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Theorem poimirlem26 33106
Description: Lemma for poimir 33113 showing an even difference between the number of admissible faces and the number of admissible simplices. Equation (6) 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))
Assertion
Ref Expression
poimirlem26 (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)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))
Distinct variable groups:   𝑓,𝑖,𝑗,𝑝,𝑠,𝑡   𝜑,𝑗   𝑗,𝑁   𝜑,𝑖,𝑝,𝑠,𝑡   𝐵,𝑓,𝑖,𝑗,𝑠,𝑡   𝑓,𝐾,𝑖,𝑗,𝑝,𝑠,𝑡   𝑓,𝑁,𝑖,𝑝,𝑠,𝑡   𝐶,𝑖,𝑝,𝑡
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑝)   𝐶(𝑓,𝑗,𝑠)

Proof of Theorem poimirlem26
Dummy variables 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzofi 12729 . . . . . 6 (0..^𝐾) ∈ Fin
2 fzfi 12727 . . . . . 6 (1...𝑁) ∈ Fin
3 mapfi 8222 . . . . . 6 (((0..^𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((0..^𝐾) ↑𝑚 (1...𝑁)) ∈ Fin)
41, 2, 3mp2an 707 . . . . 5 ((0..^𝐾) ↑𝑚 (1...𝑁)) ∈ Fin
5 mapfi 8222 . . . . . . 7 (((1...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ↑𝑚 (1...𝑁)) ∈ Fin)
62, 2, 5mp2an 707 . . . . . 6 ((1...𝑁) ↑𝑚 (1...𝑁)) ∈ Fin
7 f1of 6104 . . . . . . . 8 (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁))
87ss2abi 3659 . . . . . . 7 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓𝑓:(1...𝑁)⟶(1...𝑁)}
9 ovex 6643 . . . . . . . 8 (1...𝑁) ∈ V
109, 9mapval 7829 . . . . . . 7 ((1...𝑁) ↑𝑚 (1...𝑁)) = {𝑓𝑓:(1...𝑁)⟶(1...𝑁)}
118, 10sseqtr4i 3623 . . . . . 6 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑𝑚 (1...𝑁))
12 ssfi 8140 . . . . . 6 ((((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)
136, 11, 12mp2an 707 . . . . 5 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin
144, 13pm3.2i 471 . . . 4 (((0..^𝐾) ↑𝑚 (1...𝑁)) ∈ Fin ∧ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
15 xpfi 8191 . . . 4 ((((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)
1614, 15mp1i 13 . . 3 (𝜑 → (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin)
17 2z 11369 . . . 4 2 ∈ ℤ
1817a1i 11 . . 3 (𝜑 → 2 ∈ ℤ)
19 snfi 7998 . . . . . . 7 {𝑥} ∈ Fin
20 fzfi 12727 . . . . . . . 8 (0...𝑁) ∈ Fin
21 rabfi 8145 . . . . . . . 8 ((0...𝑁) ∈ Fin → {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} ∈ Fin)
2220, 21ax-mp 5 . . . . . . 7 {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} ∈ Fin
23 xpfi 8191 . . . . . . 7 (({𝑥} ∈ Fin ∧ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} ∈ Fin) → ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ∈ Fin)
2419, 22, 23mp2an 707 . . . . . 6 ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ∈ Fin
25 hashcl 13103 . . . . . 6 (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ∈ Fin → (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})) ∈ ℕ0)
2624, 25ax-mp 5 . . . . 5 (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})) ∈ ℕ0
2726nn0zi 11362 . . . 4 (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})) ∈ ℤ
2827a1i 11 . . 3 ((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})) ∈ ℤ)
29 poimir.0 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ)
3029ad2antrr 761 . . . . . . . 8 (((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) → 𝑁 ∈ ℕ)
31 nfv 1840 . . . . . . . . . 10 𝑗 𝑝 = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd𝑡) “ ((𝑘 + 1)...𝑁)) × {0})))
32 nfcsb1v 3535 . . . . . . . . . . 11 𝑗𝑘 / 𝑗𝑡 / 𝑠𝐶
3332nfeq2 2776 . . . . . . . . . 10 𝑗 𝐵 = 𝑘 / 𝑗𝑡 / 𝑠𝐶
3431, 33nfim 1822 . . . . . . . . 9 𝑗(𝑝 = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))) → 𝐵 = 𝑘 / 𝑗𝑡 / 𝑠𝐶)
35 oveq2 6623 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (1...𝑗) = (1...𝑘))
3635imaeq2d 5435 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → ((2nd𝑡) “ (1...𝑗)) = ((2nd𝑡) “ (1...𝑘)))
3736xpeq1d 5108 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (((2nd𝑡) “ (1...𝑗)) × {1}) = (((2nd𝑡) “ (1...𝑘)) × {1}))
38 oveq1 6622 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1))
3938oveq1d 6630 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → ((𝑗 + 1)...𝑁) = ((𝑘 + 1)...𝑁))
4039imaeq2d 5435 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → ((2nd𝑡) “ ((𝑗 + 1)...𝑁)) = ((2nd𝑡) “ ((𝑘 + 1)...𝑁)))
4140xpeq1d 5108 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (((2nd𝑡) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))
4237, 41uneq12d 3752 . . . . . . . . . . . 12 (𝑗 = 𝑘 → ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd𝑡) “ ((𝑘 + 1)...𝑁)) × {0})))
4342oveq2d 6631 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))))
4443eqeq2d 2631 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑝 = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) ↔ 𝑝 = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd𝑡) “ ((𝑘 + 1)...𝑁)) × {0})))))
45 csbeq1a 3528 . . . . . . . . . . 11 (𝑗 = 𝑘𝑡 / 𝑠𝐶 = 𝑘 / 𝑗𝑡 / 𝑠𝐶)
4645eqeq2d 2631 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝐵 = 𝑡 / 𝑠𝐶𝐵 = 𝑘 / 𝑗𝑡 / 𝑠𝐶))
4744, 46imbi12d 334 . . . . . . . . 9 (𝑗 = 𝑘 → ((𝑝 = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝑡 / 𝑠𝐶) ↔ (𝑝 = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))) → 𝐵 = 𝑘 / 𝑗𝑡 / 𝑠𝐶)))
48 nfv 1840 . . . . . . . . . . 11 𝑠 𝑝 = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑁)) × {0})))
49 nfcsb1v 3535 . . . . . . . . . . . 12 𝑠𝑡 / 𝑠𝐶
5049nfeq2 2776 . . . . . . . . . . 11 𝑠 𝐵 = 𝑡 / 𝑠𝐶
5148, 50nfim 1822 . . . . . . . . . 10 𝑠(𝑝 = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝑡 / 𝑠𝐶)
52 fveq2 6158 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → (1st𝑠) = (1st𝑡))
53 fveq2 6158 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑡 → (2nd𝑠) = (2nd𝑡))
5453imaeq1d 5434 . . . . . . . . . . . . . . 15 (𝑠 = 𝑡 → ((2nd𝑠) “ (1...𝑗)) = ((2nd𝑡) “ (1...𝑗)))
5554xpeq1d 5108 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → (((2nd𝑠) “ (1...𝑗)) × {1}) = (((2nd𝑡) “ (1...𝑗)) × {1}))
5653imaeq1d 5434 . . . . . . . . . . . . . . 15 (𝑠 = 𝑡 → ((2nd𝑠) “ ((𝑗 + 1)...𝑁)) = ((2nd𝑡) “ ((𝑗 + 1)...𝑁)))
5756xpeq1d 5108 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))
5855, 57uneq12d 3752 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑁)) × {0})))
5952, 58oveq12d 6633 . . . . . . . . . . . 12 (𝑠 = 𝑡 → ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))))
6059eqeq2d 2631 . . . . . . . . . . 11 (𝑠 = 𝑡 → (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ↔ 𝑝 = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑁)) × {0})))))
61 csbeq1a 3528 . . . . . . . . . . . 12 (𝑠 = 𝑡𝐶 = 𝑡 / 𝑠𝐶)
6261eqeq2d 2631 . . . . . . . . . . 11 (𝑠 = 𝑡 → (𝐵 = 𝐶𝐵 = 𝑡 / 𝑠𝐶))
6360, 62imbi12d 334 . . . . . . . . . 10 (𝑠 = 𝑡 → ((𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) ↔ (𝑝 = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝑡 / 𝑠𝐶)))
64 poimirlem28.1 . . . . . . . . . 10 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
6551, 63, 64chvar 2261 . . . . . . . . 9 (𝑝 = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝑡 / 𝑠𝐶)
6634, 47, 65chvar 2261 . . . . . . . 8 (𝑝 = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))) → 𝐵 = 𝑘 / 𝑗𝑡 / 𝑠𝐶)
67 poimirlem28.2 . . . . . . . . 9 ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
6867ad4ant14 1290 . . . . . . . 8 ((((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
69 xp1st 7158 . . . . . . . . . 10 (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st𝑥) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
70 elmapi 7839 . . . . . . . . . 10 ((1st𝑥) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st𝑥):(1...𝑁)⟶(0..^𝐾))
7169, 70syl 17 . . . . . . . . 9 (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st𝑥):(1...𝑁)⟶(0..^𝐾))
7271ad2antlr 762 . . . . . . . 8 (((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) → (1st𝑥):(1...𝑁)⟶(0..^𝐾))
73 xp2nd 7159 . . . . . . . . . 10 (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd𝑥) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
74 fvex 6168 . . . . . . . . . . 11 (2nd𝑥) ∈ V
75 f1oeq1 6094 . . . . . . . . . . 11 (𝑓 = (2nd𝑥) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd𝑥):(1...𝑁)–1-1-onto→(1...𝑁)))
7674, 75elab 3338 . . . . . . . . . 10 ((2nd𝑥) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd𝑥):(1...𝑁)–1-1-onto→(1...𝑁))
7773, 76sylib 208 . . . . . . . . 9 (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd𝑥):(1...𝑁)–1-1-onto→(1...𝑁))
7877ad2antlr 762 . . . . . . . 8 (((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) → (2nd𝑥):(1...𝑁)–1-1-onto→(1...𝑁))
79 nfcv 2761 . . . . . . . . . . . . 13 𝑗𝑁
80 nfcv 2761 . . . . . . . . . . . . . 14 𝑗𝑥
8180, 32nfcsb 3537 . . . . . . . . . . . . 13 𝑗𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶
8279, 81nfne 2890 . . . . . . . . . . . 12 𝑗 𝑁𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶
83 nfcv 2761 . . . . . . . . . . . . . . 15 𝑡𝐶
8483, 49, 61cbvcsb 3524 . . . . . . . . . . . . . 14 𝑥 / 𝑠𝐶 = 𝑥 / 𝑡𝑡 / 𝑠𝐶
8545csbeq2dv 3970 . . . . . . . . . . . . . 14 (𝑗 = 𝑘𝑥 / 𝑡𝑡 / 𝑠𝐶 = 𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶)
8684, 85syl5eq 2667 . . . . . . . . . . . . 13 (𝑗 = 𝑘𝑥 / 𝑠𝐶 = 𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶)
8786neeq2d 2850 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝑁𝑥 / 𝑠𝐶𝑁𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶))
8882, 87rspc 3293 . . . . . . . . . . 11 (𝑘 ∈ (0...𝑁) → (∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶𝑁𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶))
