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Theorem poimirlem27 36105
Description: Lemma for poimir 36111 showing that the difference between admissible faces in the whole cube and admissible faces on the back face is even. Equation (7) of [Kulpa] p. 548. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem28.1 (𝑝 = ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
poimirlem28.2 ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
poimirlem28.3 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝐵 < 𝑛)
poimirlem28.4 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
Assertion
Ref Expression
poimirlem27 (𝜑 → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})))
Distinct variable groups:   𝑓,𝑖,𝑗,𝑛,𝑝,𝑠,𝑡   𝜑,𝑗,𝑛   𝑗,𝑁,𝑛   𝜑,𝑖,𝑝,𝑠,𝑡   𝐵,𝑓,𝑖,𝑗,𝑛,𝑠,𝑡   𝑓,𝐾,𝑖,𝑗,𝑛,𝑝,𝑠,𝑡   𝑓,𝑁,𝑖,𝑝,𝑠,𝑡   𝐶,𝑖,𝑛,𝑝,𝑡
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑝)   𝐶(𝑓,𝑗,𝑠)

Proof of Theorem poimirlem27
Dummy variables 𝑚 𝑞 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfi 13877 . . . . . 6 (0...𝐾) ∈ Fin
2 fzfi 13877 . . . . . 6 (1...𝑁) ∈ Fin
3 mapfi 9292 . . . . . 6 (((0...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((0...𝐾) ↑m (1...𝑁)) ∈ Fin)
41, 2, 3mp2an 690 . . . . 5 ((0...𝐾) ↑m (1...𝑁)) ∈ Fin
5 fzfi 13877 . . . . 5 (0...(𝑁 − 1)) ∈ Fin
6 mapfi 9292 . . . . 5 ((((0...𝐾) ↑m (1...𝑁)) ∈ Fin ∧ (0...(𝑁 − 1)) ∈ Fin) → (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) ∈ Fin)
74, 5, 6mp2an 690 . . . 4 (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) ∈ Fin
87a1i 11 . . 3 (𝜑 → (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) ∈ Fin)
9 2z 12535 . . . 4 2 ∈ ℤ
109a1i 11 . . 3 (𝜑 → 2 ∈ ℤ)
11 fzofi 13879 . . . . . . . 8 (0..^𝐾) ∈ Fin
12 mapfi 9292 . . . . . . . 8 (((0..^𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((0..^𝐾) ↑m (1...𝑁)) ∈ Fin)
1311, 2, 12mp2an 690 . . . . . . 7 ((0..^𝐾) ↑m (1...𝑁)) ∈ Fin
14 mapfi 9292 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ↑m (1...𝑁)) ∈ Fin)
152, 2, 14mp2an 690 . . . . . . . 8 ((1...𝑁) ↑m (1...𝑁)) ∈ Fin
16 f1of 6784 . . . . . . . . . 10 (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁))
1716ss2abi 4023 . . . . . . . . 9 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓𝑓:(1...𝑁)⟶(1...𝑁)}
18 ovex 7390 . . . . . . . . . 10 (1...𝑁) ∈ V
1918, 18mapval 8777 . . . . . . . . 9 ((1...𝑁) ↑m (1...𝑁)) = {𝑓𝑓:(1...𝑁)⟶(1...𝑁)}
2017, 19sseqtrri 3981 . . . . . . . 8 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑m (1...𝑁))
21 ssfi 9117 . . . . . . . 8 ((((1...𝑁) ↑m (1...𝑁)) ∈ Fin ∧ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑m (1...𝑁))) → {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
2215, 20, 21mp2an 690 . . . . . . 7 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin
23 xpfi 9261 . . . . . . 7 ((((0..^𝐾) ↑m (1...𝑁)) ∈ Fin ∧ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) → (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin)
2413, 22, 23mp2an 690 . . . . . 6 (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin
25 fzfi 13877 . . . . . 6 (0...𝑁) ∈ Fin
26 xpfi 9261 . . . . . 6 (((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin ∧ (0...𝑁) ∈ Fin) → ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin)
2724, 25, 26mp2an 690 . . . . 5 ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin
28 rabfi 9213 . . . . 5 (((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} ∈ Fin)
2927, 28ax-mp 5 . . . 4 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} ∈ Fin
30 hashcl 14256 . . . . 5 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} ∈ Fin → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}) ∈ ℕ0)
3130nn0zd 12525 . . . 4 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} ∈ Fin → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}) ∈ ℤ)
3229, 31mp1i 13 . . 3 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}) ∈ ℤ)
33 dfrex2 3076 . . . . 5 (∃𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ↔ ¬ ∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
34 nfv 1917 . . . . . 6 𝑡(𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))))
35 nfcv 2907 . . . . . . 7 𝑡2
36 nfcv 2907 . . . . . . 7 𝑡
37 nfcv 2907 . . . . . . . 8 𝑡
38 nfrab1 3426 . . . . . . . 8 𝑡{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}
3937, 38nffv 6852 . . . . . . 7 𝑡(♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})
4035, 36, 39nfbr 5152 . . . . . 6 𝑡2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})
41 neq0 4305 . . . . . . . . . . . 12 (¬ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ ↔ ∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})
42 iddvds 16152 . . . . . . . . . . . . . . . . 17 (2 ∈ ℤ → 2 ∥ 2)
439, 42ax-mp 5 . . . . . . . . . . . . . . . 16 2 ∥ 2
44 vex 3449 . . . . . . . . . . . . . . . . . . 19 𝑠 ∈ V
45 hashsng 14269 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ V → (♯‘{𝑠}) = 1)
4644, 45ax-mp 5 . . . . . . . . . . . . . . . . . 18 (♯‘{𝑠}) = 1
4746oveq2i 7368 . . . . . . . . . . . . . . . . 17 (1 + (♯‘{𝑠})) = (1 + 1)
48 df-2 12216 . . . . . . . . . . . . . . . . 17 2 = (1 + 1)
4947, 48eqtr4i 2767 . . . . . . . . . . . . . . . 16 (1 + (♯‘{𝑠})) = 2
5043, 49breqtrri 5132 . . . . . . . . . . . . . . 15 2 ∥ (1 + (♯‘{𝑠}))
51 rabfi 9213 . . . . . . . . . . . . . . . . . . . 20 (((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∈ Fin)
52 diffi 9123 . . . . . . . . . . . . . . . . . . . 20 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∈ Fin → ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin)
5327, 51, 52mp2b 10 . . . . . . . . . . . . . . . . . . 19 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin
54 snfi 8988 . . . . . . . . . . . . . . . . . . 19 {𝑠} ∈ Fin
55 disjdifr 4432 . . . . . . . . . . . . . . . . . . 19 (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅
56 hashun 14282 . . . . . . . . . . . . . . . . . . 19 ((({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin ∧ {𝑠} ∈ Fin ∧ (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅) → (♯‘(({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠})))
5753, 54, 55, 56mp3an 1461 . . . . . . . . . . . . . . . . . 18 (♯‘(({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠}))
58 difsnid 4770 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠}) = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})
5958fveq2d 6846 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (♯‘(({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))
6057, 59eqtr3id 2790 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠})) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))
6160adantl 482 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠})) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))
62 poimir.0 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℕ)
6362ad3antrrr 728 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑁 ∈ ℕ)
64 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑢 → (2nd𝑡) = (2nd𝑢))
6564breq2d 5117 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑢 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑢)))
6665ifbid 4509 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑢 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)))
6766csbeq1d 3859 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑢if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
68 2fveq3 6847 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑢 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑢)))
69 2fveq3 6847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑡 = 𝑢 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑢)))
7069imaeq1d 6012 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑢 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑢)) “ (1...𝑗)))
7170xpeq1d 5662 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑢 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}))
7269imaeq1d 6012 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑢 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)))
