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Theorem poimirlem27 37634
Description: Lemma for poimir 37640 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 14010 . . . . . 6 (0...𝐾) ∈ Fin
2 fzfi 14010 . . . . . 6 (1...𝑁) ∈ Fin
3 mapfi 9386 . . . . . 6 (((0...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((0...𝐾) ↑m (1...𝑁)) ∈ Fin)
41, 2, 3mp2an 692 . . . . 5 ((0...𝐾) ↑m (1...𝑁)) ∈ Fin
5 fzfi 14010 . . . . 5 (0...(𝑁 − 1)) ∈ Fin
6 mapfi 9386 . . . . 5 ((((0...𝐾) ↑m (1...𝑁)) ∈ Fin ∧ (0...(𝑁 − 1)) ∈ Fin) → (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) ∈ Fin)
74, 5, 6mp2an 692 . . . 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 12647 . . . 4 2 ∈ ℤ
109a1i 11 . . 3 (𝜑 → 2 ∈ ℤ)
11 fzofi 14012 . . . . . . . 8 (0..^𝐾) ∈ Fin
12 mapfi 9386 . . . . . . . 8 (((0..^𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((0..^𝐾) ↑m (1...𝑁)) ∈ Fin)
1311, 2, 12mp2an 692 . . . . . . 7 ((0..^𝐾) ↑m (1...𝑁)) ∈ Fin
14 mapfi 9386 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ↑m (1...𝑁)) ∈ Fin)
152, 2, 14mp2an 692 . . . . . . . 8 ((1...𝑁) ↑m (1...𝑁)) ∈ Fin
16 f1of 6849 . . . . . . . . . 10 (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁))
1716ss2abi 4077 . . . . . . . . 9 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓𝑓:(1...𝑁)⟶(1...𝑁)}
18 ovex 7464 . . . . . . . . . 10 (1...𝑁) ∈ V
1918, 18mapval 8877 . . . . . . . . 9 ((1...𝑁) ↑m (1...𝑁)) = {𝑓𝑓:(1...𝑁)⟶(1...𝑁)}
2017, 19sseqtrri 4033 . . . . . . . 8 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑m (1...𝑁))
21 ssfi 9212 . . . . . . . 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 692 . . . . . . 7 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin
23 xpfi 9356 . . . . . . 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 692 . . . . . 6 (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin
25 fzfi 14010 . . . . . 6 (0...𝑁) ∈ Fin
26 xpfi 9356 . . . . . 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 692 . . . . 5 ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin
28 rabfi 9301 . . . . 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 14392 . . . . 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 12637 . . . 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 3071 . . . . 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 1912 . . . . . 6 𝑡(𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))))
35 nfcv 2903 . . . . . . 7 𝑡2
36 nfcv 2903 . . . . . . 7 𝑡
37 nfcv 2903 . . . . . . . 8 𝑡
38 nfrab1 3454 . . . . . . . 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 6917 . . . . . . 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 5195 . . . . . 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 4358 . . . . . . . . . . . 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 16304 . . . . . . . . . . . . . . . . 17 (2 ∈ ℤ → 2 ∥ 2)
439, 42ax-mp 5 . . . . . . . . . . . . . . . 16 2 ∥ 2
44 vex 3482 . . . . . . . . . . . . . . . . . . 19 𝑠 ∈ V
45 hashsng 14405 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ V → (♯‘{𝑠}) = 1)
4644, 45ax-mp 5 . . . . . . . . . . . . . . . . . 18 (♯‘{𝑠}) = 1
4746oveq2i 7442 . . . . . . . . . . . . . . . . 17 (1 + (♯‘{𝑠})) = (1 + 1)
48 df-2 12327 . . . . . . . . . . . . . . . . 17 2 = (1 + 1)
4947, 48eqtr4i 2766 . . . . . . . . . . . . . . . 16 (1 + (♯‘{𝑠})) = 2
5043, 49breqtrri 5175 . . . . . . . . . . . . . . 15 2 ∥ (1 + (♯‘{𝑠}))
51 rabfi 9301 . . . . . . . . . . . . . . . . . . . 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 9214 . . . . . . . . . . . . . . . . . . . 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 9082 . . . . . . . . . . . . . . . . . . 19 {𝑠} ∈ Fin
55 disjdifr 4479 . . . . . . . . . . . . . . . . . . 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 14418 . . . . . . . . . . . . . . . . . . 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 1460 . . . . . . . . . . . . . . . . . 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 4815 . . . . . . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . . . . 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 2789 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑢 → (2nd𝑡) = (2nd𝑢))
6564breq2d 5160 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑢 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑢)))
6665ifbid 4554 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑢 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)))
6766csbeq1d 3912 . . . . . . . . . . . . . . . . . . . . . . . 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 6912 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑢 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑢)))
69 2fveq3 6912 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑡 = 𝑢 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑢)))
7069imaeq1d 6079 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑢 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑢)) “ (1...𝑗)))
7170xpeq1d 5718 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑢 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}))
7269imaeq1d 6079 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑢 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)))
7372xpeq1d 5718 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑢 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))
7471, 73uneq12d 4179 . . . . . . . . . . . . . . . . . . . . . . . . . 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 7449 . . . . . . . . . . . . . . . . . . . . . . . . 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 3915 . . . . . . . . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . . . . . . . . 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 5250 . . . . . . . . . . . . . . . . . . . . . 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 5151 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑤 → (𝑦 < (2nd𝑢) ↔ 𝑤 < (2nd𝑢)))
80 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑤𝑦 = 𝑤)
81 oveq1 7438 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1))
8279, 80, 81ifbieq12d 4559 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑤 → if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) = if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)))
