Theorem poimirlem26 37816

Description: Lemma for poimir 37823 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	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem28.1	⊢ (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
poimirlem28.2	⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))

Assertion

Ref	Expression
poimirlem26	⊢ (𝜑 → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃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.

Step	Hyp	Ref	Expression
1		fzofi 13899	. . . . . 6 ⊢ (0..^𝐾) ∈ Fin
2		fzfi 13897	. . . . . 6 ⊢ (1...𝑁) ∈ Fin
3		mapfi 9250	. . . . . 6 ⊢ (((0..^𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((0..^𝐾) ↑m (1...𝑁)) ∈ Fin)
4	1, 2, 3	mp2an 693	. . . . 5 ⊢ ((0..^𝐾) ↑m (1...𝑁)) ∈ Fin
5		mapfi 9250	. . . . . . 7 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ↑m (1...𝑁)) ∈ Fin)
6	2, 2, 5	mp2an 693	. . . . . 6 ⊢ ((1...𝑁) ↑m (1...𝑁)) ∈ Fin
7		f1of 6773	. . . . . . . 8 ⊢ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁))
8	7	ss2abi 4017	. . . . . . 7 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}
9		ovex 7391	. . . . . . . 8 ⊢ (1...𝑁) ∈ V
10	9, 9	mapval 8777	. . . . . . 7 ⊢ ((1...𝑁) ↑m (1...𝑁)) = {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}
11	8, 10	sseqtrri 3982	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑m (1...𝑁))
12		ssfi 9099	. . . . . 6 ⊢ ((((1...𝑁) ↑m (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑m (1...𝑁))) → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
13	6, 11, 12	mp2an 693	. . . . 5 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin
14	4, 13	pm3.2i 470	. . . 4 ⊢ (((0..^𝐾) ↑m (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
15		xpfi 9222	. . . 4 ⊢ ((((0..^𝐾) ↑m (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) → (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin)
16	14, 15	mp1i 13	. . 3 ⊢ (𝜑 → (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin)
17		2z 12525	. . . 4 ⊢ 2 ∈ ℤ
18	17	a1i 11	. . 3 ⊢ (𝜑 → 2 ∈ ℤ)
19		snfi 8982	. . . . . . 7 ⊢ {𝑥} ∈ Fin
20		fzfi 13897	. . . . . . . 8 ⊢ (0...𝑁) ∈ Fin
21		rabfi 9173	. . . . . . . 8 ⊢ ((0...𝑁) ∈ Fin → {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin)
22	20, 21	ax-mp 5	. . . . . . 7 ⊢ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin
23		xpfi 9222	. . . . . . 7 ⊢ (({𝑥} ∈ Fin ∧ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin) → ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ Fin)
24	19, 22, 23	mp2an 693	. . . . . 6 ⊢ ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ Fin
25		hashcl 14281	. . . . . 6 ⊢ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ Fin → (♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ∈ ℕ0)
26	24, 25	ax-mp 5	. . . . 5 ⊢ (♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ∈ ℕ0
27	26	nn0zi 12518	. . . 4 ⊢ (♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ∈ ℤ
28	27	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ∈ ℤ)
29		poimir.0	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
30	29	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → 𝑁 ∈ ℕ)
31		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0})))
32		nfcsb1v 3872	. . . . . . . . . . 11 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
33	32	nfeq2 2915	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
34	31, 33	nfim 1898	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
35		oveq2 7366	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (1...𝑗) = (1...𝑘))
36	35	imaeq2d 6018	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((2nd ‘𝑡) “ (1...𝑗)) = ((2nd ‘𝑡) “ (1...𝑘)))
37	36	xpeq1d 5652	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (((2nd ‘𝑡) “ (1...𝑗)) × {1}) = (((2nd ‘𝑡) “ (1...𝑘)) × {1}))
38		oveq1 7365	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1))
39	38	oveq1d 7373	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → ((𝑗 + 1)...𝑁) = ((𝑘 + 1)...𝑁))
40	39	imaeq2d 6018	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)))
41	40	xpeq1d 5652	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))
42	37, 41	uneq12d 4120	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0})))
43	42	oveq2d 7374	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))))
44	43	eqeq2d 2746	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) ↔ 𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0})))))
45		csbeq1a 3862	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ⦋𝑡 / 𝑠⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
46	45	eqeq2d 2746	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝐵 = ⦋𝑡 / 𝑠⦌𝐶 ↔ 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
47	44, 46	imbi12d 344	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑡 / 𝑠⦌𝐶) ↔ (𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)))
48		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑠 𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0})))
49		nfcsb1v 3872	. . . . . . . . . . . 12 ⊢ Ⅎ𝑠⦋𝑡 / 𝑠⦌𝐶
50	49	nfeq2 2915	. . . . . . . . . . 11 ⊢ Ⅎ𝑠 𝐵 = ⦋𝑡 / 𝑠⦌𝐶
51	48, 50	nfim 1898	. . . . . . . . . 10 ⊢ Ⅎ𝑠(𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑡 / 𝑠⦌𝐶)
52		fveq2 6833	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → (1st ‘𝑠) = (1st ‘𝑡))
53		fveq2 6833	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑡 → (2nd ‘𝑠) = (2nd ‘𝑡))
54	53	imaeq1d 6017	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd ‘𝑡) “ (1...𝑗)))
55	54	xpeq1d 5652	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = (((2nd ‘𝑡) “ (1...𝑗)) × {1}))
56	53	imaeq1d 6017	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)))
57	56	xpeq1d 5652	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))
58	55, 57	uneq12d 4120	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0})))
59	52, 58	oveq12d 7376	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))))
60	59	eqeq2d 2746	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ↔ 𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0})))))
61		csbeq1a 3862	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → 𝐶 = ⦋𝑡 / 𝑠⦌𝐶)
62	61	eqeq2d 2746	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (𝐵 = 𝐶 ↔ 𝐵 = ⦋𝑡 / 𝑠⦌𝐶))
63	60, 62	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → ((𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) ↔ (𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑡 / 𝑠⦌𝐶)))
64		poimirlem28.1	. . . . . . . . . 10 ⊢ (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
65	51, 63, 64	chvarfv 2246	. . . . . . . . 9 ⊢ (𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑡 / 𝑠⦌𝐶)
66	34, 47, 65	chvarfv 2246	. . . . . . . 8 ⊢ (𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
67		poimirlem28.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
68	67	ad4ant14 753	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
69		xp1st 7965	. . . . . . . . . 10 ⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘𝑥) ∈ ((0..^𝐾) ↑m (1...𝑁)))
70		elmapi 8788	. . . . . . . . . 10 ⊢ ((1st ‘𝑥) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘𝑥):(1...𝑁)⟶(0..^𝐾))
71	69, 70	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘𝑥):(1...𝑁)⟶(0..^𝐾))
72	71	ad2antlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (1st ‘𝑥):(1...𝑁)⟶(0..^𝐾))
73		xp2nd 7966	. . . . . . . . . 10 ⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘𝑥) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
74		fvex 6846	. . . . . . . . . . 11 ⊢ (2nd ‘𝑥) ∈ V
75		f1oeq1 6761	. . . . . . . . . . 11 ⊢ (𝑓 = (2nd ‘𝑥) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘𝑥):(1...𝑁)–1-1-onto→(1...𝑁)))
76	74, 75	elab 3633	. . . . . . . . . 10 ⊢ ((2nd ‘𝑥) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘𝑥):(1...𝑁)–1-1-onto→(1...𝑁))
77	73, 76	sylib 218	. . . . . . . . 9 ⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘𝑥):(1...𝑁)–1-1-onto→(1...𝑁))
78	77	ad2antlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (2nd ‘𝑥):(1...𝑁)–1-1-onto→(1...𝑁))
79		nfcv 2897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑁
80		nfcv 2897	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑥
81	80, 32	nfcsbw 3874	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
82	79, 81	nfne 3032	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
83		nfcv 2897	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡𝐶
84	83, 49, 61	cbvcsbw 3858	. . . . . . . . . . . . . 14 ⊢ ⦋𝑥 / 𝑠⦌𝐶 = ⦋𝑥 / 𝑡⦌⦋𝑡 / 𝑠⦌𝐶
85	45	csbeq2dv 3855	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ⦋𝑥 / 𝑡⦌⦋𝑡 / 𝑠⦌𝐶 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
86	84, 85	eqtrid 2782	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ⦋𝑥 / 𝑠⦌𝐶 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
87	86	neeq2d 2991	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
88	82, 87	rspc 3563	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑁) → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
89	88	impcom 407	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
90	89	adantll 715	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
91		1st2nd2 7972	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