7372xpeq1d 5662 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑢 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))
7471, 73uneq12d 4124 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑢 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))
7568, 74oveq12d 7375 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑢 → ((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))
7675csbeq2dv 3862 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑢if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))
7767, 76eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑢if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))
7877mpteq2dv 5207 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑢 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))))
79 breq1 5108 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑤 → (𝑦 < (2nd𝑢) ↔ 𝑤 < (2nd𝑢)))
80 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑤𝑦 = 𝑤)
81 oveq1 7364 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1))
8279, 80, 81ifbieq12d 4514 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑤 → if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) = if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)))
8382csbeq1d 3859 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑤if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))
84 oveq2 7365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖))
8584imaeq2d 6013 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 𝑖 → ((2nd ‘(1st𝑢)) “ (1...𝑗)) = ((2nd ‘(1st𝑢)) “ (1...𝑖)))
8685xpeq1d 5662 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 𝑖 → (((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}))
87 oveq1 7364 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = 𝑖 → (𝑗 + 1) = (𝑖 + 1))
8887oveq1d 7372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑖 → ((𝑗 + 1)...𝑁) = ((𝑖 + 1)...𝑁))
8988imaeq2d 6013 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 𝑖 → ((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)))
9089xpeq1d 5662 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 𝑖 → (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))
9186, 90uneq12d 4124 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = 𝑖 → ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))
9291oveq2d 7373 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑖 → ((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))
9392cbvcsbv 3867 . . . . . . . . . . . . . . . . . . . . . . . 24 if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑖((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))
9483, 93eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑤if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑖((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))
9594cbvmptv 5218 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑖((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))
9678, 95eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑢 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑖((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))))
9796eqeq2d 2747 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑢 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑖((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))))
9897cbvrabv 3417 . . . . . . . . . . . . . . . . . . 19 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = {𝑢 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑖((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))}
99 elmapi 8787 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) → 𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
10099ad3antlr 729 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
101 simpr 485 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})
102 simpl 483 . . . . . . . . . . . . . . . . . . . . . 22 ((∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)
103102ralimi 3086 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)
104103ad2antlr 725 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)
105 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑝𝑛) = (𝑝𝑚))
106105neeq1d 3003 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝑝𝑛) ≠ 0 ↔ (𝑝𝑚) ≠ 0))
107106rexbidv 3175 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 0))
108 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = 𝑞 → (𝑝𝑚) = (𝑞𝑚))
109108neeq1d 3003 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = 𝑞 → ((𝑝𝑚) ≠ 0 ↔ (𝑞𝑚) ≠ 0))
110109cbvrexvw 3226 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 0)
111107, 110bitrdi 286 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 0))
112111rspccva 3580 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 0)
113104, 112sylan 580 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 0)
114 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 ((∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
115114ralimi 3086 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
116115ad2antlr 725 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
117105neeq1d 3003 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝑝𝑛) ≠ 𝐾 ↔ (𝑝𝑚) ≠ 𝐾))
118117rexbidv 3175 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 𝐾))
119108neeq1d 3003 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = 𝑞 → ((𝑝𝑚) ≠ 𝐾 ↔ (𝑞𝑚) ≠ 𝐾))
120119cbvrexvw 3226 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 𝐾)
121118, 120bitrdi 286 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 𝐾))
122121rspccva 3580 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 𝐾)
123116, 122sylan 580 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 𝐾)
12463, 98, 100, 101, 113, 123poimirlem22 36100 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∃!𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧𝑠)
125 eldifsn 4747 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ↔ (𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧𝑠))
126125eubii 2583 . . . . . . . . . . . . . . . . . . 19 (∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧𝑠))
12753elexi 3464 . . . . . . . . . . . . . . . . . . . 20 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ V
128 euhash1 14320 . . . . . . . . . . . . . . . . . . . 20 (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ V → ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})))
129127, 128ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}))
130 df-reu 3354 . . . . . . . . . . . . . . . . . . 19 (∃!𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧𝑠 ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧𝑠))
131126, 129, 1303bitr4ri 303 . . . . . . . . . . . . . . . . . 18 (∃!𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧𝑠 ↔ (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1)
132124, 131sylib 217 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1)
133132oveq1d 7372 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠})) = (1 + (♯‘{𝑠})))
13461, 133eqtr3d 2778 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (1 + (♯‘{𝑠})))
13550, 134breqtrrid 5143 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))
136135ex 413 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})))
137136exlimdv 1936 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})))
13841, 137biimtrid 241 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (¬ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})))
139 dvds0 16154 . . . . . . . . . . . . . 14 (2 ∈ ℤ → 2 ∥ 0)
1409, 139ax-mp 5 . . . . . . . . . . . . 13 2 ∥ 0
141 hash0 14267 . . . . . . . . . . . . 13 (♯‘∅) = 0
142140, 141breqtrri 5132 . . . . . . . . . . . 12 2 ∥ (♯‘∅)
143 fveq2 6842 . . . . . . . . . . . 12 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (♯‘∅))
144142, 143breqtrrid 5143 . . . . . . . . . . 11 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))
145138, 144pm2.61d2 181 . . . . . . . . . 10 (((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))