8988impcom 446 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶𝑘 ∈ (0...𝑁)) → 𝑁𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶)
9089adantll 749 . . . . . . . . 9 ((((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶)
91 1st2nd2 7165 . . . . . . . . . . 11 (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
9291csbeq1d 3526 . . . . . . . . . 10 (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → 𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶 = ⟨(1st𝑥), (2nd𝑥)⟩ / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶)
9392ad3antlr 766 . . . . . . . . 9 ((((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) ∧ 𝑘 ∈ (0...𝑁)) → 𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶 = ⟨(1st𝑥), (2nd𝑥)⟩ / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶)
9490, 93neeqtrd 2859 . . . . . . . 8 ((((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁⟨(1st𝑥), (2nd𝑥)⟩ / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶)
9530, 66, 68, 72, 78, 94poimirlem25 33105 . . . . . . 7 (((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) → 2 ∥ (#‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⟨(1st𝑥), (2nd𝑥)⟩ / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶}))
96 nfv 1840 . . . . . . . . . . . . . 14 𝑘 𝑖 = 𝑥 / 𝑠𝐶
9781nfeq2 2776 . . . . . . . . . . . . . 14 𝑗 𝑖 = 𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶
9886eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝑖 = 𝑥 / 𝑠𝐶𝑖 = 𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶))
9996, 97, 98cbvrex 3160 . . . . . . . . . . . . 13 (∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ↔ ∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶)
10092eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑖 = 𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶𝑖 = ⟨(1st𝑥), (2nd𝑥)⟩ / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶))
101100rexbidv 3047 . . . . . . . . . . . . 13 (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶 ↔ ∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⟨(1st𝑥), (2nd𝑥)⟩ / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶))
10299, 101syl5rbb 273 . . . . . . . . . . . 12 (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⟨(1st𝑥), (2nd𝑥)⟩ / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶))
103102ralbidv 2982 . . . . . . . . . . 11 (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⟨(1st𝑥), (2nd𝑥)⟩ / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶))
104 iba 524 . . . . . . . . . . 11 (∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)))
105103, 104sylan9bb 735 . . . . . . . . . 10 ((𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⟨(1st𝑥), (2nd𝑥)⟩ / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)))
106105rabbidv 3181 . . . . . . . . 9 ((𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) → {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⟨(1st𝑥), (2nd𝑥)⟩ / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶} = {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})
107106fveq2d 6162 . . . . . . . 8 ((𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) → (#‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⟨(1st𝑥), (2nd𝑥)⟩ / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶}) = (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}))
108107adantll 749 . . . . . . 7 (((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) → (#‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⟨(1st𝑥), (2nd𝑥)⟩ / 𝑡𝑘 / 𝑗𝑡 / 𝑠𝐶}) = (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}))
10995, 108breqtrd 4649 . . . . . 6 (((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) → 2 ∥ (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}))
110109ex 450 . . . . 5 ((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶 → 2 ∥ (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})))
111 dvds0 14940 . . . . . . . 8 (2 ∈ ℤ → 2 ∥ 0)
11217, 111ax-mp 5 . . . . . . 7 2 ∥ 0
113 hash0 13114 . . . . . . 7 (#‘∅) = 0
114112, 113breqtrri 4650 . . . . . 6 2 ∥ (#‘∅)
115 simpr 477 . . . . . . . . . 10 ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) → ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)
116115con3i 150 . . . . . . . . 9 (¬ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶 → ¬ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))
117116ralrimivw 2963 . . . . . . . 8 (¬ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶 → ∀𝑦 ∈ (0...𝑁) ¬ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))
118 rabeq0 3937 . . . . . . . 8 ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} = ∅ ↔ ∀𝑦 ∈ (0...𝑁) ¬ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))
119117, 118sylibr 224 . . . . . . 7 (¬ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶 → {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} = ∅)
120119fveq2d 6162 . . . . . 6 (¬ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶 → (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) = (#‘∅))
121114, 120syl5breqr 4661 . . . . 5 (¬ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶 → 2 ∥ (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}))
122110, 121pm2.61d1 171 . . . 4 ((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → 2 ∥ (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}))
123 hashxp 13177 . . . . . 6 (({𝑥} ∈ Fin ∧ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} ∈ Fin) → (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})) = ((#‘{𝑥}) · (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})))
12419, 22, 123mp2an 707 . . . . 5 (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})) = ((#‘{𝑥}) · (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}))
125 vex 3193 . . . . . . 7 𝑥 ∈ V
126 hashsng 13115 . . . . . . 7 (𝑥 ∈ V → (#‘{𝑥}) = 1)
127125, 126ax-mp 5 . . . . . 6 (#‘{𝑥}) = 1
128127oveq1i 6625 . . . . 5 ((#‘{𝑥}) · (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})) = (1 · (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}))
129 hashcl 13103 . . . . . . . 8 ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} ∈ Fin → (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ∈ ℕ0)
13022, 129ax-mp 5 . . . . . . 7 (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ∈ ℕ0
131130nn0cni 11264 . . . . . 6 (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ∈ ℂ
132131mulid2i 10003 . . . . 5 (1 · (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})) = (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})
133124, 128, 1323eqtri 2647 . . . 4 (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})) = (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})
134122, 133syl6breqr 4665 . . 3 ((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → 2 ∥ (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})))
13516, 18, 28, 134fsumdvds 14973 . 2 (𝜑 → 2 ∥ Σ𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})))
1364, 13, 15mp2an 707 . . . . . 6 (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin
137 xpfi 8191 . . . . . 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)
138136, 20, 137mp2an 707 . . . . 5 ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin
139 rabfi 8145 . . . . 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)
140138, 139ax-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
14129nncnd 10996 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℂ)
142 npcan1 10415 . . . . . . . . . . . 12 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
143141, 142syl 17 . . . . . . . . . . 11 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
144 nnm1nn0 11294 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
14529, 144syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) ∈ ℕ0)
146145nn0zd 11440 . . . . . . . . . . . 12 (𝜑 → (𝑁 − 1) ∈ ℤ)
147 uzid 11662 . . . . . . . . . . . 12 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
148 peano2uz 11701 . . . . . . . . . . . 12 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
149146, 147, 1483syl 18 . . . . . . . . . . 11 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
150143, 149eqeltrrd 2699 . . . . . . . . . 10 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
151 fzss2 12339 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
152 ssralv 3651 . . . . . . . . . 10 ((0...(𝑁 − 1)) ⊆ (0...𝑁) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶))
153150, 151, 1523syl 18 . . . . . . . . 9 (𝜑 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶))
154153adantr 481 . . . . . . . 8 ((𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶))
155 raldifb 3734 . . . . . . . . . . . 12 (∀𝑗 ∈ (0...𝑁)(𝑗 ∉ {(2nd𝑡)} → ¬ 𝑖 = (1st𝑡) / 𝑠𝐶) ↔ ∀𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ¬ 𝑖 = (1st𝑡) / 𝑠𝐶)
156 nfv 1840 . . . . . . . . . . . . . . 15 𝑗𝜑
157 nfcsb1v 3535 . . . . . . . . . . . . . . . 16 𝑗(2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶
158157nfeq2 2776 . . . . . . . . . . . . . . 15 𝑗 𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶
159156, 158nfan 1825 . . . . . . . . . . . . . 14 𝑗(𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶)
160 nfv 1840 . . . . . . . . . . . . . 14 𝑗 𝑖 ∈ (0...(𝑁 − 1))
161159, 160nfan 1825 . . . . . . . . . . . . 13 𝑗((𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1)))