8382csbeq1d 3912 . . . . . . . . . . . . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖))
8584imaeq2d 6080 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 𝑖 → ((2nd ‘(1st𝑢)) “ (1...𝑗)) = ((2nd ‘(1st𝑢)) “ (1...𝑖)))
8685xpeq1d 5718 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 𝑖 → (((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}))
87 oveq1 7438 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = 𝑖 → (𝑗 + 1) = (𝑖 + 1))
8887oveq1d 7446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑖 → ((𝑗 + 1)...𝑁) = ((𝑖 + 1)...𝑁))
8988imaeq2d 6080 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 𝑖 → ((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)))
9089xpeq1d 5718 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 𝑖 → (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))
9186, 90uneq12d 4179 . . . . . . . . . . . . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . . . . . . . . . . . . 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 3920 . . . . . . . . . . . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . . . . . . . . . . . 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 5261 . . . . . . . . . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . . . . . . . . . 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 2746 . . . . . . . . . . . . . . . . . . . 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 3444 . . . . . . . . . . . . . . . . . . 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 8888 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) → 𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
10099ad3antlr 731 . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . 22 ((∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)
103102ralimi 3081 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)
104103ad2antlr 727 . . . . . . . . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑝𝑛) = (𝑝𝑚))
106105neeq1d 2998 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝑝𝑛) ≠ 0 ↔ (𝑝𝑚) ≠ 0))
107106rexbidv 3177 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 0))
108 fveq1 6906 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = 𝑞 → (𝑝𝑚) = (𝑞𝑚))
109108neeq1d 2998 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = 𝑞 → ((𝑝𝑚) ≠ 0 ↔ (𝑞𝑚) ≠ 0))
110109cbvrexvw 3236 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 0)
111107, 110bitrdi 287 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 0))
112111rspccva 3621 . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . 22 ((∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
115114ralimi 3081 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
116115ad2antlr 727 . . . . . . . . . . . . . . . . . . . 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 2998 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝑝𝑛) ≠ 𝐾 ↔ (𝑝𝑚) ≠ 𝐾))
118117rexbidv 3177 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 𝐾))
119108neeq1d 2998 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = 𝑞 → ((𝑝𝑚) ≠ 𝐾 ↔ (𝑞𝑚) ≠ 𝐾))
120119cbvrexvw 3236 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 𝐾)
121118, 120bitrdi 287 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 𝐾))
122121rspccva 3621 . . . . . . . . . . . . . . . . . . . 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 37629 . . . . . . . . . . . . . . . . . 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 4791 . . . . . . . . . . . . . . . . . . . 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 3501 . . . . . . . . . . . . . . . . . . . 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 14456 . . . . . . . . . . . . . . . . . . . 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 3379 . . . . . . . . . . . . . . . . . . 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 304 . . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . 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 5186 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . 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 1931 . . . . . . . . . . . 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 242 . . . . . . . . . . 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 16306 . . . . . . . . . . . . . 14 (2 ∈ ℤ → 2 ∥ 0)
1409, 139ax-mp 5 . . . . . . . . . . . . 13 2 ∥ 0
141 hash0 14403 . . . . . . . . . . . . 13 (♯‘∅) = 0
142140, 141breqtrri 5175 . . . . . . . . . . . 12 2 ∥ (♯‘∅)
143 fveq2 6907 . . . . . . . . . . . 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 5186 . . . . . . . . . . 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 412 . . . . . . . . 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 490 . . . . . . . 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 527 . . . . . . . . . . 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 3441 . . . . . . . . . 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 6911 . . . . . . . . 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 5160 . . . . . . . 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 241 . . . . . . 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 3264 . . . . 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 243 . . . 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 484 . . . . . . . . 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 3081 . . . . . . 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 4394 . . . . . . 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 234 . . . . . 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 6911 . . . . 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 5186 . . . 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 16342 . 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 9301 . . . . 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 1135 . . . . . . 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 4641 . . . . . . . . . . . . 13 ((2nd𝑡) = 𝑁 → {(2nd𝑡)} = {𝑁})
169168difeq2d 4136 . . . . . . . . . . . 12 ((2nd𝑡) = 𝑁 → ((0...𝑁) ∖ {(2nd𝑡)}) = ((0...𝑁) ∖ {𝑁}))
170 difun2 4487 . . . . . . . . . . . . 13 (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((0...(𝑁 − 1)) ∖ {𝑁})
17162nnnn0d 12585 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℕ0)
172 nn0uz 12918 . . . . . . . . . . . . . . . . . 18 0 = (ℤ‘0)
173171, 172eleqtrdi 2849 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ (ℤ‘0))