92	91	csbeq1d 3852	. . . . . . . . . 10 ⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
93	92	ad3antlr 732	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ∧ 𝑘 ∈ (0...𝑁)) → ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
94	90, 93	neeqtrd 3000	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ≠ ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
95	30, 66, 68, 72, 78, 94	poimirlem25 37815	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶}))
96		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 = ⦋𝑥 / 𝑠⦌𝐶
97	81	nfeq2 2915	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗 𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
98	86	eqeq2d 2746	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
99	96, 97, 98	cbvrexw 3278	. . . . . . . . . . . . 13 ⊢ (∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ ∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
100	92	eqeq2d 2746	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ 𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
101	100	rexbidv 3159	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ ∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
102	99, 101	bitr2id 284	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶))
103	102	ralbidv 3158	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶))
104		iba 527	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))
105	103, 104	sylan9bb 509	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))
106	105	rabbidv 3405	. . . . . . . . 9 ⊢ ((𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶} = {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})
107	106	fveq2d 6837	. . . . . . . 8 ⊢ ((𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶}) = (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
108	107	adantll 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶}) = (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
109	95, 108	breqtrd 5123	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
110	109	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
111		dvds0 16200	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∥ 0)
112	17, 111	ax-mp 5	. . . . . . 7 ⊢ 2 ∥ 0
113		hash0 14292	. . . . . . 7 ⊢ (♯‘∅) = 0
114	112, 113	breqtrri 5124	. . . . . 6 ⊢ 2 ∥ (♯‘∅)
115		simpr 484	. . . . . . . . . 10 ⊢ ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)
116	115	con3i 154	. . . . . . . . 9 ⊢ (¬ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → ¬ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))
117	116	ralrimivw 3131	. . . . . . . 8 ⊢ (¬ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → ∀𝑦 ∈ (0...𝑁) ¬ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))
118		rabeq0 4339	. . . . . . . 8 ⊢ ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} = ∅ ↔ ∀𝑦 ∈ (0...𝑁) ¬ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))
119	117, 118	sylibr 234	. . . . . . 7 ⊢ (¬ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} = ∅)
120	119	fveq2d 6837	. . . . . 6 ⊢ (¬ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = (♯‘∅))
121	114, 120	breqtrrid 5135	. . . . 5 ⊢ (¬ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
122	110, 121	pm2.61d1 180	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
123		hashxp 14359	. . . . . 6 ⊢ (({𝑥} ∈ Fin ∧ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin) → (♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = ((♯‘{𝑥}) · (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
124	19, 22, 123	mp2an 693	. . . . 5 ⊢ (♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = ((♯‘{𝑥}) · (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
125		vex 3443	. . . . . . 7 ⊢ 𝑥 ∈ V
126		hashsng 14294	. . . . . . 7 ⊢ (𝑥 ∈ V → (♯‘{𝑥}) = 1)
127	125, 126	ax-mp 5	. . . . . 6 ⊢ (♯‘{𝑥}) = 1
128	127	oveq1i 7368	. . . . 5 ⊢ ((♯‘{𝑥}) · (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = (1 · (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
129		hashcl 14281	. . . . . . . 8 ⊢ ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin → (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ ℕ0)
130	22, 129	ax-mp 5	. . . . . . 7 ⊢ (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ ℕ0
131	130	nn0cni 12415	. . . . . 6 ⊢ (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ ℂ
132	131	mullidi 11139	. . . . 5 ⊢ (1 · (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})
133	124, 128, 132	3eqtri 2762	. . . 4 ⊢ (♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = (♯‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})
134	122, 133	breqtrrdi 5139	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → 2 ∥ (♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
135	16, 18, 28, 134	fsumdvds 16237	. 2 ⊢ (𝜑 → 2 ∥ Σ𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
136	4, 13, 15	mp2an 693	. . . . . 6 ⊢ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin
137		xpfi 9222	. . . . . 6 ⊢ (((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin ∧ (0...𝑁) ∈ Fin) → ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin)
138	136, 20, 137	mp2an 693	. . . . 5 ⊢ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin
139		rabfi 9173	. . . . 5 ⊢ (((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∈ Fin)
140	138, 139	ax-mp 5	. . . 4 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∈ Fin
141	29	nncnd 12163	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
142		npcan1 11564	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
143	141, 142	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
144		nnm1nn0 12444	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
145	29, 144	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
146	145	nn0zd 12515	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
147		uzid 12768	. . . . . . . . . . . 12 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)))
148		peano2uz 12816	. . . . . . . . . . . 12 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
149	146, 147, 148	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
150	143, 149	eqeltrrd 2836	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
151		fzss2 13482	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
152		ssralv 4001	. . . . . . . . . 10 ⊢ ((0...(𝑁 − 1)) ⊆ (0...𝑁) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
153	150, 151, 152	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
154	153	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
155		raldifb 4100	. . . . . . . . . . . 12 ⊢ (∀𝑗 ∈ (0...𝑁)(𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ ∀𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
156		nfv 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝜑
157		nfcsb1v 3872	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
158	157	nfeq2 2915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
159	156, 158	nfan 1901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
160		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗 𝑖 ∈ (0...(𝑁 − 1))
161	159, 160	nfan 1901	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1)))
162		nnel 3045	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑗 ∉ {(2nd ‘𝑡)} ↔ 𝑗 ∈ {(2nd ‘𝑡)})
163		velsn 4595	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ {(2nd ‘𝑡)} ↔ 𝑗 = (2nd ‘𝑡))
164	162, 163	bitri 275	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑗 ∉ {(2nd ‘𝑡)} ↔ 𝑗 = (2nd ‘𝑡))
165		csbeq1a 3862	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑡) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
166	165	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (2nd ‘𝑡) → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
167	166	biimparc 479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ 𝑗 = (2nd ‘𝑡)) → 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
168	29	nnred 12162	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑁 ∈ ℝ)
169	168	ltm1d 12076	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
170	145	nn0red 12465	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
171	170, 168	ltnled 11282	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
172	169, 171	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
173		elfzle2 13446	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ (0...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
174	172, 173	nsyl 140	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
175		eleq1 2823	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 𝑁 → (𝑖 ∈ (0...(𝑁 − 1)) ↔ 𝑁 ∈ (0...(𝑁 − 1))))
176	175	notbid 318	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑁 → (¬ 𝑖 ∈ (0...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (0...(𝑁 − 1))))
177	174, 176	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑖 = 𝑁 → ¬ 𝑖 ∈ (0...(𝑁 − 1))))
178	177	con2d 134	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑖 ∈ (0...(𝑁 − 1)) → ¬ 𝑖 = 𝑁))
179	178	imp 406	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) → ¬ 𝑖 = 𝑁)
180		eqeq2 2747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑖 = 𝑁 ↔ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