146145ex 413 . . . . . . . . 9 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})))
147146adantld 491 . . . . . . . 8 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})))
148 iba 528 . . . . . . . . . . 11 (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))))
149148rabbidv 3415 . . . . . . . . . 10 (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})
150149fveq2d 6846 . . . . . . . . 9 (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))
151150breq2d 5117 . . . . . . . 8 (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ↔ 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})))
152147, 151mpbidi 240 . . . . . . 7 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})))
153152a1d 25 . . . . . 6 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))))
15434, 40, 153rexlimd 3249 . . . . 5 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (∃𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})))
15533, 154biimtrrid 242 . . . 4 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (¬ ∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})))
156 simpr 485 . . . . . . . . 9 ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))) → ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
157156con3i 154 . . . . . . . 8 (¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))))
158157ralimi 3086 . . . . . . 7 (∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → ∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))))
159 rabeq0 4344 . . . . . . 7 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} = ∅ ↔ ∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))))
160158, 159sylibr 233 . . . . . 6 (∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} = ∅)
161160fveq2d 6846 . . . . 5 (∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}) = (♯‘∅))
162142, 161breqtrrid 5143 . . . 4 (∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))
163155, 162pm2.61d2 181 . . 3 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))
1648, 10, 32, 163fsumdvds 16190 . 2 (𝜑 → 2 ∥ Σ𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))
165 rabfi 9213 . . . . 5 (((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∈ Fin)
16627, 165ax-mp 5 . . . 4 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∈ Fin
167 simp1 1136 . . . . . . 7 ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶)
168 sneq 4596 . . . . . . . . . . . . 13 ((2nd𝑡) = 𝑁 → {(2nd𝑡)} = {𝑁})
169168difeq2d 4082 . . . . . . . . . . . 12 ((2nd𝑡) = 𝑁 → ((0...𝑁) ∖ {(2nd𝑡)}) = ((0...𝑁) ∖ {𝑁}))
170 difun2 4440 . . . . . . . . . . . . 13 (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((0...(𝑁 − 1)) ∖ {𝑁})
17162nnnn0d 12473 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℕ0)
172 nn0uz 12805 . . . . . . . . . . . . . . . . . 18 0 = (ℤ‘0)
173171, 172eleqtrdi 2848 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ (ℤ‘0))
174 fzm1 13521 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ‘0) → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
175173, 174syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
176 elun 4108 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}))
177 velsn 4602 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ {𝑁} ↔ 𝑛 = 𝑁)
178177orbi2i 911 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
179176, 178bitri 274 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
180175, 179bitr4di 288 . . . . . . . . . . . . . . 15 (𝜑 → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})))
181180eqrdv 2734 . . . . . . . . . . . . . 14 (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁}))
182181difeq1d 4081 . . . . . . . . . . . . 13 (𝜑 → ((0...𝑁) ∖ {𝑁}) = (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}))
18362nnzd 12526 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℤ)
184 uzid 12778 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
185 uznfz 13524 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ𝑁) → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
186183, 184, 1853syl 18 . . . . . . . . . . . . . 14 (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
187 disjsn 4672 . . . . . . . . . . . . . . 15 (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (0...(𝑁 − 1)))
188 disj3 4413 . . . . . . . . . . . . . . 15 (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
189187, 188bitr3i 276 . . . . . . . . . . . . . 14 𝑁 ∈ (0...(𝑁 − 1)) ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
190186, 189sylib 217 . . . . . . . . . . . . 13 (𝜑 → (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
191170, 182, 1903eqtr4a 2802 . . . . . . . . . . . 12 (𝜑 → ((0...𝑁) ∖ {𝑁}) = (0...(𝑁 − 1)))
192169, 191sylan9eqr 2798 . . . . . . . . . . 11 ((𝜑 ∧ (2nd𝑡) = 𝑁) → ((0...𝑁) ∖ {(2nd𝑡)}) = (0...(𝑁 − 1)))
193192rexeqdv 3314 . . . . . . . . . 10 ((𝜑 ∧ (2nd𝑡) = 𝑁) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
194193biimprd 247 . . . . . . . . 9 ((𝜑 ∧ (2nd𝑡) = 𝑁) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
195194ralimdv 3166 . . . . . . . 8 ((𝜑 ∧ (2nd𝑡) = 𝑁) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
196195expimpd 454 . . . . . . 7 (𝜑 → (((2nd𝑡) = 𝑁 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
197167, 196sylan2i 606 . . . . . 6 (𝜑 → (((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
198197adantr 481 . . . . 5 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
199198ss2rabdv 4033 . . . 4 (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶})
200 hashssdif 14312 . . . 4 (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∈ Fin ∧ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶}) → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})))
201166, 199, 200sylancr 587 . . 3 (𝜑 → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})))
20262adantr 481 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → 𝑁 ∈ ℕ)
203 poimirlem28.1 . . . . . . . . . 10 (𝑝 = ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
204 poimirlem28.2 . . . . . . . . . . 11 ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
205204adantlr 713 . . . . . . . . . 10 (((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
206 xp1st 7953 . . . . . . . . . . . 12 (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
207 xp1st 7953 . . . . . . . . . . . 12 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑡)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
208 elmapi 8787 . . . . . . . . . . . 12 ((1st ‘(1st𝑡)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑡)):(1...𝑁)⟶(0..^𝐾))
209206, 207, 2083syl 18 . . . . . . . . . . 11 (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘(1st𝑡)):(1...𝑁)⟶(0..^𝐾))
210209adantl 482 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (1st ‘(1st𝑡)):(1...𝑁)⟶(0..^𝐾))
211 xp2nd 7954 . . . . . . . . . . . . 13 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑡)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
212 fvex 6855 . . . . . . . . . . . . . 14 (2nd ‘(1st𝑡)) ∈ V
213 f1oeq1 6772 . . . . . . . . . . . . . 14 (𝑓 = (2nd ‘(1st𝑡)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)))
214212, 213elab 3630 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑡)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
215211, 214sylib 217 . . . . . . . . . . . 12 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
216206, 215syl 17 . . . . . . . . . . 11 (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
217216adantl 482 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
218 xp2nd 7954 . . . . . . . . . . 11 (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑡) ∈ (0...𝑁))
219218adantl 482 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (2nd𝑡) ∈ (0...𝑁))
220202, 203, 205, 210, 217, 219poimirlem24 36102 . . . . . . . . 9 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(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 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