174 fzm1 13644 . . . . . . . . . . . . . . . . 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 4163 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}))
177 velsn 4647 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ {𝑁} ↔ 𝑛 = 𝑁)
178177orbi2i 912 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
179176, 178bitri 275 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
180175, 179bitr4di 289 . . . . . . . . . . . . . . 15 (𝜑 → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})))
181180eqrdv 2733 . . . . . . . . . . . . . 14 (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁}))
182181difeq1d 4135 . . . . . . . . . . . . 13 (𝜑 → ((0...𝑁) ∖ {𝑁}) = (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}))
18362nnzd 12638 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℤ)
184 uzid 12891 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
185 uznfz 13647 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ𝑁) → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
186183, 184, 1853syl 18 . . . . . . . . . . . . . 14 (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
187 disjsn 4716 . . . . . . . . . . . . . . 15 (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (0...(𝑁 − 1)))
188 disj3 4460 . . . . . . . . . . . . . . 15 (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
189187, 188bitr3i 277 . . . . . . . . . . . . . 14 𝑁 ∈ (0...(𝑁 − 1)) ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
190186, 189sylib 218 . . . . . . . . . . . . 13 (𝜑 → (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
191170, 182, 1903eqtr4a 2801 . . . . . . . . . . . 12 (𝜑 → ((0...𝑁) ∖ {𝑁}) = (0...(𝑁 − 1)))
192169, 191sylan9eqr 2797 . . . . . . . . . . 11 ((𝜑 ∧ (2nd𝑡) = 𝑁) → ((0...𝑁) ∖ {(2nd𝑡)}) = (0...(𝑁 − 1)))
193192rexeqdv 3325 . . . . . . . . . 10 ((𝜑 ∧ (2nd𝑡) = 𝑁) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
194193biimprd 248 . . . . . . . . 9 ((𝜑 ∧ (2nd𝑡) = 𝑁) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
195194ralimdv 3167 . . . . . . . 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 453 . . . . . . 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 480 . . . . 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 4086 . . . 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 14448 . . . 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 480 . . . . . . . . . 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 715 . . . . . . . . . 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 8045 . . . . . . . . . . . 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 8045 . . . . . . . . . . . 12 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑡)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
208 elmapi 8888 . . . . . . . . . . . 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 481 . . . . . . . . . 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 8046 . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . 14 (2nd ‘(1st𝑡)) ∈ V
213 f1oeq1 6837 . . . . . . . . . . . . . 14 (𝑓 = (2nd ‘(1st𝑡)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)))
214212, 213elab 3681 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑡)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
215211, 214sylib 218 . . . . . . . . . . . 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 481 . . . . . . . . . 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 8046 . . . . . . . . . . 11 (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑡) ∈ (0...𝑁))
219218adantl 481 . . . . . . . . . 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 37631 . . . . . . . . 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 481 . . . . . . . . . 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 8052 . . . . . . . . . . . . . . 15 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st𝑡) = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩)
223222csbeq1d 3912 . . . . . . . . . . . . . 14 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st𝑡) / 𝑠𝐶 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶)
224223eqeq2d 2746 . . . . . . . . . . . . 13 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑖 = (1st𝑡) / 𝑠𝐶𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶))
225224rexbidv 3177 . . . . . . . . . . . 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 3176 . . . . . . . . . . 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 631 . . . . . . . . . 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 282 . . . . . . . 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 6745 . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . 14 ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵))
233 vex 3482 . . . . . . . . . . . . . . . 16 𝑛 ∈ V
234 eqid 2735 . . . . . . . . . . . . . . . . 17 (𝑝 ∈ ran 𝑥𝐵) = (𝑝 ∈ ran 𝑥𝐵)
235234elrnmpt 5972 . . . . . . . . . . . . . . . 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 3091 . . . . . . . . . . . . . 14 (∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
238232, 237sylbb 219 . . . . . . . . . . . . 13 ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) → ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
239 1eluzge0 12932 . . . . . . . . . . . . . . . . 17 1 ∈ (ℤ‘0)
240 fzss1 13600 . . . . . . . . . . . . . . . . 17 (1 ∈ (ℤ‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1)))
241 ssralv 4064 . . . . . . . . . . . . . . . . 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 12280 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ ℂ)
244 npcan1 11686 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
245243, 244syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
246 peano2zm 12658 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
247 uzid 12891 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
248 peano2uz 12941 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
249183, 246, 247, 2484syl 19 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
250245, 249eqeltrrd 2840 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
251 fzss2 13601 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
252250, 251syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
253252sselda 3995 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁))