181	180	notbid 318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (¬ 𝑖 = 𝑁 ↔ ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
182	179, 181	syl5ibcom 245	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
183	167, 182	syl5 34	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) → ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ 𝑗 = (2nd ‘𝑡)) → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
184	183	expdimp 452	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (𝑗 = (2nd ‘𝑡) → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
185	184	an32s 653	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (𝑗 = (2nd ‘𝑡) → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
186	164, 185	biimtrid 242	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (¬ 𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
187		idd 24	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
188	186, 187	jad 187	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → ((𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
189	188	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
190	161, 189	ralimdaa 3236	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (∀𝑗 ∈ (0...𝑁)(𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
191	155, 190	biimtrrid 243	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (∀𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
192	191	con3d 152	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (¬ ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ¬ ∀𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
193		dfrex2 3062	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
194		dfrex2 3062	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∀𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
195	192, 193, 194	3imtr4g 296	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
196	195	ralimdva 3147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
197	154, 196	syld 47	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
198	197	expimpd 453	. . . . . 6 ⊢ (𝜑 → ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
199	198	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
200	199	ss2rabdv 4026	. . . 4 ⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶})
201		hashssdif 14337	. . . 4 ⊢ (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∈ Fin ∧ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})))
202	140, 200, 201	sylancr 588	. . 3 ⊢ (𝜑 → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})))
203		xp2nd 7966	. . . . . . . 8 ⊢ (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑡) ∈ (0...𝑁))
204		df-ne 2932	. . . . . . . . . . . 12 ⊢ (𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
205	204	ralbii 3081	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁) ¬ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
206		ralnex 3061	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ (0...𝑁) ¬ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
207	205, 206	bitri 275	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
208	29	nnnn0d 12464	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
209		nn0uz 12791	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
210	208, 209	eleqtrdi 2845	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
211	143, 210	eqeltrd 2835	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘0))
212		fzsplit2 13467	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
213	211, 150, 212	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
214	143	oveq1d 7373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
215	29	nnzd 12516	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
216		fzsn 13484	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
217	215, 216	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
218	214, 217	eqtrd 2770	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
219	218	uneq2d 4119	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((0...(𝑁 − 1)) ∪ {𝑁}))
220	213, 219	eqtrd 2770	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁}))
221	220	raleqdv 3295	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
222		ralunb 4148	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
223		difss 4087	. . . . . . . . . . . . . . . . . 18 ⊢ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ⊆ (0...𝑁)
224		ssrexv 4002	. . . . . . . . . . . . . . . . . 18 ⊢ (((0...𝑁) ∖ {(2nd ‘𝑡)}) ⊆ (0...𝑁) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
225	223, 224	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
226	225	ralimi 3072	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
227	226	biantrurd 532	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
228	222, 227	bitr4id 290	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (∀𝑖 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
229	221, 228	sylan9bb 509	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
230	229	adantlr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
231		nn0fz0 13543	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁))
232	208, 231	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
233	232	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑁 ∈ (0...𝑁))
234		eqeq1 2739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑁 → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
235	234	rexbidv 3159	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑁 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
236	235	rspcva 3573	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
237		nfv 1916	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗(𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁))
238		nfcv 2897	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗(0...(𝑁 − 1))
239		nfre1 3260	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
240	238, 239	nfralw 3282	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
241	237, 240	nfan 1901	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
242		eleq1 2823	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑁 ∈ (0...(𝑁 − 1)) ↔ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
243	242	notbid 318	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (¬ 𝑁 ∈ (0...(𝑁 − 1)) ↔ ¬ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
244	174, 243	syl5ibcom 245	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ¬ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
245	244	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ¬ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
246		eldifsn 4741	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ↔ (𝑗 ∈ (0...𝑁) ∧ 𝑗 ≠ (2nd ‘𝑡)))
247		diffi 9101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0...𝑁) ∈ Fin → ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin)
248	20, 247	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin
249		ssrab2 4031	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ⊆ ((0...𝑁) ∖ {(2nd ‘𝑡)})
250		ssdomg 8939	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin → ({𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ⊆ ((0...𝑁) ∖ {(2nd ‘𝑡)}) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≼ ((0...𝑁) ∖ {(2nd ‘𝑡)})))
251	248, 249, 250	mp2 9	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≼ ((0...𝑁) ∖ {(2nd ‘𝑡)})
252		hashdifsn 14339	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((0...𝑁) ∈ Fin ∧ (2nd ‘𝑡) ∈ (0...𝑁)) → (♯‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = ((♯‘(0...𝑁)) − 1))
253	20, 252	mpan 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((2nd ‘𝑡) ∈ (0...𝑁) → (♯‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = ((♯‘(0...𝑁)) − 1))
254		1cnd 11129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → 1 ∈ ℂ)
255	141, 254, 254	addsubd 11515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑁 + 1) − 1) = ((𝑁 − 1) + 1))
256		hashfz0 14357	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑁 ∈ ℕ0 → (♯‘(0...𝑁)) = (𝑁 + 1))
257	208, 256	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (♯‘(0...𝑁)) = (𝑁 + 1))
258	257	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((♯‘(0...𝑁)) − 1) = ((𝑁 + 1) − 1))
259		hashfz0 14357	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑁 − 1) ∈ ℕ0 → (♯‘(0...(𝑁 − 1))) = ((𝑁 − 1) + 1))
260	145, 259	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (♯‘(0...(𝑁 − 1))) = ((𝑁 − 1) + 1))
261	255, 258, 260	3eqtr4d 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((♯‘(0...𝑁)) − 1) = (♯‘(0...(𝑁 − 1))))
262	253, 261	sylan9eqr 2792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) → (♯‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = (♯‘(0...(𝑁 − 1))))
263		fzfi 13897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0...(𝑁 − 1)) ∈ Fin
264		hashen 14272	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin ∧ (0...(𝑁 − 1)) ∈ Fin) → ((♯‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = (♯‘(0...(𝑁 − 1))) ↔ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≈ (0...(𝑁 − 1))))