221206adantl 482 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
222 1st2nd2 7960 . . . . . . . . . . . . . . 15 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st𝑡) = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩)
223222csbeq1d 3859 . . . . . . . . . . . . . 14 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st𝑡) / 𝑠𝐶 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶)
224223eqeq2d 2747 . . . . . . . . . . . . 13 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑖 = (1st𝑡) / 𝑠𝐶𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶))
225224rexbidv 3175 . . . . . . . . . . . 12 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶))
226225ralbidv 3174 . . . . . . . . . . 11 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶))
227226anbi1d 630 . . . . . . . . . 10 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
228221, 227syl 17 . . . . . . . . 9 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
229220, 228bitr4d 281 . . . . . . . 8 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(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 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
23099frnd 6676 . . . . . . . . . . . . . 14 (𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) → ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)))
231230anim2i 617 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))))
232 dfss3 3932 . . . . . . . . . . . . . 14 ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵))
233 vex 3449 . . . . . . . . . . . . . . . 16 𝑛 ∈ V
234 eqid 2736 . . . . . . . . . . . . . . . . 17 (𝑝 ∈ ran 𝑥𝐵) = (𝑝 ∈ ran 𝑥𝐵)
235234elrnmpt 5911 . . . . . . . . . . . . . . . 16 (𝑛 ∈ V → (𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
236233, 235ax-mp 5 . . . . . . . . . . . . . . 15 (𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
237236ralbii 3096 . . . . . . . . . . . . . 14 (∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
238232, 237sylbb 218 . . . . . . . . . . . . 13 ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) → ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
239 1eluzge0 12817 . . . . . . . . . . . . . . . . 17 1 ∈ (ℤ‘0)
240 fzss1 13480 . . . . . . . . . . . . . . . . 17 (1 ∈ (ℤ‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1)))
241 ssralv 4010 . . . . . . . . . . . . . . . . 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))
242239, 240, 241mp2b 10 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
24362nncnd 12169 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ ℂ)
244 npcan1 11580 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
245243, 244syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
246 peano2zm 12546 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
247183, 246syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑁 − 1) ∈ ℤ)
248 uzid 12778 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
249 peano2uz 12826 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
250247, 248, 2493syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
251245, 250eqeltrrd 2839 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
252 fzss2 13481 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
253251, 252syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
254253sselda 3944 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁))
255254adantlr 713 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁))
256 simplr 767 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)))
257 ssel2 3939 . . . . . . . . . . . . . . . . . . . . . . 23 ((ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝 ∈ ((0...𝐾) ↑m (1...𝑁)))
258 elmapi 8787 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ∈ ((0...𝐾) ↑m (1...𝑁)) → 𝑝:(1...𝑁)⟶(0...𝐾))
259257, 258syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾))
260256, 259sylan 580 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾))
261 poimirlem28.3 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝐵 < 𝑛)
262 elfzelz 13441 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
263262zred 12607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
264263ltnrd 11289 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 ∈ (1...𝑁) → ¬ 𝑛 < 𝑛)
265 breq1 5108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 = 𝐵 → (𝑛 < 𝑛𝐵 < 𝑛))
266265notbid 317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = 𝐵 → (¬ 𝑛 < 𝑛 ↔ ¬ 𝐵 < 𝑛))
267264, 266syl5ibcom 244 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (1...𝑁) → (𝑛 = 𝐵 → ¬ 𝐵 < 𝑛))
268267necon2ad 2958 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (1...𝑁) → (𝐵 < 𝑛𝑛𝐵))
2692683ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0) → (𝐵 < 𝑛𝑛𝐵))
270269adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → (𝐵 < 𝑛𝑛𝐵))
271261, 270mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝑛𝐵)
2722713exp2 1354 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝𝑛) = 0 → 𝑛𝐵))))
273272imp31 418 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝𝑛) = 0 → 𝑛𝐵))
274273necon2d 2966 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝𝑛) ≠ 0))
275274adantllr 717 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝𝑛) ≠ 0))
276260, 275syldan 591 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → (𝑛 = 𝐵 → (𝑝𝑛) ≠ 0))
277276reximdva 3165 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
278255, 277syldan 591 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
279278ralimdva 3164 . . . . . . . . . . . . . . . . 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(𝑝𝑛) ≠ 0))
280279imp 407 . . . . . . . . . . . . . . . 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(𝑝𝑛) ≠ 0)
281242, 280sylan2 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(𝑝𝑛) ≠ 0)
282281biantrurd 533 . . . . . . . . . . . . . 14 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)))
283 nnuz 12806 . . . . . . . . . . . . . . . . . . . . . 22 ℕ = (ℤ‘1)
28462, 283eleqtrdi 2848 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ (ℤ‘1))
285 fzm1 13521 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘1) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
286284, 285syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
287 elun 4108 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}))
288177orbi2i 911 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
289287, 288bitri 274 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
290286, 289bitr4di 288 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑛 ∈ (1...𝑁) ↔ 𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
291290eqrdv 2734 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
292291raleqdv 3313 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∀𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
293 ralunb 4151 . . . . . . . . . . . . . . . . 17 (∀𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
294292, 293bitrdi 286 . . . . . . . . . . . . . . . 16 (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)))
295 fveq2 6842 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑁 → (𝑝𝑛) = (𝑝𝑁))
296295neeq1d 3003 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑁 → ((𝑝𝑛) ≠ 0 ↔ (𝑝𝑁) ≠ 0))
297296rexbidv 3175 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑁 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0))
298297ralsng 4634 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0))
29962, 298syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0))
300299anbi2d 629 . . . . . . . . . . . . . . . 16 (𝜑 → ((∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0) ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)))
301294, 300bitrd 278 . . . . . . . . . . . . . . 15 (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)))
302301ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)))
303 0z 12510 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℤ
304 1z 12533 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℤ
305 fzshftral 13529 . . . . . . . . . . . . . . . . . . . . . . 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))
306303, 304, 305mp3an13 1452 . . . . . . . . . . . . . . . . . . . . . 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))
307183, 246, 3063syl 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))
308 0p1e1 12275 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 + 1) = 1
309308a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (0 + 1) = 1)
310309, 245oveq12d 7375 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((0 + 1)...((𝑁 − 1) + 1)) = (1...𝑁))
311310raleqdv 3313 . . . . . . . . . . . . . . . . . . . . 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))
312307, 311bitrd 278 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
313 ovex 7390 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 − 1) ∈ V
314 eqeq1 2740 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = (𝑚 − 1) → (𝑛 = 𝐵 ↔ (𝑚 − 1) = 𝐵))
315314rexbidv 3175 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = (𝑚 − 1) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵))
316313, 315sbcie 3782 . . . . . . . . . . . . . . . . . . . . . 22 ([(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)
317316ralbii 3096 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)
318 oveq1 7364 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
319318eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑛 → ((𝑚 − 1) = 𝐵 ↔ (𝑛 − 1) = 𝐵))
320319rexbidv 3175 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑛 → (∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵))
321320cbvralvw 3225 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
322317, 321bitri 274 . . . . . . . . . . . . . . . . . . . 20 (∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
323312, 322bitrdi 286 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵))
324323biimpa 477 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
325324adantlr 713 . . . . . . . . . . . . . . . . 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)
326 poimirlem28.4 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
327326necomd 2999 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → (𝑛 − 1) ≠ 𝐵)
3283273exp2 1354 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵))))
329328imp31 418 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵))
330329necon2d 2966 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝𝑛) ≠ 𝐾))
331330adantllr 717 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝𝑛) ≠ 𝐾))
332260, 331syldan 591 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → ((𝑛 − 1) = 𝐵 → (𝑝𝑛) ≠ 𝐾))
333332reximdva 3165 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))
334333ralimdva 3164 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))
335334imp 407 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
336325, 335syldan 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(𝑝𝑛) ≠ 𝐾)
337336biantrud 532 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
338 r19.26 3114 . . . . . . . . . . . . . . 15 (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))
339337, 338bitr4di 288 . . . . . . . . . . . . . 14 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
340282, 302, 3393bitr2d 306 . . . . . . . . . . . . 13 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
341231, 238, 340syl2an 596 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ (0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵)) → (∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
342341pm5.32da 579 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0) ↔ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))))
343342anbi2d 629 . . . . . . . . . 10 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)) ↔ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))))
344343rexbidva 3173 . . . . . . . . 9 (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))))
345344adantr 481 . . . . . . . 8 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(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 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))))
346191rexeqdv 3314 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
347346biimpd 228 . . . . . . . . . . . . . . . . . 18 (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 → ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
348347ralimdv 3166 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
349169rexeqdv 3314 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑡) = 𝑁 → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶))
350349ralbidv 3174 . . . . . . . . . . . . . . . . . 18 ((2nd𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶))
351350imbi1d 341 . . . . . . . . . . . . . . . . 17 ((2nd𝑡) = 𝑁 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶)))
352348, 351syl5ibrcom 246 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶)))
353352com23 86 . . . . . . . . . . . . . . 15 (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → ((2nd𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶)))
354353imp 407 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → ((2nd𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
355354adantrd 492 . . . . . . . . . . . . 13 ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
356355pm4.71rd 563 . . . . . . . . . . . 12 ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
357 an12 643 . . . . . . . . . . . . 13 ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))) ↔ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))))
358 3anass 1095 . . . . . . . . . . . . . 14 ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))
359358anbi2i 623 . . . . . . . . . . . . 13 (((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) ↔ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))))
360357, 359bitr4i 277 . . . . . . . . . . . 12 ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))) ↔ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))
361356, 360bitrdi 286 . . . . . . . . . . 11 ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) ↔ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))))
362361notbid 317 . . . . . . . . . 10 ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) ↔ ¬ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))))
363362pm5.32da 579 . . . . . . . . 9 (𝜑 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
364363adantr 481 . . . . . . . 8 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
365229, 345, 3643bitr3d 308 . . . . . . 7 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(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 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
366365rabbidva 3414 . . . . . 6 (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(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 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))})
367 iunrab 5012 . . . . . 6 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(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 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}
368 difrab 4268 . . . . . 6 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))}
369366, 367, 3683eqtr4g 2801 . . . . 5 (𝜑 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} = ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}))
370369fveq2d 6846 . . . 4 (𝜑 → (♯‘ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}) = (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})))
37127, 28mp1i 13 . . . . 5 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} ∈ Fin)
372 simpl 483 . . . . . . . . . . . 12 ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))) → 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))))
373372a1i 11 . . . . . . . . . . 11 (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))) → 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
374373ss2rabi 4034 . . . . . . . . . 10 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