254253adantlr 715 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁))
255 simplr 769 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)))
256 ssel2 3990 . . . . . . . . . . . . . . . . . . . . . . 23 ((ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝 ∈ ((0...𝐾) ↑m (1...𝑁)))
257 elmapi 8888 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ∈ ((0...𝐾) ↑m (1...𝑁)) → 𝑝:(1...𝑁)⟶(0...𝐾))
258256, 257syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾))
259255, 258sylan 580 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾))
260 poimirlem28.3 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝐵 < 𝑛)
261 elfzelz 13561 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
262261zred 12720 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
263262ltnrd 11393 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 ∈ (1...𝑁) → ¬ 𝑛 < 𝑛)
264 breq1 5151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 = 𝐵 → (𝑛 < 𝑛𝐵 < 𝑛))
265264notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = 𝐵 → (¬ 𝑛 < 𝑛 ↔ ¬ 𝐵 < 𝑛))
266263, 265syl5ibcom 245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (1...𝑁) → (𝑛 = 𝐵 → ¬ 𝐵 < 𝑛))
267266necon2ad 2953 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (1...𝑁) → (𝐵 < 𝑛𝑛𝐵))
2682673ad2ant1 1132 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0) → (𝐵 < 𝑛𝑛𝐵))
269268adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → (𝐵 < 𝑛𝑛𝐵))
270260, 269mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝑛𝐵)
2712703exp2 1353 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝𝑛) = 0 → 𝑛𝐵))))
272271imp31 417 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝𝑛) = 0 → 𝑛𝐵))
273272necon2d 2961 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝𝑛) ≠ 0))
274273adantllr 719 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝𝑛) ≠ 0))
275259, 274syldan 591 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → (𝑛 = 𝐵 → (𝑝𝑛) ≠ 0))
276275reximdva 3166 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
277254, 276syldan 591 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
278277ralimdva 3165 . . . . . . . . . . . . . . . . 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))
279278imp 406 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)
280242, 279sylan2 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)
281280biantrurd 532 . . . . . . . . . . . . . 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)))
282 nnuz 12919 . . . . . . . . . . . . . . . . . . . . . 22 ℕ = (ℤ‘1)
28362, 282eleqtrdi 2849 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ (ℤ‘1))
284 fzm1 13644 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘1) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
285283, 284syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
286 elun 4163 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}))
287177orbi2i 912 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
288286, 287bitri 275 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
289285, 288bitr4di 289 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑛 ∈ (1...𝑁) ↔ 𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
290289eqrdv 2733 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
291290raleqdv 3324 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∀𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
292 ralunb 4207 . . . . . . . . . . . . . . . . 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))
293291, 292bitrdi 287 . . . . . . . . . . . . . . . 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)))
294 fveq2 6907 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑁 → (𝑝𝑛) = (𝑝𝑁))
295294neeq1d 2998 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑁 → ((𝑝𝑛) ≠ 0 ↔ (𝑝𝑁) ≠ 0))
296295rexbidv 3177 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑁 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0))
297296ralsng 4680 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0))
29862, 297syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0))
299298anbi2d 630 . . . . . . . . . . . . . . . 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)))
300293, 299bitrd 279 . . . . . . . . . . . . . . 15 (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)))
301300ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)))
302 0z 12622 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℤ
303 1z 12645 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℤ
304 fzshftral 13652 . . . . . . . . . . . . . . . . . . . . . . 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))
305302, 303, 304mp3an13 1451 . . . . . . . . . . . . . . . . . . . . . 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))
306183, 246, 3053syl 18 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
307 0p1e1 12386 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 + 1) = 1
308307a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (0 + 1) = 1)
309308, 245oveq12d 7449 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((0 + 1)...((𝑁 − 1) + 1)) = (1...𝑁))
310309raleqdv 3324 . . . . . . . . . . . . . . . . . . . . 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))
311306, 310bitrd 279 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
312 ovex 7464 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 − 1) ∈ V
313 eqeq1 2739 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = (𝑚 − 1) → (𝑛 = 𝐵 ↔ (𝑚 − 1) = 𝐵))
314313rexbidv 3177 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = (𝑚 − 1) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵))
315312, 314sbcie 3835 . . . . . . . . . . . . . . . . . . . . . 22 ([(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)
316315ralbii 3091 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)
317 oveq1 7438 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
318317eqeq1d 2737 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑛 → ((𝑚 − 1) = 𝐵 ↔ (𝑛 − 1) = 𝐵))
319318rexbidv 3177 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑛 → (∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵))
320319cbvralvw 3235 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
321316, 320bitri 275 . . . . . . . . . . . . . . . . . . . 20 (∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