265	248, 263, 264	mp2an 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((♯‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = (♯‘(0...(𝑁 − 1))) ↔ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≈ (0...(𝑁 − 1)))
266	262, 265	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) → ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≈ (0...(𝑁 − 1)))
267		rabfi 9173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∈ Fin)
268	248, 267	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∈ Fin
269		eleq1 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑖 ∈ (0...(𝑁 − 1)) ↔ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
270	269	biimpac 478	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
271		rabid 3419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↔ (𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∧ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
272	271	simplbi2com 502	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) → (𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}))
273	270, 272	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}))
274	273	impancom 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}))
275	274	ancrd 551	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
276	275	expimpd 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑖 ∈ (0...(𝑁 − 1)) → ((𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
277	276	reximdv2 3145	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
278	271	simplbi 497	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} → 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}))
279	274	pm4.71rd 562	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
280		df-mpt 5179	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) = {⟨𝑘, 𝑖⟩ ∣ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)}
281		nfv 1916	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ Ⅎ𝑘(𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
282		nfrab1 3418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ Ⅎ𝑗{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}
283	282	nfcri 2889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ Ⅎ𝑗 𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}
284		nfcsb1v 3872	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
285	284	nfeq2 2915	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ Ⅎ𝑗 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
286	283, 285	nfan 1901	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ Ⅎ𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
287		eleq1 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↔ 𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}))
288		csbeq1a 3862	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑗 = 𝑘 → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
289	288	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑗 = 𝑘 → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
290	287, 289	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑗 = 𝑘 → ((𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
291	281, 286, 290	cbvopab1 5171	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ {⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} = {⟨𝑘, 𝑖⟩ ∣ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)}
292	280, 291	eqtr4i 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) = {⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}
293	292	breqi 5103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖 ↔ 𝑗{⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}𝑖)
294		df-br 5098	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑗{⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}𝑖 ↔ ⟨𝑗, 𝑖⟩ ∈ {⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})
295		opabidw 5471	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (⟨𝑗, 𝑖⟩ ∈ {⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ↔ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
296	293, 294, 295	3bitri 297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖 ↔ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
297	279, 296	bitr4di 289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
298	278, 297	sylan2 594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
299	298	rexbidva 3157	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
300		nfcv 2897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ Ⅎ𝑝{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}
301		nfv 1916	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ Ⅎ𝑝 𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖
302		nfcv 2897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ Ⅎ𝑗𝑝
303	282, 284	nfmpt 5195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ Ⅎ𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
304		nfcv 2897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ Ⅎ𝑗𝑖
305	302, 303, 304	nfbr 5144	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ Ⅎ𝑗 𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖
306		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑗 = 𝑝 → (𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖 ↔ 𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
307	282, 300, 301, 305, 306	cbvrexfw 3276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖 ↔ ∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖)
308	299, 307	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
309	277, 308	sylibd 239	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
310	309	ralimia 3069	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖)
311		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) = (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
312		nfcv 2897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ Ⅎ𝑗𝑘
313		nfcv 2897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ Ⅎ𝑗((0...𝑁) ∖ {(2nd ‘𝑡)})
314	284	nfel1 2914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))
315	288	eleq1d 2820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑗 = 𝑘 → (⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ↔ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
316	312, 313, 314, 315	elrabf 3642	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↔ (𝑘 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∧ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
317	316	simprbi 496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} → ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
318	311, 317	fmpti 7057	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}⟶(0...(𝑁 − 1))
319	310, 318	jctil 519	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ((𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}⟶(0...(𝑁 − 1)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
320		dffo4 7048	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}–onto→(0...(𝑁 − 1)) ↔ ((𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}⟶(0...(𝑁 − 1)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
321	319, 320	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}–onto→(0...(𝑁 − 1)))
322		fodomfi 9214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (({𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∈ Fin ∧ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}–onto→(0...(𝑁 − 1))) → (0...(𝑁 − 1)) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})
323	268, 321, 322	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (0...(𝑁 − 1)) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})
324		endomtr 8951	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((0...𝑁) ∖ {(2nd ‘𝑡)}) ≈ (0...(𝑁 − 1)) ∧ (0...(𝑁 − 1)) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) → ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})
325	266, 323, 324	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})
326		sbth 9027	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (({𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≼ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∧ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≈ ((0...𝑁) ∖ {(2nd ‘𝑡)}))
327	251, 325, 326	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≈ ((0...𝑁) ∖ {(2nd ‘𝑡)}))
328		fisseneq 9165	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin ∧ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ⊆ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∧ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} = ((0...𝑁) ∖ {(2nd ‘𝑡)}))
329	248, 249, 327, 328	mp3an12i 1468	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} = ((0...𝑁) ∖ {(2nd ‘𝑡)}))
330	329	eleq2d 2821	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↔ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})))
331	330	biimpar 477	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})
332	288	equcoms 2022	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑗 → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
333	332	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑗 → ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