375374sseli 3940 . . . . . . . . 9 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})
376 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑠 → (2nd𝑡) = (2nd𝑠))
377376breq2d 5117 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑠 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑠)))
378377ifbid 4509 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑠 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)))
379378csbeq1d 3859 . . . . . . . . . . . . . . 15 (𝑡 = 𝑠if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
380 2fveq3 6847 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑠 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑠)))
381 2fveq3 6847 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑠 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑠)))
382381imaeq1d 6012 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑠 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑠)) “ (1...𝑗)))
383382xpeq1d 5662 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑠 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}))
384381imaeq1d 6012 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑠 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)))
385384xpeq1d 5662 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑠 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))
386383, 385uneq12d 4124 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑠 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))
387380, 386oveq12d 7375 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑠 → ((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))
388387csbeq2dv 3862 . . . . . . . . . . . . . . 15 (𝑡 = 𝑠if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))
389379, 388eqtrd 2776 . . . . . . . . . . . . . 14 (𝑡 = 𝑠if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))
390389mpteq2dv 5207 . . . . . . . . . . . . 13 (𝑡 = 𝑠 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))))
391390eqeq2d 2747 . . . . . . . . . . . 12 (𝑡 = 𝑠 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
392 eqcom 2743 . . . . . . . . . . . 12 (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)
393391, 392bitrdi 286 . . . . . . . . . . 11 (𝑡 = 𝑠 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥))
394393elrab 3645 . . . . . . . . . 10 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ↔ (𝑠 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥))
395394simprbi 497 . . . . . . . . 9 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)
396375, 395syl 17 . . . . . . . 8 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)
397396rgen 3066 . . . . . . 7 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥
398397rgenw 3068 . . . . . 6 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))∀𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥
399 invdisj 5089 . . . . . 6 (∀𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))∀𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥Disj 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})
400398, 399mp1i 13 . . . . 5 (𝜑Disj 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})
4018, 371, 400hashiun 15707 . . . 4 (𝜑 → (♯‘ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}) = Σ𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))
402370, 401eqtr3d 2778 . . 3 (𝜑 → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})) = Σ𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))
403 fo1st 7941 . . . . . . . . . . . . 13 1st :V–onto→V
404 fofun 6757 . . . . . . . . . . . . 13 (1st :V–onto→V → Fun 1st )
405403, 404ax-mp 5 . . . . . . . . . . . 12 Fun 1st
406 ssv 3968 . . . . . . . . . . . . 13 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ⊆ V
407 fof 6756 . . . . . . . . . . . . . . 15 (1st :V–onto→V → 1st :V⟶V)
408403, 407ax-mp 5 . . . . . . . . . . . . . 14 1st :V⟶V
409408fdmi 6680 . . . . . . . . . . . . 13 dom 1st = V
410406, 409sseqtrri 3981 . . . . . . . . . . . 12 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st
411 fores 6766 . . . . . . . . . . . 12 ((Fun 1st ∧ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st ) → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}))
412405, 410, 411mp2an 690 . . . . . . . . . . 11 (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})
413 fveqeq2 6851 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑥 → ((2nd𝑡) = 𝑁 ↔ (2nd𝑥) = 𝑁))
414 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑥 → (1st𝑡) = (1st𝑥))
415414csbeq1d 3859 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑥(1st𝑡) / 𝑠𝐶 = (1st𝑥) / 𝑠𝐶)
416415eqeq2d 2747 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 → (𝑖 = (1st𝑡) / 𝑠𝐶𝑖 = (1st𝑥) / 𝑠𝐶))
417416rexbidv 3175 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶))
418417ralbidv 3174 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶))
419 2fveq3 6847 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑥)))
420419fveq1d 6844 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥 → ((1st ‘(1st𝑡))‘𝑁) = ((1st ‘(1st𝑥))‘𝑁))
421420eqeq1d 2738 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥 → (((1st ‘(1st𝑡))‘𝑁) = 0 ↔ ((1st ‘(1st𝑥))‘𝑁) = 0))
422 2fveq3 6847 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑥)))
423422fveq1d 6844 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥 → ((2nd ‘(1st𝑡))‘𝑁) = ((2nd ‘(1st𝑥))‘𝑁))
424423eqeq1d 2738 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥 → (((2nd ‘(1st𝑡))‘𝑁) = 𝑁 ↔ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁))
425418, 421, 4243anbi123d 1436 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑥 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)))
426413, 425anbi12d 631 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑥 → (((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) ↔ ((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁))))
427426rexrab 3654 . . . . . . . . . . . . . . 15 (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠 ↔ ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠))
428 xp1st 7953 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
429428anim1i 615 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) → ((1st𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)))
430 eleq1 2825 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑥) = 𝑠 → ((1st𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})))
431 csbeq1a 3869 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑠 = (1st𝑥) → 𝐶 = (1st𝑥) / 𝑠𝐶)
432431eqcoms 2744 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1st𝑥) = 𝑠𝐶 = (1st𝑥) / 𝑠𝐶)
433432eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑥) = 𝑠(1st𝑥) / 𝑠𝐶 = 𝐶)
434433eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑥) = 𝑠 → (𝑖 = (1st𝑥) / 𝑠𝐶𝑖 = 𝐶))
435434rexbidv 3175 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑥) = 𝑠 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
436435ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑥) = 𝑠 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
437 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑥) = 𝑠 → (1st ‘(1st𝑥)) = (1st𝑠))
438437fveq1d 6844 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑥) = 𝑠 → ((1st ‘(1st𝑥))‘𝑁) = ((1st𝑠)‘𝑁))
439438eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑥) = 𝑠 → (((1st ‘(1st𝑥))‘𝑁) = 0 ↔ ((1st𝑠)‘𝑁) = 0))
440 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑥) = 𝑠 → (2nd ‘(1st𝑥)) = (2nd𝑠))
441440fveq1d 6844 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑥) = 𝑠 → ((2nd ‘(1st𝑥))‘𝑁) = ((2nd𝑠)‘𝑁))
442441eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑥) = 𝑠 → (((2nd ‘(1st𝑥))‘𝑁) = 𝑁 ↔ ((2nd𝑠)‘𝑁) = 𝑁))
443436, 439, 4423anbi123d 1436 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑥) = 𝑠 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)))
444430, 443anbi12d 631 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥) = 𝑠 → (((1st𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))))