322311, 321bitrdi 287 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵))
323322biimpa 476 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
324323adantlr 715 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
325 poimirlem28.4 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
326325necomd 2994 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → (𝑛 − 1) ≠ 𝐵)
3273263exp2 1353 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵))))
328327imp31 417 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵))
329328necon2d 2961 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝𝑛) ≠ 𝐾))
330329adantllr 719 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝𝑛) ≠ 𝐾))
331259, 330syldan 591 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → ((𝑛 − 1) = 𝐵 → (𝑝𝑛) ≠ 𝐾))
332331reximdva 3166 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))
333332ralimdva 3165 . . . . . . . . . . . . . . . . . 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))
334333imp 406 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
335324, 334syldan 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)
336335biantrud 531 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
337 r19.26 3109 . . . . . . . . . . . . . . 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))
338336, 337bitr4di 289 . . . . . . . . . . . . . 14 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
339281, 301, 3383bitr2d 307 . . . . . . . . . . . . 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)))
340231, 238, 339syl2an 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)))
341340pm5.32da 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))))
342341anbi2d 630 . . . . . . . . . 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)))))
343342rexbidva 3175 . . . . . . . . 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)))))
344343adantr 480 . . . . . . . 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)))))
345191rexeqdv 3325 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
346345biimpd 229 . . . . . . . . . . . . . . . . . 18 (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 → ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
347346ralimdv 3167 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
348169rexeqdv 3325 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑡) = 𝑁 → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶))
349348ralbidv 3176 . . . . . . . . . . . . . . . . . 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))
350349imbi1d 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)))
351347, 350syl5ibrcom 247 . . . . . . . . . . . . . . . 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)))
352351com23 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)))
353352imp 406 . . . . . . . . . . . . . 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))
354353adantrd 491 . . . . . . . . . . . . 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))
355354pm4.71rd 562 . . . . . . . . . . . 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)))))
356 an12 645 . . . . . . . . . . . . 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))))
357 3anass 1094 . . . . . . . . . . . . . 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)))
358357anbi2i 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))))
359356, 358bitr4i 278 . . . . . . . . . . . 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)))
360355, 359bitrdi 287 . . . . . . . . . . 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))))
361360notbid 318 . . . . . . . . . 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))))
362361pm5.32da 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)))))
363362adantr 480 . . . . . . . 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)))))
364229, 344, 3633bitr3d 309 . . . . . . 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)))))
365364rabbidva 3440 . . . . . 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)))})
366 iunrab 5057 . . . . . 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)))}
367 difrab 4324 . . . . . 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)))}
368365, 366, 3673eqtr4g 2800 . . . . 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))}))
369368fveq2d 6911 . . . 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))})))
37027, 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)
371 simpl 482 . . . . . . . . . . . 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})))))
372371a1i 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}))))))
373372ss2rabi 4087 . . . . . . . . . 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}))))}
374373sseli 3991 . . . . . . . . 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}))))})
375 fveq2 6907 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑠 → (2nd𝑡) = (2nd𝑠))
376375breq2d 5160 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑠 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑠)))
377376ifbid 4554 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑠 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)))
378377csbeq1d 3912 . . . . . . . . . . . . . . 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}))))
379 2fveq3 6912 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑠 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑠)))
380 2fveq3 6912 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑠 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑠)))
381380imaeq1d 6079 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑠 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑠)) “ (1...𝑗)))
382381xpeq1d 5718 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑠 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}))
383380imaeq1d 6079 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑠 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)))
384383xpeq1d 5718 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑠 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))
385382, 384uneq12d 4179 . . . . . . . . . . . . . . . . 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})))
386379, 385oveq12d 7449 . . . . . . . . . . . . . . . 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}))))