334	333	eleq1d 2820	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 = 𝑗 → (⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ↔ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
335	334, 317	vtoclga 3531	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
336	331, 335	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
337	246, 336	sylan2br 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑗 ≠ (2nd ‘𝑡))) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
338	337	expr 456	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (𝑗 ≠ (2nd ‘𝑡) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
339	338	necon1bd 2949	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (¬ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) → 𝑗 = (2nd ‘𝑡)))
340	245, 339	syld 47	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑗 = (2nd ‘𝑡)))
341	340	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑗 = (2nd ‘𝑡))
342	341, 165	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
343		eqtr 2755	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
344	343	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
345	344	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
346	342, 345	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
347	346	exp31 419	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (𝑗 ∈ (0...𝑁) → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
348	241, 158, 347	rexlimd 3242	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
349	236, 348	syl5 34	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
350	233, 349	mpand 696	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
351	350	pm4.71rd 562	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
352	235	ralsng 4631	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
353	29, 352	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
354	353	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
355	230, 351, 354	3bitr3rd 310	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
356	355	notbid 318	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (¬ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
357	207, 356	bitrid 283	. . . . . . . . 9 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
358	357	pm5.32da 579	. . . . . . . 8 ⊢ ((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))))
359	203, 358	sylan2 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))))
360	359	rabbidva 3404	. . . . . 6 ⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))})
361		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝑡 = ⟨𝑥, 𝑘⟩
362		nfv 1916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
363		nfrab1 3418	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}
364	363	nfcri 2889	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}
365	362, 364	nfan 1901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})
366	361, 365	nfan 1901	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
367		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))
368		opeq2 4829	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → ⟨𝑥, 𝑘⟩ = ⟨𝑥, 𝑦⟩)
369	368	eqeq2d 2746	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑦 → (𝑡 = ⟨𝑥, 𝑘⟩ ↔ 𝑡 = ⟨𝑥, 𝑦⟩))
370		eleq1 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑦 → (𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ↔ 𝑦 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
371		rabid 3419	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ↔ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))
372	370, 371	bitrdi 287	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑦 → (𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ↔ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))
373	372	anbi2d 631	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → ((𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))))
374		3anass 1095	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)) ↔ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))
375	373, 374	bitr4di 289	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑦 → ((𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))
376	369, 375	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑦 → ((𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ↔ (𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))))
377	366, 367, 376	cbvexv1 2345	. . . . . . . . . 10 ⊢ (∃𝑘(𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ↔ ∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))
378	377	exbii 1850	. . . . . . . . 9 ⊢ (∃𝑥∃𝑘(𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ↔ ∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))
379		eliunxp 5785	. . . . . . . . 9 ⊢ (𝑡 ∈ ∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ ∃𝑥∃𝑘(𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
380		elopab 5474	. . . . . . . . 9 ⊢ (𝑡 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))} ↔ ∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))
381	378, 379, 380	3bitr4i 303	. . . . . . . 8 ⊢ (𝑡 ∈ ∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ 𝑡 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))})
382	381	eqriv 2732	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))}
383		vex 3443	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
384	125, 383	op2ndd 7944	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑡) = 𝑦)
385	384	sneqd 4591	. . . . . . . . . . . 12 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → {(2nd ‘𝑡)} = {𝑦})
386	385	difeq2d 4077	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → ((0...𝑁) ∖ {(2nd ‘𝑡)}) = ((0...𝑁) ∖ {𝑦}))
387	125, 383	op1std 7943	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑡) = 𝑥)
388	387	csbeq1d 3852	. . . . . . . . . . . 12 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋𝑥 / 𝑠⦌𝐶)
389	388	eqeq2d 2746	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑥 / 𝑠⦌𝐶))
390	386, 389	rexeqbidv 3316	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶))
391	390	ralbidv 3158	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶))
392	388	neeq2d 2991	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))
393	392	ralbidv 3158	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))
394	391, 393	anbi12d 633	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))
395	394	rabxp 5671	. . . . . . 7 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))}
396	382, 395	eqtr4i 2761	. . . . . 6 ⊢ ∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}
397		difrab 4269	. . . . . 6 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))}
398	360, 396, 397	3eqtr4g 2795	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}))
399	398	fveq2d 6837	. . . 4 ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})))
400	24	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ Fin)
401		inxp 5779	. . . . . . . . . 10 ⊢ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = (({𝑥} ∩ {𝑡}) × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)}))
402		df-ne 2932	. . . . . . . . . . . . 13 ⊢ (𝑥 ≠ 𝑡 ↔ ¬ 𝑥 = 𝑡)
403		disjsn2 4668	. . . . . . . . . . . . 13 ⊢ (𝑥 ≠ 𝑡 → ({𝑥} ∩ {𝑡}) = ∅)
404	402, 403	sylbir 235	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = 𝑡 → ({𝑥} ∩ {𝑡}) = ∅)
405	404	xpeq1d 5652	. . . . . . . . . . 11 ⊢ (¬ 𝑥 = 𝑡 → (({𝑥} ∩ {𝑡}) × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = (∅ × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})))
406		0xp 5722	. . . . . . . . . . 11 ⊢ (∅ × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅
407	405, 406	eqtrdi 2786	. . . . . . . . . 10 ⊢ (¬ 𝑥 = 𝑡 → (({𝑥} ∩ {𝑡}) × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅)
408	401, 407	eqtrid 2782	. . . . . . . . 9 ⊢ (¬ 𝑥 = 𝑡 → (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅)
409	408	orri 863	. . . . . . . 8 ⊢ (𝑥 = 𝑡 ∨ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅)
410	409	rgen2w 3055	. . . . . . 7 ⊢ ∀𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑡 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(𝑥 = 𝑡 ∨ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅)
411		sneq 4589	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → {𝑥} = {𝑡})
412		csbeq1 3851	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → ⦋𝑥 / 𝑠⦌𝐶 = ⦋𝑡 / 𝑠⦌𝐶)
413	412	eqeq2d 2746	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑡 / 𝑠⦌𝐶))
414	413	rexbidv 3159	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶))
415	414	ralbidv 3158	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶))
416	412	neeq2d 2991	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶))
417	416	ralbidv 3158	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶))
418	415, 417	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)))
419	418	rabbidv 3405	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} = {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})
420	411, 419	xpeq12d 5654	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)}))
421	420	disjor 5079	. . . . . . 7 ⊢ (Disj 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ ∀𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑡 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(𝑥 = 𝑡 ∨ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅))