162 nnel 2902 . . . . . . . . . . . . . . . . 17 𝑗 ∉ {(2nd𝑡)} ↔ 𝑗 ∈ {(2nd𝑡)})
163 velsn 4171 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ {(2nd𝑡)} ↔ 𝑗 = (2nd𝑡))
164162, 163bitri 264 . . . . . . . . . . . . . . . 16 𝑗 ∉ {(2nd𝑡)} ↔ 𝑗 = (2nd𝑡))
165 csbeq1a 3528 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (2nd𝑡) → (1st𝑡) / 𝑠𝐶 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶)
166165eqeq2d 2631 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (2nd𝑡) → (𝑁 = (1st𝑡) / 𝑠𝐶𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶))
167166biimparc 504 . . . . . . . . . . . . . . . . . . 19 ((𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶𝑗 = (2nd𝑡)) → 𝑁 = (1st𝑡) / 𝑠𝐶)
16829nnred 10995 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑁 ∈ ℝ)
169168ltm1d 10916 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑁 − 1) < 𝑁)
170145nn0red 11312 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑁 − 1) ∈ ℝ)
171170, 168ltnled 10144 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
172169, 171mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
173 elfzle2 12303 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ (0...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
174172, 173nsyl 135 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
175 eleq1 2686 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑁 → (𝑖 ∈ (0...(𝑁 − 1)) ↔ 𝑁 ∈ (0...(𝑁 − 1))))
176175notbid 308 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑁 → (¬ 𝑖 ∈ (0...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (0...(𝑁 − 1))))
177174, 176syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑖 = 𝑁 → ¬ 𝑖 ∈ (0...(𝑁 − 1))))
178177con2d 129 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑖 ∈ (0...(𝑁 − 1)) → ¬ 𝑖 = 𝑁))
179178imp 445 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (0...(𝑁 − 1))) → ¬ 𝑖 = 𝑁)
180 eqeq2 2632 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 = (1st𝑡) / 𝑠𝐶 → (𝑖 = 𝑁𝑖 = (1st𝑡) / 𝑠𝐶))
181180notbid 308 . . . . . . . . . . . . . . . . . . . 20 (𝑁 = (1st𝑡) / 𝑠𝐶 → (¬ 𝑖 = 𝑁 ↔ ¬ 𝑖 = (1st𝑡) / 𝑠𝐶))
182179, 181syl5ibcom 235 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0...(𝑁 − 1))) → (𝑁 = (1st𝑡) / 𝑠𝐶 → ¬ 𝑖 = (1st𝑡) / 𝑠𝐶))
183167, 182syl5 34 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0...(𝑁 − 1))) → ((𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶𝑗 = (2nd𝑡)) → ¬ 𝑖 = (1st𝑡) / 𝑠𝐶))
184183expdimp 453 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0...(𝑁 − 1))) ∧ 𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) → (𝑗 = (2nd𝑡) → ¬ 𝑖 = (1st𝑡) / 𝑠𝐶))
185184an32s 845 . . . . . . . . . . . . . . . 16 (((𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (𝑗 = (2nd𝑡) → ¬ 𝑖 = (1st𝑡) / 𝑠𝐶))
186164, 185syl5bi 232 . . . . . . . . . . . . . . 15 (((𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (¬ 𝑗 ∉ {(2nd𝑡)} → ¬ 𝑖 = (1st𝑡) / 𝑠𝐶))
187 idd 24 . . . . . . . . . . . . . . 15 (((𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (¬ 𝑖 = (1st𝑡) / 𝑠𝐶 → ¬ 𝑖 = (1st𝑡) / 𝑠𝐶))
188186, 187jad 174 . . . . . . . . . . . . . 14 (((𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → ((𝑗 ∉ {(2nd𝑡)} → ¬ 𝑖 = (1st𝑡) / 𝑠𝐶) → ¬ 𝑖 = (1st𝑡) / 𝑠𝐶))
189188adantr 481 . . . . . . . . . . . . 13 ((((𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑗 ∉ {(2nd𝑡)} → ¬ 𝑖 = (1st𝑡) / 𝑠𝐶) → ¬ 𝑖 = (1st𝑡) / 𝑠𝐶))
190161, 189ralimdaa 2954 . . . . . . . . . . . 12 (((𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (∀𝑗 ∈ (0...𝑁)(𝑗 ∉ {(2nd𝑡)} → ¬ 𝑖 = (1st𝑡) / 𝑠𝐶) → ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = (1st𝑡) / 𝑠𝐶))
191155, 190syl5bir 233 . . . . . . . . . . 11 (((𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (∀𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ¬ 𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = (1st𝑡) / 𝑠𝐶))
192191con3d 148 . . . . . . . . . 10 (((𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (¬ ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = (1st𝑡) / 𝑠𝐶 → ¬ ∀𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ¬ 𝑖 = (1st𝑡) / 𝑠𝐶))
193 dfrex2 2992 . . . . . . . . . 10 (∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ¬ ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = (1st𝑡) / 𝑠𝐶)
194 dfrex2 2992 . . . . . . . . . 10 (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ¬ ∀𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ¬ 𝑖 = (1st𝑡) / 𝑠𝐶)
195192, 193, 1943imtr4g 285 . . . . . . . . 9 (((𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
196195ralimdva 2958 . . . . . . . 8 ((𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
197154, 196syld 47 . . . . . . 7 ((𝜑𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
198197expimpd 628 . . . . . 6 (𝜑 → ((𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
199198adantr 481 . . . . 5 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
200199ss2rabdv 3668 . . . 4 (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)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})
201 hashssdif 13156 . . . 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)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)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)) ∣ ∀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)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)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)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)})))
202140, 200, 201sylancr 694 . . 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)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)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)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)})))
203 xp2nd 7159 . . . . . . . 8 (𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑡) ∈ (0...𝑁))
204 df-ne 2791 . . . . . . . . . . . 12 (𝑁(1st𝑡) / 𝑠𝐶 ↔ ¬ 𝑁 = (1st𝑡) / 𝑠𝐶)
205204ralbii 2976 . . . . . . . . . . 11 (∀𝑗 ∈ (0...𝑁)𝑁(1st𝑡) / 𝑠𝐶 ↔ ∀𝑗 ∈ (0...𝑁) ¬ 𝑁 = (1st𝑡) / 𝑠𝐶)
206 ralnex 2988 . . . . . . . . . . 11 (∀𝑗 ∈ (0...𝑁) ¬ 𝑁 = (1st𝑡) / 𝑠𝐶 ↔ ¬ ∃𝑗 ∈ (0...𝑁)𝑁 = (1st𝑡) / 𝑠𝐶)
207205, 206bitri 264 . . . . . . . . . 10 (∀𝑗 ∈ (0...𝑁)𝑁(1st𝑡) / 𝑠𝐶 ↔ ¬ ∃𝑗 ∈ (0...𝑁)𝑁 = (1st𝑡) / 𝑠𝐶)
20829nnnn0d 11311 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℕ0)
209 nn0uz 11682 . . . . . . . . . . . . . . . . . . 19 0 = (ℤ‘0)
210208, 209syl6eleq 2708 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (ℤ‘0))
211143, 210eqeltrd 2698 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘0))
212 fzsplit2 12324 . . . . . . . . . . . . . . . . 17 ((((𝑁 − 1) + 1) ∈ (ℤ‘0) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
213211, 150, 212syl2anc 692 . . . . . . . . . . . . . . . 16 (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
214143oveq1d 6630 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
21529nnzd 11441 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℤ)
216 fzsn 12341 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
217215, 216syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁...𝑁) = {𝑁})
218214, 217eqtrd 2655 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
219218uneq2d 3751 . . . . . . . . . . . . . . . 16 (𝜑 → ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((0...(𝑁 − 1)) ∪ {𝑁}))
220213, 219eqtrd 2655 . . . . . . . . . . . . . . 15 (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁}))
221220raleqdv 3137 . . . . . . . . . . . . . 14 (𝜑 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∀𝑖 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶))
222 difss 3721 . . . . . . . . . . . . . . . . . 18 ((0...𝑁) ∖ {(2nd𝑡)}) ⊆ (0...𝑁)
223 ssrexv 3652 . . . . . . . . . . . . . . . . . 18 (((0...𝑁) ∖ {(2nd𝑡)}) ⊆ (0...𝑁) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → ∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶))
224222, 223ax-mp 5 . . . . . . . . . . . . . . . . 17 (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → ∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)
225224ralimi 2948 . . . . . . . . . . . . . . . 16 (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)
226225biantrurd 529 . . . . . . . . . . . . . . 15 (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)))
227 ralunb 3778 . . . . . . . . . . . . . . 15 (∀𝑖 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶))
228226, 227syl6rbbr 279 . . . . . . . . . . . . . 14 (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → (∀𝑖 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶))
229221, 228sylan9bb 735 . . . . . . . . . . . . 13 ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶))
230229adantlr 750 . . . . . . . . . . . 12 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶))
231 nn0fz0 12394 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
232208, 231sylib 208 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (0...𝑁))
233232ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → 𝑁 ∈ (0...𝑁))
234 eqeq1 2625 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑁 → (𝑖 = (1st𝑡) / 𝑠𝐶𝑁 = (1st𝑡) / 𝑠𝐶))
235234rexbidv 3047 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑁 → (∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = (1st𝑡) / 𝑠𝐶))
236235rspcva 3297 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶) → ∃𝑗 ∈ (0...𝑁)𝑁 = (1st𝑡) / 𝑠𝐶)
237 nfv 1840 . . . . . . . . . . . . . . . . 17 𝑗(𝜑 ∧ (2nd𝑡) ∈ (0...𝑁))