387386csbeq2dv 3915 . . . . . . . . . . . . . . 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}))))
388378, 387eqtrd 2775 . . . . . . . . . . . . . 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}))))
389388mpteq2dv 5250 . . . . . . . . . . . . 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})))))
390389eqeq2d 2746 . . . . . . . . . . . 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}))))))
391 eqcom 2742 . . . . . . . . . . . 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)
392390, 391bitrdi 287 . . . . . . . . . . 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))
393392elrab 3695 . . . . . . . . . 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))
394393simprbi 496 . . . . . . . . 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)
395374, 394syl 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)
396395rgen 3061 . . . . . . 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
397396rgenw 3063 . . . . . 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
398 invdisj 5134 . . . . . 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)))})
399397, 398mp1i 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)))})
4008, 370, 399hashiun 15855 . . . 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)))}))
401369, 400eqtr3d 2777 . . 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)))}))
402 fo1st 8033 . . . . . . . . . . . . 13 1st :V–onto→V
403 fofun 6822 . . . . . . . . . . . . 13 (1st :V–onto→V → Fun 1st )
404402, 403ax-mp 5 . . . . . . . . . . . 12 Fun 1st
405 ssv 4020 . . . . . . . . . . . . 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
406 fof 6821 . . . . . . . . . . . . . . 15 (1st :V–onto→V → 1st :V⟶V)
407402, 406ax-mp 5 . . . . . . . . . . . . . 14 1st :V⟶V
408407fdmi 6748 . . . . . . . . . . . . 13 dom 1st = V
409405, 408sseqtrri 4033 . . . . . . . . . . . 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
410 fores 6831 . . . . . . . . . . . 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))}))
411404, 409, 410mp2an 692 . . . . . . . . . . 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))})
412 fveqeq2 6916 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑥 → ((2nd𝑡) = 𝑁 ↔ (2nd𝑥) = 𝑁))
413 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑥 → (1st𝑡) = (1st𝑥))
414413csbeq1d 3912 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑥(1st𝑡) / 𝑠𝐶 = (1st𝑥) / 𝑠𝐶)
415414eqeq2d 2746 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 → (𝑖 = (1st𝑡) / 𝑠𝐶𝑖 = (1st𝑥) / 𝑠𝐶))
416415rexbidv 3177 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶))
417416ralbidv 3176 . . . . . . . . . . . . . . . . . 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))
418 2fveq3 6912 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑥)))
419418fveq1d 6909 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥 → ((1st ‘(1st𝑡))‘𝑁) = ((1st ‘(1st𝑥))‘𝑁))
420419eqeq1d 2737 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥 → (((1st ‘(1st𝑡))‘𝑁) = 0 ↔ ((1st ‘(1st𝑥))‘𝑁) = 0))
421 2fveq3 6912 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑥)))
422421fveq1d 6909 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥 → ((2nd ‘(1st𝑡))‘𝑁) = ((2nd ‘(1st𝑥))‘𝑁))
423422eqeq1d 2737 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥 → (((2nd ‘(1st𝑡))‘𝑁) = 𝑁 ↔ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁))
424417, 420, 4233anbi123d 1435 . . . . . . . . . . . . . . . . 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)))
425412, 424anbi12d 632 . . . . . . . . . . . . . . . 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))))
426425rexrab 3705 . . . . . . . . . . . . . . 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))
427 xp1st 8045 . . . . . . . . . . . . . . . . . . . . 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)}))
428427anim1i 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)))
429 eleq1 2827 . . . . . . . . . . . . . . . . . . . . 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)})))
430 csbeq1a 3922 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑠 = (1st𝑥) → 𝐶 = (1st𝑥) / 𝑠𝐶)
431430eqcoms 2743 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1st𝑥) = 𝑠𝐶 = (1st𝑥) / 𝑠𝐶)
432431eqcomd 2741 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑥) = 𝑠(1st𝑥) / 𝑠𝐶 = 𝐶)
433432eqeq2d 2746 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑥) = 𝑠 → (𝑖 = (1st𝑥) / 𝑠𝐶𝑖 = 𝐶))
434433rexbidv 3177 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑥) = 𝑠 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
435434ralbidv 3176 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑥) = 𝑠 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
436 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑥) = 𝑠 → (1st ‘(1st𝑥)) = (1st𝑠))
437436fveq1d 6909 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑥) = 𝑠 → ((1st ‘(1st𝑥))‘𝑁) = ((1st𝑠)‘𝑁))
438437eqeq1d 2737 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑥) = 𝑠 → (((1st ‘(1st𝑥))‘𝑁) = 0 ↔ ((1st𝑠)‘𝑁) = 0))
439 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑥) = 𝑠 → (2nd ‘(1st𝑥)) = (2nd𝑠))
440439fveq1d 6909 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑥) = 𝑠 → ((2nd ‘(1st𝑥))‘𝑁) = ((2nd𝑠)‘𝑁))
441440eqeq1d 2737 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑥) = 𝑠 → (((2nd ‘(1st𝑥))‘𝑁) = 𝑁 ↔ ((2nd𝑠)‘𝑁) = 𝑁))
442435, 438, 4413anbi123d 1435 . . . . . . . . . . . . . . . . . . . . 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)))
443429, 442anbi12d 632 . . . . . . . . . . . . . . . . . . . 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))))
444428, 443syl5ibcom 245 . . . . . . . . . . . . . . . . . . 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))))
445444adantrl 716 . . . . . . . . . . . . . . . . . 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))))
446445expimpd 453 . . . . . . . . . . . . . . . . 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))))
447446rexlimiv 3146 . . . . . . . . . . . . . . . 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)))
448 nn0fz0 13662 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
449171, 448sylib 218 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ (0...𝑁))
450 opelxpi 5726 . . . . . . . . . . . . . . . . . . . 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)))