422	410, 421	mpbir 231	. . . . . 6 ⊢ Disj 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})
423	422	a1i 11	. . . . 5 ⊢ (𝜑 → Disj 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
424	16, 400, 423	hashiun 15747	. . . 4 ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = Σ𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
425	399, 424	eqtr3d 2772	. . 3 ⊢ (𝜑 → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) = Σ𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
426		fo1st 7953	. . . . . . . . . . . 12 ⊢ 1st :V–onto→V
427		fofun 6746	. . . . . . . . . . . 12 ⊢ (1st :V–onto→V → Fun 1st )
428	426, 427	ax-mp 5	. . . . . . . . . . 11 ⊢ Fun 1st
429		ssv 3957	. . . . . . . . . . . 12 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ V
430		fof 6745	. . . . . . . . . . . . . 14 ⊢ (1st :V–onto→V → 1st :V⟶V)
431	426, 430	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 1st :V⟶V
432	431	fdmi 6672	. . . . . . . . . . . 12 ⊢ dom 1st = V
433	429, 432	sseqtrri 3982	. . . . . . . . . . 11 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ dom 1st
434		fores 6755	. . . . . . . . . . 11 ⊢ ((Fun 1st ∧ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ dom 1st ) → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}))
435	428, 433, 434	mp2an 693	. . . . . . . . . 10 ⊢ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})
436		fveq2 6833	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → (2nd ‘𝑡) = (2nd ‘𝑥))
437	436	csbeq1d 3852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
438		fveq2 6833	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑥 → (1st ‘𝑡) = (1st ‘𝑥))
439	438	csbeq1d 3852	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
440	439	csbeq2dv 3855	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
441	437, 440	eqtrd 2770	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑥 → ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
442	441	eqeq2d 2746	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑥 → (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
443	439	eqeq2d 2746	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
444	443	rexbidv 3159	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑥 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
445	444	ralbidv 3158	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑥 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
446	442, 445	anbi12d 633	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑥 → ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
447	446	rexrab 3653	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠 ↔ ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠))
448		xp1st 7965	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
449	448	anim1i 616	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
450		eleq1 2823	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑥) = 𝑠 → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})))
451		csbeq1a 3862	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 = (1st ‘𝑥) → 𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
452	451	eqcoms 2743	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1st ‘𝑥) = 𝑠 → 𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
453	452	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1st ‘𝑥) = 𝑠 → ⦋(1st ‘𝑥) / 𝑠⦌𝐶 = 𝐶)
454	453	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝑥) = 𝑠 → (𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶))
455	454	rexbidv 3159	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑥) = 𝑠 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
456	455	ralbidv 3158	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑥) = 𝑠 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
457	450, 456	anbi12d 633	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑥) = 𝑠 → (((1st ‘𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
458	449, 457	syl5ibcom 245	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
459	458	adantrl 717	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
460	459	expimpd 453	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
461	460	rexlimiv 3129	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
462		simplr 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
463		ovex 7391	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0...𝑁) ∈ V
464	463	enref 8924	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0...𝑁) ≈ (0...𝑁)
465		phpreu 37774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((0...𝑁) ∈ Fin ∧ (0...𝑁) ≈ (0...𝑁)) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
466	20, 464, 465	mp2an 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
467	466	biimpi 216	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 → ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
468		eqeq1 2739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑁 → (𝑖 = 𝐶 ↔ 𝑁 = 𝐶))
469	468	reubidv 3365	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑁 → (∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶))
470	469	rspcva 3573	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶)
471	232, 467, 470	syl2an 597	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶)
472		riotacl 7332	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶 → (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁))
473	471, 472	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁))
474	473	adantlr 716	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁))
475		opelxpi 5660	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(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)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
476	462, 474, 475	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(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)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
477		riotasbc 7333	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶 → [(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶)
478	471, 477	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → [(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶)
479		riotaex 7319	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ V
480		sbceq2g 4370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ V → ([(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶 ↔ 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶))
481	479, 480	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ([(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶 ↔ 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶)
482	478, 481	sylib 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶)
483	482	expcom 413	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 → (𝜑 → 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶))
484	483	imdistanri 569	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
485	484	adantlr 716	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
486		vex 3443	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑠 ∈ V
487	486, 479	op2ndd 7944	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (2nd ‘𝑥) = (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶))
488	487	csbeq1d 3852	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ⦋(2nd ‘𝑥) / 𝑗⦌𝐶 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶)
489		nfcv 2897	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑗𝑠
490		nfriota1 7322	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑗(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)
491	489, 490	nfop 4844	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑗⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩
492	491	nfeq2 2915	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑗 𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩
493	486, 479	op1std 7943	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (1st ‘𝑥) = 𝑠)
494	493	eqcomd 2741	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → 𝑠 = (1st ‘𝑥))
495	494, 451	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → 𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
496	492, 495	csbeq2d 3854	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ⦋(2nd ‘𝑥) / 𝑗⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
497	488, 496	eqtr3d 2772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
498	497	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
499	495	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (𝑖 = 𝐶 ↔ 𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
500	492, 499	rexbid 3249	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
501	500	ralbidv 3158	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
502	498, 501	anbi12d 633	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ((𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) ↔ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
503	493	biantrud 531	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠)))