238 nfcv 2761 . . . . . . . . . . . . . . . . . 18 𝑗(0...(𝑁 − 1))
239 nfre1 3001 . . . . . . . . . . . . . . . . . 18 𝑗𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶
240238, 239nfral 2941 . . . . . . . . . . . . . . . . 17 𝑗𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶
241237, 240nfan 1825 . . . . . . . . . . . . . . . 16 𝑗((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶)
242 eleq1 2686 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 = (1st𝑡) / 𝑠𝐶 → (𝑁 ∈ (0...(𝑁 − 1)) ↔ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))))
243242notbid 308 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 = (1st𝑡) / 𝑠𝐶 → (¬ 𝑁 ∈ (0...(𝑁 − 1)) ↔ ¬ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))))
244174, 243syl5ibcom 235 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑁 = (1st𝑡) / 𝑠𝐶 → ¬ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))))
245244ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (𝑁 = (1st𝑡) / 𝑠𝐶 → ¬ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))))
246 eldifsn 4294 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ↔ (𝑗 ∈ (0...𝑁) ∧ 𝑗 ≠ (2nd𝑡)))
247 diffi 8152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((0...𝑁) ∈ Fin → ((0...𝑁) ∖ {(2nd𝑡)}) ∈ Fin)
24820, 247ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((0...𝑁) ∖ {(2nd𝑡)}) ∈ Fin
249 ssrab2 3672 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ⊆ ((0...𝑁) ∖ {(2nd𝑡)})
250 ssdomg 7961 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((0...𝑁) ∖ {(2nd𝑡)}) ∈ Fin → ({𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ⊆ ((0...𝑁) ∖ {(2nd𝑡)}) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ≼ ((0...𝑁) ∖ {(2nd𝑡)})))
251248, 249, 250mp2 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ≼ ((0...𝑁) ∖ {(2nd𝑡)})
252 hashdifsn 13158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((0...𝑁) ∈ Fin ∧ (2nd𝑡) ∈ (0...𝑁)) → (#‘((0...𝑁) ∖ {(2nd𝑡)})) = ((#‘(0...𝑁)) − 1))
25320, 252mpan 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((2nd𝑡) ∈ (0...𝑁) → (#‘((0...𝑁) ∖ {(2nd𝑡)})) = ((#‘(0...𝑁)) − 1))
254 1cnd 10016 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → 1 ∈ ℂ)
255141, 254, 254addsubd 10373 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → ((𝑁 + 1) − 1) = ((𝑁 − 1) + 1))
256 hashfz0 13175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑁 ∈ ℕ0 → (#‘(0...𝑁)) = (𝑁 + 1))
257208, 256syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → (#‘(0...𝑁)) = (𝑁 + 1))
258257oveq1d 6630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → ((#‘(0...𝑁)) − 1) = ((𝑁 + 1) − 1))
259 hashfz0 13175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑁 − 1) ∈ ℕ0 → (#‘(0...(𝑁 − 1))) = ((𝑁 − 1) + 1))
260145, 259syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → (#‘(0...(𝑁 − 1))) = ((𝑁 − 1) + 1))
261255, 258, 2603eqtr4d 2665 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑 → ((#‘(0...𝑁)) − 1) = (#‘(0...(𝑁 − 1))))
262253, 261sylan9eqr 2677 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) → (#‘((0...𝑁) ∖ {(2nd𝑡)})) = (#‘(0...(𝑁 − 1))))
263 fzfi 12727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (0...(𝑁 − 1)) ∈ Fin
264 hashen 13091 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((0...𝑁) ∖ {(2nd𝑡)}) ∈ Fin ∧ (0...(𝑁 − 1)) ∈ Fin) → ((#‘((0...𝑁) ∖ {(2nd𝑡)})) = (#‘(0...(𝑁 − 1))) ↔ ((0...𝑁) ∖ {(2nd𝑡)}) ≈ (0...(𝑁 − 1))))
265248, 263, 264mp2an 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘((0...𝑁) ∖ {(2nd𝑡)})) = (#‘(0...(𝑁 − 1))) ↔ ((0...𝑁) ∖ {(2nd𝑡)}) ≈ (0...(𝑁 − 1)))
266262, 265sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) → ((0...𝑁) ∖ {(2nd𝑡)}) ≈ (0...(𝑁 − 1)))
267 rabfi 8145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((0...𝑁) ∖ {(2nd𝑡)}) ∈ Fin → {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∈ Fin)
268248, 267ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∈ Fin
269 eleq1 2686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑖 = (1st𝑡) / 𝑠𝐶 → (𝑖 ∈ (0...(𝑁 − 1)) ↔ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))))
270269biimpac 503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑖 = (1st𝑡) / 𝑠𝐶) → (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1)))
271 rabid 3110 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↔ (𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∧ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))))
272271simplbi2com 656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1)) → (𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}))
273270, 272syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑖 = (1st𝑡) / 𝑠𝐶) → (𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}))
274273impancom 456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})) → (𝑖 = (1st𝑡) / 𝑠𝐶𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}))
275274ancrd 576 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})) → (𝑖 = (1st𝑡) / 𝑠𝐶 → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = (1st𝑡) / 𝑠𝐶)))
276275expimpd 628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑖 ∈ (0...(𝑁 − 1)) → ((𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∧ 𝑖 = (1st𝑡) / 𝑠𝐶) → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = (1st𝑡) / 𝑠𝐶)))
277276reximdv2 3010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → ∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}𝑖 = (1st𝑡) / 𝑠𝐶))
278271simplbi 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} → 𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}))
279274pm4.71rd 666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})) → (𝑖 = (1st𝑡) / 𝑠𝐶 ↔ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = (1st𝑡) / 𝑠𝐶)))
280 df-mpt 4685 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶) = {⟨𝑘, 𝑖⟩ ∣ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)}
281 nfv 1840 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 𝑘(𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = (1st𝑡) / 𝑠𝐶)
282 nfrab1 3115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 𝑗{𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}
283282nfcri 2755 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 𝑗 𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}
284 nfcsb1v 3535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 𝑗𝑘 / 𝑗(1st𝑡) / 𝑠𝐶
285284nfeq2 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 𝑗 𝑖 = 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶
286283, 285nfan 1825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)
287 eleq1 2686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑗 = 𝑘 → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↔ 𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}))
288 csbeq1a 3528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (𝑗 = 𝑘(1st𝑡) / 𝑠𝐶 = 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)
289288eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑗 = 𝑘 → (𝑖 = (1st𝑡) / 𝑠𝐶𝑖 = 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶))
290287, 289anbi12d 746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑗 = 𝑘 → ((𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = (1st𝑡) / 𝑠𝐶) ↔ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)))
291281, 286, 290cbvopab1 4695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 {⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = (1st𝑡) / 𝑠𝐶)} = {⟨𝑘, 𝑖⟩ ∣ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)}
292280, 291eqtr4i 2646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶) = {⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = (1st𝑡) / 𝑠𝐶)}
293292breqi 4629 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖𝑗{⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = (1st𝑡) / 𝑠𝐶)}𝑖)
294 df-br 4624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑗{⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = (1st𝑡) / 𝑠𝐶)}𝑖 ↔ ⟨𝑗, 𝑖⟩ ∈ {⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = (1st𝑡) / 𝑠𝐶)})
295 opabid 4952 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (⟨𝑗, 𝑖⟩ ∈ {⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = (1st𝑡) / 𝑠𝐶)} ↔ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = (1st𝑡) / 𝑠𝐶))
296293, 294, 2953bitri 286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖 ↔ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = (1st𝑡) / 𝑠𝐶))
297279, 296syl6bbr 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})) → (𝑖 = (1st𝑡) / 𝑠𝐶𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖))
298278, 297sylan2 491 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}) → (𝑖 = (1st𝑡) / 𝑠𝐶𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖))
299298rexbidva 3044 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖))
300 nfcv 2761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑝{𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}
301 nfv 1840 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑝 𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖
302 nfcv 2761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 𝑗𝑝
303282, 284nfmpt 4716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)
304 nfcv 2761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 𝑗𝑖
305302, 303, 304nfbr 4669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑗 𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖
306 breq1 4626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑗 = 𝑝 → (𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖))
307282, 300, 301, 305, 306cbvrexf 3158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖 ↔ ∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖)
308299, 307syl6bb 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖))
309277, 308sylibd 229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → ∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖))
310309ralimia 2946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖)
311 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶) = (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)
312 nfcv 2761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑗𝑘
313 nfcv 2761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑗((0...𝑁) ∖ {(2nd𝑡)})
314284nfel1 2775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑗𝑘 / 𝑗(1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))
315288eleq1d 2683 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 = 𝑘 → ((1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1)) ↔ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))))
316312, 313, 314, 315elrabf 3348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↔ (𝑘 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∧ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))))
317316simprbi 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} → 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1)))
318311, 317fmpti 6349 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}⟶(0...(𝑁 − 1))
319310, 318jctil 559 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → ((𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}⟶(0...(𝑁 − 1)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖))
320 dffo4 6341 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}–onto→(0...(𝑁 − 1)) ↔ ((𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}⟶(0...(𝑁 − 1)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)𝑖))
321319, 320sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}–onto→(0...(𝑁 − 1)))
322 fodomfi 8199 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (({𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ∈ Fin ∧ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↦ 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}–onto→(0...(𝑁 − 1))) → (0...(𝑁 − 1)) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))})
323268, 321, 322sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → (0...(𝑁 − 1)) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))})
324 endomtr 7974 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((0...𝑁) ∖ {(2nd𝑡)}) ≈ (0...(𝑁 − 1)) ∧ (0...(𝑁 − 1)) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}) → ((0...𝑁) ∖ {(2nd𝑡)}) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))})
325266, 323, 324syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → ((0...𝑁) ∖ {(2nd𝑡)}) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))})
326 sbth 8040 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (({𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ≼ ((0...𝑁) ∖ {(2nd𝑡)}) ∧ ((0...𝑁) ∖ {(2nd𝑡)}) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))}) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ≈ ((0...𝑁) ∖ {(2nd𝑡)}))
327251, 325, 326sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ≈ ((0...𝑁) ∖ {(2nd𝑡)}))
328 fisseneq 8131 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((0...𝑁) ∖ {(2nd𝑡)}) ∈ Fin ∧ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ⊆ ((0...𝑁) ∖ {(2nd𝑡)}) ∧ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ≈ ((0...𝑁) ∖ {(2nd𝑡)})) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} = ((0...𝑁) ∖ {(2nd𝑡)}))
329248, 249, 327, 328mp3an12i 1425 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} = ((0...𝑁) ∖ {(2nd𝑡)}))
330329eleq2d 2684 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} ↔ 𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})))
331330biimpar 502 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})) → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))})
332288equcoms 1944 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑗(1st𝑡) / 𝑠𝐶 = 𝑘 / 𝑗(1st𝑡) / 𝑠𝐶)
333332eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 = 𝑗𝑘 / 𝑗(1st𝑡) / 𝑠𝐶 = (1st𝑡) / 𝑠𝐶)
334333eleq1d 2683 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 𝑗 → (𝑘 / 𝑗(1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1)) ↔ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))))
335334, 317vtoclga 3262 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)}) ∣ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))} → (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1)))
336331, 335syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})) → (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1)))
337246, 336sylan2br 493 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑗 ≠ (2nd𝑡))) → (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1)))
338337expr 642 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (𝑗 ≠ (2nd𝑡) → (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1))))
339338necon1bd 2808 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (¬ (1st𝑡) / 𝑠𝐶 ∈ (0...(𝑁 − 1)) → 𝑗 = (2nd𝑡)))
340245, 339syld 47 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (𝑁 = (1st𝑡) / 𝑠𝐶𝑗 = (2nd𝑡)))
341340imp 445 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = (1st𝑡) / 𝑠𝐶) → 𝑗 = (2nd𝑡))
342341, 165syl 17 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = (1st𝑡) / 𝑠𝐶) → (1st𝑡) / 𝑠𝐶 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶)
343 eqtr 2640 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 = (1st𝑡) / 𝑠𝐶(1st𝑡) / 𝑠𝐶 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶) → 𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶)
344343ex 450 . . . . . . . . . . . . . . . . . . 19 (𝑁 = (1st𝑡) / 𝑠𝐶 → ((1st𝑡) / 𝑠𝐶 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶))
345344adantl 482 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = (1st𝑡) / 𝑠𝐶) → ((1st𝑡) / 𝑠𝐶 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶))
346342, 345mpd 15 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = (1st𝑡) / 𝑠𝐶) → 𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶)
347346exp31 629 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (𝑗 ∈ (0...𝑁) → (𝑁 = (1st𝑡) / 𝑠𝐶𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶)))
348241, 158, 347rexlimd 3021 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (∃𝑗 ∈ (0...𝑁)𝑁 = (1st𝑡) / 𝑠𝐶𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶))
349236, 348syl5 34 . . . . . . . . . . . . . 14 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶) → 𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶))
350233, 349mpand 710 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶))
351350pm4.71rd 666 . . . . . . . . . . . 12 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)))
352235ralsng 4196 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = (1st𝑡) / 𝑠𝐶))
35329, 352syl 17 . . . . . . . . . . . . 13 (𝜑 → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = (1st𝑡) / 𝑠𝐶))
354353ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = (1st𝑡) / 𝑠𝐶))
355230, 351, 3543bitr3rd 299 . . . . . . . . . . 11 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (∃𝑗 ∈ (0...𝑁)𝑁 = (1st𝑡) / 𝑠𝐶 ↔ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)))
356355notbid 308 . . . . . . . . . 10 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (¬ ∃𝑗 ∈ (0...𝑁)𝑁 = (1st𝑡) / 𝑠𝐶 ↔ ¬ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)))
357207, 356syl5bb 272 . . . . . . . . 9 (((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (∀𝑗 ∈ (0...𝑁)𝑁(1st𝑡) / 𝑠𝐶 ↔ ¬ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)))
358357pm5.32da 672 . . . . . . . 8 ((𝜑 ∧ (2nd𝑡) ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁(1st𝑡) / 𝑠𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶))))
359203, 358sylan2 491 . . . . . . 7 ((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 ∧ ∀𝑗 ∈ (0...𝑁)𝑁(1st𝑡) / 𝑠𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶))))
360359rabbidva 3180 . . . . . 6 (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 ∧ ∀𝑗 ∈ (0...𝑁)𝑁(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 ∧ ¬ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶))})
361 nfv 1840 . . . . . . . . . . . 12 𝑦 𝑡 = ⟨𝑥, 𝑘
362 nfv 1840 . . . . . . . . . . . . 13 𝑦 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
363 nfrab1 3115 . . . . . . . . . . . . . 14 𝑦{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}
364363nfcri 2755 . . . . . . . . . . . . 13 𝑦 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}
365362, 364nfan 1825 . . . . . . . . . . . 12 𝑦(𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})
366361, 365nfan 1825 . . . . . . . . . . 11 𝑦(𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}))
367 nfv 1840 . . . . . . . . . . 11 𝑘(𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)))
368 opeq2 4378 . . . . . . . . . . . . 13 (𝑘 = 𝑦 → ⟨𝑥, 𝑘⟩ = ⟨𝑥, 𝑦⟩)
369368eqeq2d 2631 . . . . . . . . . . . 12 (𝑘 = 𝑦 → (𝑡 = ⟨𝑥, 𝑘⟩ ↔ 𝑡 = ⟨𝑥, 𝑦⟩))
370 eleq1 2686 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → (𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} ↔ 𝑦 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}))