445429, 444syl5ibcom 244 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) → ((1st𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))))
446445adantrl 714 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁))) → ((1st𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))))
447446expimpd 454 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))))
448447rexlimiv 3145 . . . . . . . . . . . . . . . 16 (∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)))
449 nn0fz0 13539 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
450171, 449sylib 217 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ (0...𝑁))
451 opelxpi 5670 . . . . . . . . . . . . . . . . . . . 20 ((𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑁 ∈ (0...𝑁)) → ⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
452450, 451sylan2 593 . . . . . . . . . . . . . . . . . . 19 ((𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝜑) → ⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
453452ancoms 459 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → ⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
454 opelxp2 5675 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → 𝑁 ∈ (0...𝑁))
455 op2ndg 7934 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd ‘⟨𝑠, 𝑁⟩) = 𝑁)
456455biantrurd 533 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((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⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁))))
457 op1stg 7933 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)
458 csbeq1a 3869 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑠 = (1st ‘⟨𝑠, 𝑁⟩) → 𝐶 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶)
459458eqcoms 2744 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((1st ‘⟨𝑠, 𝑁⟩) = 𝑠𝐶 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶)
460459eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((1st ‘⟨𝑠, 𝑁⟩) = 𝑠(1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 = 𝐶)
461457, 460syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 = 𝐶)
462461eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶𝑖 = 𝐶))
463462rexbidv 3175 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
464463ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
465457fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘(1st ‘⟨𝑠, 𝑁⟩)) = (1st𝑠))
466465fveq1d 6844 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = ((1st𝑠)‘𝑁))
467466eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ↔ ((1st𝑠)‘𝑁) = 0))
468457fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd ‘(1st ‘⟨𝑠, 𝑁⟩)) = (2nd𝑠))
469468fveq1d 6844 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = ((2nd𝑠)‘𝑁))
470469eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁 ↔ ((2nd𝑠)‘𝑁) = 𝑁))
471464, 467, 4703anbi123d 1436 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)))
472457biantrud 532 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((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⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)))
473456, 471, 4723bitr3d 308 . . . . . . . . . . . . . . . . . . . . 21 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁) ↔ (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)))
47444, 454, 473sylancr 587 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁) ↔ (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)))
475474biimpa 477 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)) → (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠))
476 fveqeq2 6851 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = ⟨𝑠, 𝑁⟩ → ((2nd𝑥) = 𝑁 ↔ (2nd ‘⟨𝑠, 𝑁⟩) = 𝑁))
477 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = ⟨𝑠, 𝑁⟩ → (1st𝑥) = (1st ‘⟨𝑠, 𝑁⟩))
478477csbeq1d 3859 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ⟨𝑠, 𝑁⟩ → (1st𝑥) / 𝑠𝐶 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶)
479478eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑠, 𝑁⟩ → (𝑖 = (1st𝑥) / 𝑠𝐶𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶))
480479rexbidv 3175 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑠, 𝑁⟩ → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶))
481480ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑠, 𝑁⟩ → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶))
482 2fveq3 6847 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑠, 𝑁⟩ → (1st ‘(1st𝑥)) = (1st ‘(1st ‘⟨𝑠, 𝑁⟩)))
483482fveq1d 6844 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑠, 𝑁⟩ → ((1st ‘(1st𝑥))‘𝑁) = ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁))
484483eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑠, 𝑁⟩ → (((1st ‘(1st𝑥))‘𝑁) = 0 ↔ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0))
485 2fveq3 6847 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑠, 𝑁⟩ → (2nd ‘(1st𝑥)) = (2nd ‘(1st ‘⟨𝑠, 𝑁⟩)))
486485fveq1d 6844 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑠, 𝑁⟩ → ((2nd ‘(1st𝑥))‘𝑁) = ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁))
487486eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑠, 𝑁⟩ → (((2nd ‘(1st𝑥))‘𝑁) = 𝑁 ↔ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁))
488481, 484, 4873anbi123d 1436 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = ⟨𝑠, 𝑁⟩ → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)))
489476, 488anbi12d 631 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = ⟨𝑠, 𝑁⟩ → (((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ↔ ((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁))))
490 fveqeq2 6851 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = ⟨𝑠, 𝑁⟩ → ((1st𝑥) = 𝑠 ↔ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠))
491489, 490anbi12d 631 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = ⟨𝑠, 𝑁⟩ → ((((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠) ↔ (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)))
492491rspcev 3581 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(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⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠))
493475, 492syldan 591 . . . . . . . . . . . . . . . . . 18 ((⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠))
494453, 493sylan 580 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠))
495494expl 458 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠)))
496448, 495impbid2 225 . . . . . . . . . . . . . . 15 (𝜑 → (∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠) ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))))
497427, 496bitrid 282 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠 ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))))
498497abbidv 2805 . . . . . . . . . . . . 13 (𝜑 → {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))})
499 dfimafn 6905 . . . . . . . . . . . . . . 15 ((Fun 1st ∧ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st ) → (1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑦})
500405, 410, 499mp2an 690 . . . . . . . . . . . . . 14 (1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑦}
501 nfv 1917 . . . . . . . . . . . . . . . . . 18 𝑠(2nd𝑡) = 𝑁
502 nfcv 2907 . . . . . . . . . . . . . . . . . . . 20 𝑠(0...(𝑁 − 1))
503 nfcsb1v 3880 . . . . . . . . . . . . . . . . . . . . . 22 𝑠(1st𝑡) / 𝑠𝐶
504503nfeq2 2924 . . . . . . . . . . . . . . . . . . . . 21 𝑠 𝑖 = (1st𝑡) / 𝑠𝐶
505502, 504nfrexw 3296 . . . . . . . . . . . . . . . . . . . 20 𝑠𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶
506502, 505nfralw 3294 . . . . . . . . . . . . . . . . . . 19 𝑠𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶
507 nfv 1917 . . . . . . . . . . . . . . . . . . 19 𝑠((1st ‘(1st𝑡))‘𝑁) = 0
508 nfv 1917 . . . . . . . . . . . . . . . . . . 19 𝑠((2nd ‘(1st𝑡))‘𝑁) = 𝑁
509506, 507, 508nf3an 1904 . . . . . . . . . . . . . . . . . 18 𝑠(∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)
510501, 509nfan 1902 . . . . . . . . . . . . . . . . 17 𝑠((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))