504	502, 503	bitr2d 280	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠) ↔ (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
505	504	rspcev 3575	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠))
506	476, 485, 505	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠))
507	506	expl 457	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠)))
508	461, 507	impbid2 226	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠) ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
509	447, 508	bitrid 283	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠 ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
510	509	abbidv 2801	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)})
511		dfimafn 6895	. . . . . . . . . . . . . 14 ⊢ ((Fun 1st ∧ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ dom 1st ) → (1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦})
512	428, 433, 511	mp2an 693	. . . . . . . . . . . . 13 ⊢ (1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦}
513		nfcv 2897	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠(2nd ‘𝑡)
514		nfcsb1v 3872	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠⦋(1st ‘𝑡) / 𝑠⦌𝐶
515	513, 514	nfcsbw 3874	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑠⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
516	515	nfeq2 2915	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑠 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
517		nfcv 2897	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑠(0...𝑁)
518	514	nfeq2 2915	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
519	517, 518	nfrexw 3283	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑠∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
520	517, 519	nfralw 3282	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑠∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
521	516, 520	nfan 1901	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑠(𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
522		nfcv 2897	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑠((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))
523	521, 522	nfrabw 3435	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑠{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}
524		nfv 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑠(1st ‘𝑥) = 𝑦
525	523, 524	nfrexw 3283	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑠∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦
526		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠
527		eqeq2 2747	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑠 → ((1st ‘𝑥) = 𝑦 ↔ (1st ‘𝑥) = 𝑠))
528	527	rexbidv 3159	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑠 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠))
529	525, 526, 528	cbvabw 2806	. . . . . . . . . . . . 13 ⊢ {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦} = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠}
530	512, 529	eqtri 2758	. . . . . . . . . . . 12 ⊢ (1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠}
531		df-rab 3399	. . . . . . . . . . . 12 ⊢ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)}
532	510, 530, 531	3eqtr4g 2795	. . . . . . . . . . 11 ⊢ (𝜑 → (1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
533		foeq3 6743	. . . . . . . . . . 11 ⊢ ((1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
534	532, 533	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
535	435, 534	mpbii 233	. . . . . . . . 9 ⊢ (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
536		fof 6745	. . . . . . . . 9 ⊢ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}⟶{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
537	535, 536	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}⟶{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
538		fvres 6852	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = (1st ‘𝑥))
539		fvres 6852	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑦) = (1st ‘𝑦))
540	538, 539	eqeqan12d 2749	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑦) ↔ (1st ‘𝑥) = (1st ‘𝑦)))
541	540	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑦) ↔ (1st ‘𝑥) = (1st ‘𝑦)))
542	446	elrab 3645	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ↔ (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
543		xp2nd 7966	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑥) ∈ (0...𝑁))
544	543	anim1i 616	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) → ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
545	542, 544	sylbi 217	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
546		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
547	546	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
548	547	ss2rabi 4027	. . . . . . . . . . . . . . . 16 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶}
549	548	sseli 3928	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶})
550		fveq2 6833	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑦 → (2nd ‘𝑡) = (2nd ‘𝑦))
551	550	csbeq1d 3852	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑦 → ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
552		fveq2 6833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑦 → (1st ‘𝑡) = (1st ‘𝑦))
553	552	csbeq1d 3852	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑦 → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(1st ‘𝑦) / 𝑠⦌𝐶)
554	553	csbeq2dv 3855	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑦 → ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)
555	551, 554	eqtrd 2770	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑦 → ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)
556	555	eqeq2d 2746	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑦 → (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
557	556	elrab 3645	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶} ↔ (𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
558		xp2nd 7966	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑦) ∈ (0...𝑁))
559	558	anim1i 616	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶) → ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
560	557, 559	sylbi 217	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶} → ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
561	549, 560	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
562	545, 561	anim12i 614	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) → (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)))
563		an4 657	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)) ↔ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)))
564	563	anbi2i 624	. . . . . . . . . . . . . 14 ⊢ ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))))
565		anass 468	. . . . . . . . . . . . . . . . 17 ⊢ ((((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
566		ancom 460	. . . . . . . . . . . . . . . . 17 ⊢ ((((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
567	565, 566	bitr3i 277	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
568	567	anbi1i 625	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)) ↔ ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)))
569		anass 468	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))))
570	568, 569	bitri 275	. . . . . . . . . . . . . 14 ⊢ ((((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))))
571		anass 468	. . . . . . . . . . . . . 14 ⊢ (((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁))) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))))
572	564, 570, 571	3bitr4i 303	. . . . . . . . . . . . 13 ⊢ ((((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)) ↔ ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁))) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)))
573	562, 572	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) → ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁))) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)))
574		phpreu 37774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((0...𝑁) ∈ Fin ∧ (0...𝑁) ≈ (0...𝑁)) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
575	20, 464, 574	mp2an 693	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
576		reurmo 3352	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃!𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 → ∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
577	576	ralimi 3072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...𝑁)∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
578	575, 577	sylbi 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...𝑁)∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
579		eqeq1 2739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑁 → (𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
580	579	rmobidv 3364	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑁 → (∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∃*𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
581	580	rspcva 3573	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → ∃*𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