451449, 450sylan2 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)))
452451ancoms 458 . . . . . . . . . . . . . . . . . 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)))
453 opelxp2 5732 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → 𝑁 ∈ (0...𝑁))
454 op2ndg 8026 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd ‘⟨𝑠, 𝑁⟩) = 𝑁)
455454biantrurd 532 . . . . . . . . . . . . . . . . . . . . . 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))))
456 op1stg 8025 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)
457 csbeq1a 3922 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑠 = (1st ‘⟨𝑠, 𝑁⟩) → 𝐶 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶)
458457eqcoms 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((1st ‘⟨𝑠, 𝑁⟩) = 𝑠𝐶 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶)
459458eqcomd 2741 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((1st ‘⟨𝑠, 𝑁⟩) = 𝑠(1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 = 𝐶)
460456, 459syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 = 𝐶)
461460eqeq2d 2746 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶𝑖 = 𝐶))
462461rexbidv 3177 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
463462ralbidv 3176 . . . . . . . . . . . . . . . . . . . . . . 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))
464456fveq2d 6911 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘(1st ‘⟨𝑠, 𝑁⟩)) = (1st𝑠))
465464fveq1d 6909 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = ((1st𝑠)‘𝑁))
466465eqeq1d 2737 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ↔ ((1st𝑠)‘𝑁) = 0))
467456fveq2d 6911 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd ‘(1st ‘⟨𝑠, 𝑁⟩)) = (2nd𝑠))
468467fveq1d 6909 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = ((2nd𝑠)‘𝑁))
469468eqeq1d 2737 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁 ↔ ((2nd𝑠)‘𝑁) = 𝑁))
470463, 466, 4693anbi123d 1435 . . . . . . . . . . . . . . . . . . . . . 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)))
471456biantrud 531 . . . . . . . . . . . . . . . . . . . . . 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)))
472455, 470, 4713bitr3d 309 . . . . . . . . . . . . . . . . . . . . 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)))
47344, 453, 472sylancr 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)))
474473biimpa 476 . . . . . . . . . . . . . . . . . . 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))
475 fveqeq2 6916 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = ⟨𝑠, 𝑁⟩ → ((2nd𝑥) = 𝑁 ↔ (2nd ‘⟨𝑠, 𝑁⟩) = 𝑁))
476 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = ⟨𝑠, 𝑁⟩ → (1st𝑥) = (1st ‘⟨𝑠, 𝑁⟩))
477476csbeq1d 3912 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ⟨𝑠, 𝑁⟩ → (1st𝑥) / 𝑠𝐶 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶)
478477eqeq2d 2746 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑠, 𝑁⟩ → (𝑖 = (1st𝑥) / 𝑠𝐶𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶))
479478rexbidv 3177 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑠, 𝑁⟩ → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶))
480479ralbidv 3176 . . . . . . . . . . . . . . . . . . . . . . 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))
481 2fveq3 6912 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑠, 𝑁⟩ → (1st ‘(1st𝑥)) = (1st ‘(1st ‘⟨𝑠, 𝑁⟩)))
482481fveq1d 6909 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑠, 𝑁⟩ → ((1st ‘(1st𝑥))‘𝑁) = ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁))
483482eqeq1d 2737 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑠, 𝑁⟩ → (((1st ‘(1st𝑥))‘𝑁) = 0 ↔ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0))
484 2fveq3 6912 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑠, 𝑁⟩ → (2nd ‘(1st𝑥)) = (2nd ‘(1st ‘⟨𝑠, 𝑁⟩)))
485484fveq1d 6909 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑠, 𝑁⟩ → ((2nd ‘(1st𝑥))‘𝑁) = ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁))
486485eqeq1d 2737 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑠, 𝑁⟩ → (((2nd ‘(1st𝑥))‘𝑁) = 𝑁 ↔ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁))
487480, 483, 4863anbi123d 1435 . . . . . . . . . . . . . . . . . . . . . 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)))
488475, 487anbi12d 632 . . . . . . . . . . . . . . . . . . . . 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))))
489 fveqeq2 6916 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = ⟨𝑠, 𝑁⟩ → ((1st𝑥) = 𝑠 ↔ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠))
490488, 489anbi12d 632 . . . . . . . . . . . . . . . . . . . 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)))
491490rspcev 3622 . . . . . . . . . . . . . . . . . . 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))
492474, 491syldan 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))
493452, 492sylan 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))
494493expl 457 . . . . . . . . . . . . . . . 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)))
495447, 494impbid2 226 . . . . . . . . . . . . . . 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))))
496426, 495bitrid 283 . . . . . . . . . . . . . 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))))
497496abbidv 2806 . . . . . . . . . . . . 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))})
498 dfimafn 6971 . . . . . . . . . . . . . . 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})
499404, 409, 498mp2an 692 . . . . . . . . . . . . . 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}
500 nfv 1912 . . . . . . . . . . . . . . . . . 18 𝑠(2nd𝑡) = 𝑁
501 nfcv 2903 . . . . . . . . . . . . . . . . . . . 20 𝑠(0...(𝑁 − 1))
502 nfcsb1v 3933 . . . . . . . . . . . . . . . . . . . . . 22 𝑠(1st𝑡) / 𝑠𝐶
503502nfeq2 2921 . . . . . . . . . . . . . . . . . . . . 21 𝑠 𝑖 = (1st𝑡) / 𝑠𝐶
504501, 503nfrexw 3311 . . . . . . . . . . . . . . . . . . . 20 𝑠𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶
505501, 504nfralw 3309 . . . . . . . . . . . . . . . . . . 19 𝑠𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶
506 nfv 1912 . . . . . . . . . . . . . . . . . . 19 𝑠((1st ‘(1st𝑡))‘𝑁) = 0
507 nfv 1912 . . . . . . . . . . . . . . . . . . 19 𝑠((2nd ‘(1st𝑡))‘𝑁) = 𝑁
508505, 506, 507nf3an 1899 . . . . . . . . . . . . . . . . . 18 𝑠(∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)
