . . . 15
⊢
Ⅎ𝑠∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦 | 
| 514 |  | nfv 1913 | . . . . . . . . . . . . . . 15
⊢
Ⅎ𝑦∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠 | 
| 515 |  | eqeq2 2748 | . . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑠 → ((1st ‘𝑥) = 𝑦 ↔ (1st ‘𝑥) = 𝑠)) | 
| 516 | 515 | rexbidv 3178 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑠 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠)) | 
| 517 | 513, 514,
516 | cbvabw 2812 | . . . . . . . . . . . . . 14
⊢ {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦} = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠} | 
| 518 | 499, 517 | eqtri 2764 | . . . . . . . . . . . . 13
⊢
(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠} | 
| 519 |  | df-rab 3436 | . . . . . . . . . . . . 13
⊢ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))} | 
| 520 | 497, 518,
519 | 3eqtr4g 2801 | . . . . . . . . . . . 12
⊢ (𝜑 → (1st “
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) | 
| 521 |  | foeq3 6817 | . . . . . . . . . . . 12
⊢
((1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})) | 
| 522 | 520, 521 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → ((1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})) | 
| 523 | 411, 522 | mpbii 233 | . . . . . . . . . 10
⊢ (𝜑 → (1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) | 
| 524 |  | fof 6819 | . . . . . . . . . 10
⊢
((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) | 
| 525 | 523, 524 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → (1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) | 
| 526 |  | fvres 6924 | . . . . . . . . . . . 12
⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = (1st ‘𝑥)) | 
| 527 |  | fvres 6924 | . . . . . . . . . . . 12
⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) = (1st ‘𝑦)) | 
| 528 | 526, 527 | eqeqan12d 2750 | . . . . . . . . . . 11
⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) ↔ (1st ‘𝑥) = (1st ‘𝑦))) | 
| 529 |  | simpl 482 | . . . . . . . . . . . . . . . 16
⊢
(((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) → (2nd ‘𝑡) = 𝑁) | 
| 530 | 529 | a1i 11 | . . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) → (2nd ‘𝑡) = 𝑁)) | 
| 531 | 530 | ss2rabi 4076 | . . . . . . . . . . . . . 14
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁} | 
| 532 | 531 | sseli 3978 | . . . . . . . . . . . . 13
⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} → 𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁}) | 
| 533 | 412 | elrab 3691 | . . . . . . . . . . . . 13
⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁} ↔ (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑥) = 𝑁)) | 
| 534 | 532, 533 | sylib 218 | . . . . . . . . . . . 12
⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} → (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑥) = 𝑁)) | 
| 535 | 531 | sseli 3978 | . . . . . . . . . . . . 13
⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} → 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁}) | 
| 536 |  | fveqeq2 6914 | . . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑦 → ((2nd ‘𝑡) = 𝑁 ↔ (2nd ‘𝑦) = 𝑁)) | 
| 537 | 536 | elrab 3691 | . . . . . . . . . . . . 13
⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁} ↔ (𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑦) = 𝑁)) | 
| 538 | 535, 537 | sylib 218 | . . . . . . . . . . . 12
⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} → (𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑦) = 𝑁)) | 
| 539 |  | eqtr3 2762 | . . . . . . . . . . . . . 14
⊢
(((2nd ‘𝑥) = 𝑁 ∧ (2nd ‘𝑦) = 𝑁) → (2nd ‘𝑥) = (2nd ‘𝑦)) | 
| 540 |  | xpopth 8056 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd
‘𝑥) = (2nd
‘𝑦)) ↔ 𝑥 = 𝑦)) | 
| 541 | 540 | biimpd 229 | . . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd
‘𝑥) = (2nd
‘𝑦)) → 𝑥 = 𝑦)) | 
| 542 | 541 | ancomsd 465 | . . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((2nd ‘𝑥) = (2nd ‘𝑦) ∧ (1st
‘𝑥) = (1st
‘𝑦)) → 𝑥 = 𝑦)) | 
| 543 | 542 | expdimp 452 | . . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → ((1st
‘𝑥) = (1st
‘𝑦) → 𝑥 = 𝑦)) | 
| 544 | 539, 543 | sylan2 593 | . . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ ((2nd ‘𝑥) = 𝑁 ∧ (2nd ‘𝑦) = 𝑁)) → ((1st ‘𝑥) = (1st ‘𝑦) → 𝑥 = 𝑦)) | 
| 545 | 544 | an4s 660 | . . . . . . . . . . . 12
⊢ (((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑥) = 𝑁) ∧ (𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑦) = 𝑁)) → ((1st ‘𝑥) = (1st ‘𝑦) → 𝑥 = 𝑦)) | 
| 546 | 534, 538,
545 | syl2an 596 | . . . . . . . . . . 11
⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) → ((1st ‘𝑥) = (1st ‘𝑦) → 𝑥 = 𝑦)) | 
| 547 | 528, 546 | sylbid 240 | . . . . . . . . . 10
⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦)) | 
| 548 | 547 | rgen2 3198 | . . . . . . . . 9
⊢
∀𝑥 ∈
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦) | 
| 549 | 525, 548 | jctir 520 | . . . . . . . 8
⊢ (𝜑 → ((1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∧ ∀𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦))) | 
| 550 |  | dff13 7276 | . . . . . . . 8
⊢
((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ↔ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∧ ∀𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦))) | 
| 551 | 549, 550 | sylibr 234 | . . . . . . 7
⊢ (𝜑 → (1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) | 
| 552 |  | df-f1o 6567 | . . . . . . 7
⊢
((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ↔ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∧ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})) | 
| 553 | 551, 523,
552 | sylanbrc 583 | . . . . . 6
⊢ (𝜑 → (1st ↾
{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) | 
| 554 |  | rabfi 9304 | . . . . . . . . 9
⊢
(((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∈ Fin) | 
| 555 | 27, 554 | ax-mp 5 | . . . . . . . 8
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∈ Fin | 
| 556 | 555 | elexi 3502 | . . . . . . 7
⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∈ V | 
| 557 | 556 | f1oen 9014 | . . . . . 6
⊢
((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) | 
| 558 | 553, 557 | syl 17 | . . . . 5
⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) | 
| 559 |  | rabfi 9304 | . . . . . . 7
⊢
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∈ Fin) | 
| 560 | 24, 559 | ax-mp 5 | . . . . . 6
⊢ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∈ Fin | 
| 561 |  | hashen 14387 | . . . . . 6
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∈ Fin ∧ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∈ Fin) → ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) ↔ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})) | 
| 562 | 555, 560,
561 | mp2an 692 | . . . . 5
⊢
((♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) ↔ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) | 
| 563 | 558, 562 | sylibr 234 | . . . 4
⊢ (𝜑 → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})) | 
| 564 | 563 | oveq2d 7448 | . . 3
⊢ (𝜑 → ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}))) | 
| 565 | 201, 401,
564 | 3eqtr3d 2784 | . 2
⊢ (𝜑 → Σ𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 −
1)))(♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}))) | 
| 566 | 164, 565 | breqtrd 5168 | 1
⊢ (𝜑 → 2 ∥
((♯‘{𝑡 ∈
((((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}))) |