371 rabid 3110 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} ↔ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)))
372370, 371syl6bb 276 . . . . . . . . . . . . . 14 (𝑘 = 𝑦 → (𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} ↔ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))))
373372anbi2d 739 . . . . . . . . . . . . 13 (𝑘 = 𝑦 → ((𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ↔ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)))))
374 3anass 1040 . . . . . . . . . . . . 13 ((𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)) ↔ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))))
375373, 374syl6bbr 278 . . . . . . . . . . . 12 (𝑘 = 𝑦 → ((𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ↔ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))))
376369, 375anbi12d 746 . . . . . . . . . . 11 (𝑘 = 𝑦 → ((𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})) ↔ (𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)))))
377366, 367, 376cbvex 2271 . . . . . . . . . 10 (∃𝑘(𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})) ↔ ∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))))
378377exbii 1771 . . . . . . . . 9 (∃𝑥𝑘(𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})) ↔ ∃𝑥𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))))
379 eliunxp 5229 . . . . . . . . 9 (𝑡 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ↔ ∃𝑥𝑘(𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})))
380 elopab 4953 . . . . . . . . 9 (𝑡 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))} ↔ ∃𝑥𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))))
381378, 379, 3803bitr4i 292 . . . . . . . 8 (𝑡 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ↔ 𝑡 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))})
382381eqriv 2618 . . . . . . 7 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))}
383 vex 3193 . . . . . . . . . . . . . 14 𝑦 ∈ V
384125, 383op2ndd 7139 . . . . . . . . . . . . 13 (𝑡 = ⟨𝑥, 𝑦⟩ → (2nd𝑡) = 𝑦)
385384sneqd 4167 . . . . . . . . . . . 12 (𝑡 = ⟨𝑥, 𝑦⟩ → {(2nd𝑡)} = {𝑦})
386385difeq2d 3712 . . . . . . . . . . 11 (𝑡 = ⟨𝑥, 𝑦⟩ → ((0...𝑁) ∖ {(2nd𝑡)}) = ((0...𝑁) ∖ {𝑦}))
387125, 383op1std 7138 . . . . . . . . . . . . 13 (𝑡 = ⟨𝑥, 𝑦⟩ → (1st𝑡) = 𝑥)
388387csbeq1d 3526 . . . . . . . . . . . 12 (𝑡 = ⟨𝑥, 𝑦⟩ → (1st𝑡) / 𝑠𝐶 = 𝑥 / 𝑠𝐶)
389388eqeq2d 2631 . . . . . . . . . . 11 (𝑡 = ⟨𝑥, 𝑦⟩ → (𝑖 = (1st𝑡) / 𝑠𝐶𝑖 = 𝑥 / 𝑠𝐶))
390386, 389rexeqbidv 3146 . . . . . . . . . 10 (𝑡 = ⟨𝑥, 𝑦⟩ → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶))
391390ralbidv 2982 . . . . . . . . 9 (𝑡 = ⟨𝑥, 𝑦⟩ → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶))
392388neeq2d 2850 . . . . . . . . . 10 (𝑡 = ⟨𝑥, 𝑦⟩ → (𝑁(1st𝑡) / 𝑠𝐶𝑁𝑥 / 𝑠𝐶))
393392ralbidv 2982 . . . . . . . . 9 (𝑡 = ⟨𝑥, 𝑦⟩ → (∀𝑗 ∈ (0...𝑁)𝑁(1st𝑡) / 𝑠𝐶 ↔ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))
394391, 393anbi12d 746 . . . . . . . 8 (𝑡 = ⟨𝑥, 𝑦⟩ → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁(1st𝑡) / 𝑠𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)))
395394rabxp 5124 . . . . . . 7 {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 ∧ ∀𝑗 ∈ (0...𝑁)𝑁(1st𝑡) / 𝑠𝐶)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶))}
396382, 395eqtr4i 2646 . . . . . 6 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 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 ∧ ∀𝑗 ∈ (0...𝑁)𝑁(1st𝑡) / 𝑠𝐶)}
397 difrab 3883 . . . . . 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)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)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 ∧ ¬ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶))}
398360, 396, 3973eqtr4g 2680 . . . . 5 (𝜑 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 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)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}))
399398fveq2d 6162 . . . 4 (𝜑 → (#‘ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 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)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)})))
40024a1i 11 . . . . 5 ((𝜑𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ∈ Fin)
401 inxp 5224 . . . . . . . . . 10 (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶)})) = (({𝑥} ∩ {𝑡}) × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶)}))
402 df-ne 2791 . . . . . . . . . . . . 13 (𝑥𝑡 ↔ ¬ 𝑥 = 𝑡)
403 disjsn2 4224 . . . . . . . . . . . . 13 (𝑥𝑡 → ({𝑥} ∩ {𝑡}) = ∅)
404402, 403sylbir 225 . . . . . . . . . . . 12 𝑥 = 𝑡 → ({𝑥} ∩ {𝑡}) = ∅)
405404xpeq1d 5108 . . . . . . . . . . 11 𝑥 = 𝑡 → (({𝑥} ∩ {𝑡}) × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶)})) = (∅ × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶)})))
406 0xp 5170 . . . . . . . . . . 11 (∅ × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶)})) = ∅
407405, 406syl6eq 2671 . . . . . . . . . 10 𝑥 = 𝑡 → (({𝑥} ∩ {𝑡}) × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶)})) = ∅)
408401, 407syl5eq 2667 . . . . . . . . 9 𝑥 = 𝑡 → (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶)})) = ∅)
409408orri 391 . . . . . . . 8 (𝑥 = 𝑡 ∨ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶)})) = ∅)
410409rgen2w 2921 . . . . . . 7 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(𝑥 = 𝑡 ∨ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶)})) = ∅)
411 sneq 4165 . . . . . . . . 9 (𝑥 = 𝑡 → {𝑥} = {𝑡})
412 csbeq1 3522 . . . . . . . . . . . . . 14 (𝑥 = 𝑡𝑥 / 𝑠𝐶 = 𝑡 / 𝑠𝐶)
413412eqeq2d 2631 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝑖 = 𝑥 / 𝑠𝐶𝑖 = 𝑡 / 𝑠𝐶))
414413rexbidv 3047 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶))
415414ralbidv 2982 . . . . . . . . . . 11 (𝑥 = 𝑡 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶))
416412neeq2d 2850 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (𝑁𝑥 / 𝑠𝐶𝑁𝑡 / 𝑠𝐶))
417416ralbidv 2982 . . . . . . . . . . 11 (𝑥 = 𝑡 → (∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶 ↔ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶))
418415, 417anbi12d 746 . . . . . . . . . 10 (𝑥 = 𝑡 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶)))
419418rabbidv 3181 . . . . . . . . 9 (𝑥 = 𝑡 → {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)} = {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶)})
420411, 419xpeq12d 5110 . . . . . . . 8 (𝑥 = 𝑡 → ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) = ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶)}))
421420disjor 4607 . . . . . . 7 (Disj 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ↔ ∀𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(𝑥 = 𝑡 ∨ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑡 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑡 / 𝑠𝐶)})) = ∅))
422410, 421mpbir 221 . . . . . 6 Disj 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})
423422a1i 11 . . . . 5 (𝜑Disj 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)}))
42416, 400, 423hashiun 14500 . . . 4 (𝜑 → (#‘ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})) = Σ𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})))
425399, 424eqtr3d 2657 . . 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)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)})) = Σ𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑥 / 𝑠𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁𝑥 / 𝑠𝐶)})))
426 fo1st 7148 . . . . . . . . . . . 12 1st :V–onto→V
427 fofun 6083 . . . . . . . . . . . 12 (1st :V–onto→V → Fun 1st )
428426, 427ax-mp 5 . . . . . . . . . . 11 Fun 1st
429 ssv 3610 . . . . . . . . . . . 12 {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} ⊆ V
430 fof 6082 . . . . . . . . . . . . . 14 (1st :V–onto→V → 1st :V⟶V)
431426, 430ax-mp 5 . . . . . . . . . . . . 13 1st :V⟶V
432431fdmi 6019 . . . . . . . . . . . 12 dom 1st = V
433429, 432sseqtr4i 3623 . . . . . . . . . . 11 {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} ⊆ dom 1st
434 fores 6091 . . . . . . . . . . 11 ((Fun 1st ∧ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} ⊆ dom 1st ) → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}))
435428, 433, 434mp2an 707 . . . . . . . . . 10 (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)})
436 fveq2 6158 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥 → (2nd𝑡) = (2nd𝑥))
437436csbeq1d 3526 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥(2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 = (2nd𝑥) / 𝑗(1st𝑡) / 𝑠𝐶)
438 fveq2 6158 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 → (1st𝑡) = (1st𝑥))
439438csbeq1d 3526 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥(1st𝑡) / 𝑠𝐶 = (1st𝑥) / 𝑠𝐶)
440439csbeq2dv 3970 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥(2nd𝑥) / 𝑗(1st𝑡) / 𝑠𝐶 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶)
441437, 440eqtrd 2655 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑥(2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶)
442441eqeq2d 2631 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑥 → (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶))
443439eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥 → (𝑖 = (1st𝑡) / 𝑠𝐶𝑖 = (1st𝑥) / 𝑠𝐶))
444443rexbidv 3047 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑥 → (∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶))