509500, 508nfan 1897 . . . . . . . . . . . . . . . . 17 𝑠((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))
510 nfcv 2903 . . . . . . . . . . . . . . . . 17 𝑠((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))
511509, 510nfrabw 3473 . . . . . . . . . . . . . . . 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))}
512 nfv 1912 . . . . . . . . . . . . . . . 16 𝑠(1st𝑥) = 𝑦
513511, 512nfrexw 3311 . . . . . . . . . . . . . . 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
514 nfv 1912 . . . . . . . . . . . . . . 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
515 eqeq2 2747 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑠 → ((1st𝑥) = 𝑦 ↔ (1st𝑥) = 𝑠))
516515rexbidv 3177 . . . . . . . . . . . . . . 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))
517513, 514, 516cbvabw 2811 . . . . . . . . . . . . . 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}
518499, 517eqtri 2763 . . . . . . . . . . . . 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}
519 df-rab 3434 . . . . . . . . . . . . 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))}
520497, 518, 5193eqtr4g 2800 . . . . . . . . . . . 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)})
521 foeq3 6819 . . . . . . . . . . . 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)}))
522520, 521syl 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)}))
523411, 522mpbii 233 . . . . . . . . . 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)})
524 fof 6821 . . . . . . . . . 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)})
525523, 524syl 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)})
526 fvres 6926 . . . . . . . . . . . 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))
527 fvres 6926 . . . . . . . . . . . 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))
528526, 527eqeqan12d 2749 . . . . . . . . . . 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)))
529 simpl 482 . . . . . . . . . . . . . . . 16 (((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) → (2nd𝑡) = 𝑁)
530529a1i 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))
531530ss2rabi 4087 . . . . . . . . . . . . . 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}
532531sseli 3991 . . . . . . . . . . . . 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})
533412elrab 3695 . . . . . . . . . . . . 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))
534532, 533sylib 218 . . . . . . . . . . . 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))
535531sseli 3991 . . . . . . . . . . . . 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})
536 fveqeq2 6916 . . . . . . . . . . . . . 14 (𝑡 = 𝑦 → ((2nd𝑡) = 𝑁 ↔ (2nd𝑦) = 𝑁))
537536elrab 3695 . . . . . . . . . . . . 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))
538535, 537sylib 218 . . . . . . . . . . . 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))
539 eqtr3 2761 . . . . . . . . . . . . . 14 (((2nd𝑥) = 𝑁 ∧ (2nd𝑦) = 𝑁) → (2nd𝑥) = (2nd𝑦))
540 xpopth 8054 . . . . . . . . . . . . . . . . 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))
541540biimpd 229 . . . . . . . . . . . . . . . 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))
542541ancomsd 465 . . . . . . . . . . . . . . 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))
543542expdimp 452 . . . . . . . . . . . . . 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))
544539, 543sylan2 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))
545544an4s 660 . . . . . . . . . . . 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))
546534, 538, 545syl2an 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))
547528, 546sylbid 240 . . . . . . . . . 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))
548547rgen2 3197 . . . . . . . . 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)
549525, 548jctir 520 . . . . . . . 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)))
550 dff13 7275 . . . . . . . 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)))
551549, 550sylibr 234 . . . . . . 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)})
552 df-f1o 6570 . . . . . . 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)}))
553551, 523, 552sylanbrc 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)})
554 rabfi 9301 . . . . . . . . 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)
55527, 554ax-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
556555elexi 3501 . . . . . . 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
557556f1oen 9012 . . . . . 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)})
558553, 557syl 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)})
559 rabfi 9301 . . . . . . 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)
56024, 559ax-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
561 hashen 14383 . . . . . 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)}))
562555, 560, 561mp2an 692 . . . . 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)})
563558, 562sylibr 234 . . . 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)}))
564563oveq2d 7447 . . 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)})))
565201, 401, 5643eqtr3d 2783 . 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)})))
566164, 565breqtrd 5174 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 206  wa 395  wo 847  w3a 1086   = wceq 1537  wex 1776  wcel 2106  ∃!weu 2566  {cab 2712  wne 2938  wral 3059  wrex 3068  ∃!wreu 3376  {crab 3433  Vcvv 3478  [wsbc 3791  csb 3908  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  ifcif 4531  {csn 4631  cop 4637   ciun 4996  Disj wdisj 5115   class class class wbr 5148  cmpt 5231   × cxp 5687  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Fun wfun 6557  wf 6559  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  f cof 7695  1st c1st 8011  2nd c2nd 8012  m cmap 8865  cen 8981  Fincfn 8984  cc 11151  0cc0 11153  1c1 11154   + caddc 11156   < clt 11293  cmin 11490  cn 12264  2c2 12319  0cn0 12524  cz 12611  cuz 12876  ...cfz 13544  ..^cfzo 13691  chash 14366  Σcsu 15719  cdvds 16287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-dvds 16288
This theorem is referenced by:  poimirlem28  37635
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