511 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑠((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))
512510, 511nfrabw 3440 . . . . . . . . . . . . . . . 16 𝑠{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}
513 nfv 1917 . . . . . . . . . . . . . . . 16 𝑠(1st𝑥) = 𝑦
514512, 513nfrexw 3296 . . . . . . . . . . . . . . 15 𝑠𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑦
515 nfv 1917 . . . . . . . . . . . . . . 15 𝑦𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠
516 eqeq2 2748 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑠 → ((1st𝑥) = 𝑦 ↔ (1st𝑥) = 𝑠))
517516rexbidv 3175 . . . . . . . . . . . . . . 15 (𝑦 = 𝑠 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑦 ↔ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠))
518514, 515, 517cbvabw 2810 . . . . . . . . . . . . . 14 {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑦} = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠}
519500, 518eqtri 2764 . . . . . . . . . . . . 13 (1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠}
520 df-rab 3408 . . . . . . . . . . . . 13 {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))}
521498, 519, 5203eqtr4g 2801 . . . . . . . . . . . 12 (𝜑 → (1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})
522 foeq3 6754 . . . . . . . . . . . 12 ((1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)}))
523521, 522syl 17 . . . . . . . . . . 11 (𝜑 → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)}))
524412, 523mpbii 232 . . . . . . . . . 10 (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})
525 fof 6756 . . . . . . . . . 10 ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})
526524, 525syl 17 . . . . . . . . 9 (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})
527 fvres 6861 . . . . . . . . . . . 12 (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})‘𝑥) = (1st𝑥))
528 fvres 6861 . . . . . . . . . . . 12 (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})‘𝑦) = (1st𝑦))
529527, 528eqeqan12d 2750 . . . . . . . . . . 11 ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})‘𝑦) ↔ (1st𝑥) = (1st𝑦)))
530 simpl 483 . . . . . . . . . . . . . . . 16 (((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) → (2nd𝑡) = 𝑁)
531530a1i 11 . . . . . . . . . . . . . . 15 (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) → (2nd𝑡) = 𝑁))
532531ss2rabi 4034 . . . . . . . . . . . . . 14 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd𝑡) = 𝑁}
533532sseli 3940 . . . . . . . . . . . . 13 (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} → 𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd𝑡) = 𝑁})
534413elrab 3645 . . . . . . . . . . . . 13 (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd𝑡) = 𝑁} ↔ (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd𝑥) = 𝑁))
535533, 534sylib 217 . . . . . . . . . . . 12 (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} → (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd𝑥) = 𝑁))
536532sseli 3940 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} → 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd𝑡) = 𝑁})
537 fveqeq2 6851 . . . . . . . . . . . . . 14 (𝑡 = 𝑦 → ((2nd𝑡) = 𝑁 ↔ (2nd𝑦) = 𝑁))
538537elrab 3645 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd𝑡) = 𝑁} ↔ (𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd𝑦) = 𝑁))
539536, 538sylib 217 . . . . . . . . . . . 12 (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} → (𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd𝑦) = 𝑁))
540 eqtr3 2762 . . . . . . . . . . . . . 14 (((2nd𝑥) = 𝑁 ∧ (2nd𝑦) = 𝑁) → (2nd𝑥) = (2nd𝑦))
541 xpopth 7962 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)) ↔ 𝑥 = 𝑦))
542541biimpd 228 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)) → 𝑥 = 𝑦))
543542ancomsd 466 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((2nd𝑥) = (2nd𝑦) ∧ (1st𝑥) = (1st𝑦)) → 𝑥 = 𝑦))
544543expdimp 453 . . . . . . . . . . . . . 14 (((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ (2nd𝑥) = (2nd𝑦)) → ((1st𝑥) = (1st𝑦) → 𝑥 = 𝑦))
545540, 544sylan2 593 . . . . . . . . . . . . 13 (((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ ((2nd𝑥) = 𝑁 ∧ (2nd𝑦) = 𝑁)) → ((1st𝑥) = (1st𝑦) → 𝑥 = 𝑦))
546545an4s 658 . . . . . . . . . . . 12 (((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd𝑥) = 𝑁) ∧ (𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd𝑦) = 𝑁)) → ((1st𝑥) = (1st𝑦) → 𝑥 = 𝑦))
547535, 539, 546syl2an 596 . . . . . . . . . . 11 ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) → ((1st𝑥) = (1st𝑦) → 𝑥 = 𝑦))
548529, 547sylbid 239 . . . . . . . . . 10 ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦))
549548rgen2 3194 . . . . . . . . 9 𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦)
550526, 549jctir 521 . . . . . . . 8 (𝜑 → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} ∧ ∀𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦)))
551 dff13 7202 . . . . . . . 8 ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–1-1→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} ↔ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} ∧ ∀𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦)))
552550, 551sylibr 233 . . . . . . 7 (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–1-1→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})
553 df-f1o 6503 . . . . . . 7 ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} ↔ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–1-1→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} ∧ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)}))
554552, 524, 553sylanbrc 583 . . . . . 6 (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})
555 rabfi 9213 . . . . . . . . 9 (((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ∈ Fin)
55627, 555ax-mp 5 . . . . . . . 8 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ∈ Fin
557556elexi 3464 . . . . . . 7 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ∈ V
558557f1oen 8913 . . . . . 6 ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})
559554, 558syl 17 . . . . 5 (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})
560 rabfi 9213 . . . . . . 7 ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} ∈ Fin)
56124, 560ax-mp 5 . . . . . 6 {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} ∈ Fin
562 hashen 14247 . . . . . 6 (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ∈ Fin ∧ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} ∈ Fin) → ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)}) ↔ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)}))
563556, 561, 562mp2an 690 . . . . 5 ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)}) ↔ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})
564559, 563sylibr 233 . . . 4 (𝜑 → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)}))
565564oveq2d 7373 . . 3 (𝜑 → ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})))
566201, 402, 5653eqtr3d 2784 . 2 (𝜑 → Σ𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})))
567164, 566breqtrd 5131 1 (𝜑 → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wex 1781  wcel 2106  ∃!weu 2566  {cab 2713  wne 2943  wral 3064  wrex 3073  ∃!wreu 3351  {crab 3407  Vcvv 3445  [wsbc 3739  csb 3855  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  ifcif 4486  {csn 4586  cop 4592   ciun 4954  Disj wdisj 5070   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Fun wfun 6490  wf 6492  1-1wf1 6493  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  f cof 7615  1st c1st 7919  2nd c2nd 7920  m cmap 8765  cen 8880  Fincfn 8883  cc 11049  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cmin 11385  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424  ..^cfzo 13567  chash 14230  Σcsu 15570  cdvds 16136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-dvds 16137
This theorem is referenced by:  poimirlem28  36106
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