445444ralbidv 2982 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑥 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶))
446442, 445anbi12d 746 . . . . . . . . . . . . . . 15 (𝑡 = 𝑥 → ((𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶) ↔ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)))
447446rexrab 3357 . . . . . . . . . . . . . 14 (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} (1st𝑥) = 𝑠 ↔ ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) ∧ (1st𝑥) = 𝑠))
448 xp1st 7158 . . . . . . . . . . . . . . . . . . . 20 (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)}))
449448anim1i 591 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) → ((1st𝑥) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶))
450 eleq1 2686 . . . . . . . . . . . . . . . . . . . 20 ((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)})))
451 csbeq1a 3528 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑠 = (1st𝑥) → 𝐶 = (1st𝑥) / 𝑠𝐶)
452451eqcoms 2629 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑥) = 𝑠𝐶 = (1st𝑥) / 𝑠𝐶)
453452eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑥) = 𝑠(1st𝑥) / 𝑠𝐶 = 𝐶)
454453eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑥) = 𝑠 → (𝑖 = (1st𝑥) / 𝑠𝐶𝑖 = 𝐶))
455454rexbidv 3047 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑥) = 𝑠 → (∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
456455ralbidv 2982 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥) = 𝑠 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
457450, 456anbi12d 746 . . . . . . . . . . . . . . . . . . 19 ((1st𝑥) = 𝑠 → (((1st𝑥) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
458449, 457syl5ibcom 235 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) → ((1st𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
459458adantrl 751 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)) → ((1st𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
460459expimpd 628 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) ∧ (1st𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
461460rexlimiv 3022 . . . . . . . . . . . . . . 15 (∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) ∧ (1st𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
462 simplr 791 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
463 ovex 6643 . . . . . . . . . . . . . . . . . . . . . . . 24 (0...𝑁) ∈ V
464463enref 7948 . . . . . . . . . . . . . . . . . . . . . . 23 (0...𝑁) ≈ (0...𝑁)
465 phpreu 33064 . . . . . . . . . . . . . . . . . . . . . . 23 (((0...𝑁) ∈ Fin ∧ (0...𝑁) ≈ (0...𝑁)) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
46620, 464, 465mp2an 707 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
467466biimpi 206 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 → ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
468 eqeq1 2625 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑁 → (𝑖 = 𝐶𝑁 = 𝐶))
469468reubidv 3119 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑁 → (∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶))
470469rspcva 3297 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶)
471232, 467, 470syl2an 494 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶)
472 riotacl 6590 . . . . . . . . . . . . . . . . . . . 20 (∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶 → (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁))
473471, 472syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁))
474473adantlr 750 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁))
475 opelxpi 5118 . . . . . . . . . . . . . . . . . 18 ((𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁)) → ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
476462, 474, 475syl2anc 692 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
477 riotasbc 6591 . . . . . . . . . . . . . . . . . . . . . 22 (∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶[(𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶)
478471, 477syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → [(𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶)
479 riotaex 6580 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ V
480 sbceq2g 3968 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ V → ([(𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶𝑁 = (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗𝐶))
481479, 480ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 ([(𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶𝑁 = (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗𝐶)
482478, 481sylib 208 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → 𝑁 = (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗𝐶)
483482expcom 451 . . . . . . . . . . . . . . . . . . 19 (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 → (𝜑𝑁 = (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗𝐶))
484483imdistanri 726 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (𝑁 = (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
485484adantlr 750 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (𝑁 = (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
486 vex 3193 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑠 ∈ V
487486, 479op2ndd 7139 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (2nd𝑥) = (𝑗 ∈ (0...𝑁)𝑁 = 𝐶))
488487csbeq1d 3526 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (2nd𝑥) / 𝑗𝐶 = (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗𝐶)
489 nfcv 2761 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑗𝑠
490 nfriota1 6583 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑗(𝑗 ∈ (0...𝑁)𝑁 = 𝐶)
491489, 490nfop 4393 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑗𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩
492491nfeq2 2776 . . . . . . . . . . . . . . . . . . . . . . 23 𝑗 𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩
493486, 479op1std 7138 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (1st𝑥) = 𝑠)
494493eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → 𝑠 = (1st𝑥))
495494, 451syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → 𝐶 = (1st𝑥) / 𝑠𝐶)
496492, 495csbeq2d 3969 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (2nd𝑥) / 𝑗𝐶 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶)
497488, 496eqtr3d 2657 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗𝐶 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶)
498497eqeq2d 2631 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (𝑁 = (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗𝐶𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶))
499495eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (𝑖 = 𝐶𝑖 = (1st𝑥) / 𝑠𝐶))
500492, 499rexbid 3046 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶))
501500ralbidv 2982 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶))
502498, 501anbi12d 746 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ((𝑁 = (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) ↔ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)))
503493biantrud 528 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ((𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) ↔ ((𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) ∧ (1st𝑥) = 𝑠)))
504502, 503bitr2d 269 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (((𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) ∧ (1st𝑥) = 𝑠) ↔ (𝑁 = (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
505504rspcev 3299 . . . . . . . . . . . . . . . . 17 ((⟨𝑠, (𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = (𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) ∧ (1st𝑥) = 𝑠))
506476, 485, 505syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) ∧ (1st𝑥) = 𝑠))
507506expl 647 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) ∧ (1st𝑥) = 𝑠)))
508461, 507impbid2 216 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) ∧ (1st𝑥) = 𝑠) ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
509447, 508syl5bb 272 . . . . . . . . . . . . 13 (𝜑 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} (1st𝑥) = 𝑠 ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
510509abbidv 2738 . . . . . . . . . . . 12 (𝜑 → {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} (1st𝑥) = 𝑠} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)})
511 dfimafn 6212 . . . . . . . . . . . . . 14 ((Fun 1st ∧ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} ⊆ dom 1st ) → (1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} (1st𝑥) = 𝑦})
512428, 433, 511mp2an 707 . . . . . . . . . . . . 13 (1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} (1st𝑥) = 𝑦}
513 nfcv 2761 . . . . . . . . . . . . . . . . . . 19 𝑠(2nd𝑡)
514 nfcsb1v 3535 . . . . . . . . . . . . . . . . . . 19 𝑠(1st𝑡) / 𝑠𝐶
515513, 514nfcsb 3537 . . . . . . . . . . . . . . . . . 18 𝑠(2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶
516515nfeq2 2776 . . . . . . . . . . . . . . . . 17 𝑠 𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶
517 nfcv 2761 . . . . . . . . . . . . . . . . . 18 𝑠(0...𝑁)
518514nfeq2 2776 . . . . . . . . . . . . . . . . . . 19 𝑠 𝑖 = (1st𝑡) / 𝑠𝐶
519517, 518nfrex 3003 . . . . . . . . . . . . . . . . . 18 𝑠𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶
520517, 519nfral 2941 . . . . . . . . . . . . . . . . 17 𝑠𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶
521516, 520nfan 1825 . . . . . . . . . . . . . . . 16 𝑠(𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)
522 nfcv 2761 . . . . . . . . . . . . . . . 16 𝑠((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))
523521, 522nfrab 3116 . . . . . . . . . . . . . . 15 𝑠{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}
524 nfv 1840 . . . . . . . . . . . . . . 15 𝑠(1st𝑥) = 𝑦
525523, 524nfrex 3003 . . . . . . . . . . . . . 14 𝑠𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} (1st𝑥) = 𝑦
526 nfv 1840 . . . . . . . . . . . . . 14 𝑦𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} (1st𝑥) = 𝑠
527 eqeq2 2632 . . . . . . . . . . . . . . 15 (𝑦 = 𝑠 → ((1st𝑥) = 𝑦 ↔ (1st𝑥) = 𝑠))
528527rexbidv 3047 . . . . . . . . . . . . . 14 (𝑦 = 𝑠 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} (1st𝑥) = 𝑦 ↔ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} (1st𝑥) = 𝑠))
529525, 526, 528cbvab 2743 . . . . . . . . . . . . 13 {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} (1st𝑥) = 𝑦} = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} (1st𝑥) = 𝑠}
530512, 529eqtri 2643 . . . . . . . . . . . 12 (1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}) = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} (1st𝑥) = 𝑠}
531 df-rab 2917 . . . . . . . . . . . 12 {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)}
532510, 530, 5313eqtr4g 2680 . . . . . . . . . . 11 (𝜑 → (1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}) = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
533 foeq3 6080 . . . . . . . . . . 11 ((1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}) = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
534532, 533syl 17 . . . . . . . . . 10 (𝜑 → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
535435, 534mpbii 223 . . . . . . . . 9 (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
536 fof 6082 . . . . . . . . 9 ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}⟶{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
537535, 536syl 17 . . . . . . . 8 (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}⟶{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
538 fvres 6174 . . . . . . . . . . . 12 (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)})‘𝑥) = (1st𝑥))
539 fvres 6174 . . . . . . . . . . . 12 (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)})‘𝑦) = (1st𝑦))
540538, 539eqeqan12d 2637 . . . . . . . . . . 11 ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)})‘𝑦) ↔ (1st𝑥) = (1st𝑦)))
541540adantl 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)})) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)})‘𝑦) ↔ (1st𝑥) = (1st𝑦)))
542446elrab 3351 . . . . . . . . . . . . . . 15 (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} ↔ (𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)))
543 xp2nd 7159 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑥) ∈ (0...𝑁))
544543anim1i 591 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)) → ((2nd𝑥) ∈ (0...𝑁) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)))
545542, 544sylbi 207 . . . . . . . . . . . . . 14 (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} → ((2nd𝑥) ∈ (0...𝑁) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)))
546 simpl 473 . . . . . . . . . . . . . . . . . 18 ((𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶) → 𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶)
547546a1i 11 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶) → 𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶))
548547ss2rabi 3669 . . . . . . . . . . . . . . . 16 {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶}
549548sseli 3584 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} → 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶})
550 fveq2 6158 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑦 → (2nd𝑡) = (2nd𝑦))
551550csbeq1d 3526 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑦(2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 = (2nd𝑦) / 𝑗(1st𝑡) / 𝑠𝐶)
552 fveq2 6158 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑦 → (1st𝑡) = (1st𝑦))
553552csbeq1d 3526 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑦(1st𝑡) / 𝑠𝐶 = (1st𝑦) / 𝑠𝐶)
554553csbeq2dv 3970 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑦(2nd𝑦) / 𝑗(1st𝑡) / 𝑠𝐶 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶)
555551, 554eqtrd 2655 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑦(2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶)
556555eqeq2d 2631 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑦 → (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶))
557556elrab 3351 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶} ↔ (𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶))
558 xp2nd 7159 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑦) ∈ (0...𝑁))
559558anim1i 591 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶) → ((2nd𝑦) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶))
560557, 559sylbi 207 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶} → ((2nd𝑦) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶))
561549, 560syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} → ((2nd𝑦) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶))
562545, 561anim12i 589 . . . . . . . . . . . . 13 ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}) → (((2nd𝑥) ∈ (0...𝑁) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)) ∧ ((2nd𝑦) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶)))
563 an4 864 . . . . . . . . . . . . . . 15 ((((2nd𝑥) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶) ∧ ((2nd𝑦) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶)) ↔ (((2nd𝑥) ∈ (0...𝑁) ∧ (2nd𝑦) ∈ (0...𝑁)) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶)))
564563anbi2i 729 . . . . . . . . . . . . . 14 ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ∧ (((2nd𝑥) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶) ∧ ((2nd𝑦) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶))) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ∧ (((2nd𝑥) ∈ (0...𝑁) ∧ (2nd𝑦) ∈ (0...𝑁)) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶))))
565 anass 680 . . . . . . . . . . . . . . . . 17 ((((2nd𝑥) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) ↔ ((2nd𝑥) ∈ (0...𝑁) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)))
566 ancom 466 . . . . . . . . . . . . . . . . 17 ((((2nd𝑥) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((2nd𝑥) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶)))
567565, 566bitr3i 266 . . . . . . . . . . . . . . . 16 (((2nd𝑥) ∈ (0...𝑁) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((2nd𝑥) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶)))
568567anbi1i 730 . . . . . . . . . . . . . . 15 ((((2nd𝑥) ∈ (0...𝑁) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)) ∧ ((2nd𝑦) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶)) ↔ ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((2nd𝑥) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶)) ∧ ((2nd𝑦) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶)))
569 anass 680 . . . . . . . . . . . . . . 15 (((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((2nd𝑥) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶)) ∧ ((2nd𝑦) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ∧ (((2nd𝑥) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶) ∧ ((2nd𝑦) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶))))
570568, 569bitri 264 . . . . . . . . . . . . . 14 ((((2nd𝑥) ∈ (0...𝑁) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)) ∧ ((2nd𝑦) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ∧ (((2nd𝑥) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶) ∧ ((2nd𝑦) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶))))
571 anass 680 . . . . . . . . . . . . . 14 (((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((2nd𝑥) ∈ (0...𝑁) ∧ (2nd𝑦) ∈ (0...𝑁))) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ∧ (((2nd𝑥) ∈ (0...𝑁) ∧ (2nd𝑦) ∈ (0...𝑁)) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶))))
572564, 570, 5713bitr4i 292 . . . . . . . . . . . . 13 ((((2nd𝑥) ∈ (0...𝑁) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)) ∧ ((2nd𝑦) ∈ (0...𝑁) ∧ 𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶)) ↔ ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((2nd𝑥) ∈ (0...𝑁) ∧ (2nd𝑦) ∈ (0...𝑁))) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶)))
573562, 572sylib 208 . . . . . . . . . . . 12 ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = (2nd𝑡) / 𝑗(1st𝑡) / 𝑠𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑡) / 𝑠𝐶)}) → ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((2nd𝑥) ∈ (0...𝑁) ∧ (2nd𝑦) ∈ (0...𝑁))) ∧ (𝑁 = (2nd𝑥) / 𝑗(1st𝑥) / 𝑠𝐶𝑁 = (2nd𝑦) / 𝑗(1st𝑦) / 𝑠𝐶)))
574 phpreu 33064 . . . . . . . . . . . . . . . . . . . . 21 (((0...𝑁) ∈ Fin ∧ (0...𝑁) ≈ (0...𝑁)) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶))
57520, 464, 574mp2an 707 . . . . . . . . . . . . . . . . . . . 20 (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)
576 reurmo 3154 . . . . . . . . . . . . . . . . . . . . 21 (∃!𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶 → ∃*𝑗 ∈ (0...𝑁)𝑖 = (1st𝑥) / 𝑠𝐶)
577576ralimi