582	232, 578, 581	syl2an 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → ∃*𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
583		nfv 1916	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘 𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶
584	583	rmo3 3838	. . . . . . . . . . . . . . . . . 18 ⊢ (∃*𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁)∀𝑘 ∈ (0...𝑁)((𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → 𝑗 = 𝑘))
585	582, 584	sylib 218	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → ∀𝑗 ∈ (0...𝑁)∀𝑘 ∈ (0...𝑁)((𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → 𝑗 = 𝑘))
586		nfcsb1v 3872	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑗⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶
587	586	nfeq2 2915	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶
588		nfs1v 2162	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗[𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶
589	587, 588	nfan 1901	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗(𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
590		nfv 1916	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗(2nd ‘𝑥) = 𝑘
591	589, 590	nfim 1898	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = 𝑘)
592		nfv 1916	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦))
593		csbeq1a 3862	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑥) → ⦋(1st ‘𝑥) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
594	593	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (2nd ‘𝑥) → (𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
595	594	anbi1d 632	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (2nd ‘𝑥) → ((𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
596		eqeq1 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (2nd ‘𝑥) → (𝑗 = 𝑘 ↔ (2nd ‘𝑥) = 𝑘))
597	595, 596	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (2nd ‘𝑥) → (((𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → 𝑗 = 𝑘) ↔ ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = 𝑘)))
598		sbsbc 3743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
599		vex 3443	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑘 ∈ V
600		sbceq2g 4370	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ V → ([𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
601	599, 600	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
602	598, 601	bitri 275	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ([𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
603		csbeq1 3851	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = (2nd ‘𝑦) → ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
604	603	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = (2nd ‘𝑦) → (𝑁 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
605	602, 604	bitrid 283	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = (2nd ‘𝑦) → ([𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
606	605	anbi2d 631	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (2nd ‘𝑦) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
607		eqeq2 2747	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (2nd ‘𝑦) → ((2nd ‘𝑥) = 𝑘 ↔ (2nd ‘𝑥) = (2nd ‘𝑦)))
608	606, 607	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (2nd ‘𝑦) → (((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = 𝑘) ↔ ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦))))
609	591, 592, 597, 608	rspc2 3584	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)) → (∀𝑗 ∈ (0...𝑁)∀𝑘 ∈ (0...𝑁)((𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → 𝑗 = 𝑘) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦))))
610	585, 609	syl5com 31	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦))))
611	610	impr 454	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)))) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦)))
612		csbeq1 3851	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑥) = (1st ‘𝑦) → ⦋(1st ‘𝑥) / 𝑠⦌𝐶 = ⦋(1st ‘𝑦) / 𝑠⦌𝐶)
613	612	csbeq2dv 3855	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑥) = (1st ‘𝑦) → ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)
614	613	eqeq2d 2746	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑥) = (1st ‘𝑦) → (𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
615	614	anbi2d 631	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑥) = (1st ‘𝑦) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)))
616	615	imbi1d 341	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑥) = (1st ‘𝑦) → (((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦)) ↔ ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦))))
617	611, 616	syl5ibcom 245	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)))) → ((1st ‘𝑥) = (1st ‘𝑦) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦))))
618	617	com23 86	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)))) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶) → ((1st ‘𝑥) = (1st ‘𝑦) → (2nd ‘𝑥) = (2nd ‘𝑦))))
619	618	impr 454	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁))) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))) → ((1st ‘𝑥) = (1st ‘𝑦) → (2nd ‘𝑥) = (2nd ‘𝑦)))
620	573, 619	sylan2 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) → ((1st ‘𝑥) = (1st ‘𝑦) → (2nd ‘𝑥) = (2nd ‘𝑦)))
621		elrabi 3641	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → 𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
622		elrabi 3641	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
623		xpopth 7974	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) ↔ 𝑥 = 𝑦))
624	623	biimpd 229	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → 𝑥 = 𝑦))
625	624	expd 415	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((1st ‘𝑥) = (1st ‘𝑦) → ((2nd ‘𝑥) = (2nd ‘𝑦) → 𝑥 = 𝑦)))
626	621, 622, 625	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) → ((1st ‘𝑥) = (1st ‘𝑦) → ((2nd ‘𝑥) = (2nd ‘𝑦) → 𝑥 = 𝑦)))
627	626	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) → ((1st ‘𝑥) = (1st ‘𝑦) → ((2nd ‘𝑥) = (2nd ‘𝑦) → 𝑥 = 𝑦)))
628	620, 627	mpdd 43	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) → ((1st ‘𝑥) = (1st ‘𝑦) → 𝑥 = 𝑦))
629	541, 628	sylbid 240	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑦) → 𝑥 = 𝑦))
630	629	ralrimivva 3178	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑦) → 𝑥 = 𝑦))
631		dff13 7200	. . . . . . . 8 ⊢ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–1-1→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ↔ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}⟶{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ∧ ∀𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑦) → 𝑥 = 𝑦)))
632	537, 630, 631	sylanbrc 584	. . . . . . 7 ⊢ (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–1-1→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
633		df-f1o 6498	. . . . . . 7 ⊢ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ↔ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–1-1→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ∧ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
634	632, 535, 633	sylanbrc 584	. . . . . 6 ⊢ (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
635		rabfi 9173	. . . . . . . . 9 ⊢ (((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∈ Fin)
636	138, 635	ax-mp 5	. . . . . . . 8 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∈ Fin
637	636	elexi 3462	. . . . . . 7 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∈ V
638	637	f1oen 8911	. . . . . 6 ⊢ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
639	634, 638	syl 17	. . . . 5 ⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
640		rabfi 9173	. . . . . . 7 ⊢ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ∈ Fin)
641	136, 640	ax-mp 5	. . . . . 6 ⊢ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ∈ Fin
642		hashen 14272	. . . . . 6 ⊢ (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∈ Fin ∧ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ∈ Fin) → ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) ↔ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
643	636, 641, 642	mp2an 693	. . . . 5 ⊢ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) ↔ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
644	639, 643	sylibr 234	. . . 4 ⊢ (𝜑 → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
645	644	oveq2d 7374	. . 3 ⊢ (𝜑 → ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))
646	202, 425, 645	3eqtr3d 2778	. 2 ⊢ (𝜑 → Σ𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(♯‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))
647	135, 646	breqtrd 5123	1 ⊢ (𝜑 → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542  ∃wex 1781  [wsb 2068   ∈ wcel 2114  {cab 2713   ≠ wne 2931   ∉ wnel 3035  ∀wral 3050  ∃wrex 3059  ∃!wreu 3347  ∃*wrmo 3348  {crab 3398  Vcvv 3439  [wsbc 3739  ⦋csb 3848   ∖ cdif 3897   ∪ cun 3898   ∩ cin 3899   ⊆ wss 3900  ∅c0 4284  {csn 4579  ⟨cop 4585  ∪ ciun 4945  Disj wdisj 5064   class class class wbr 5097  {copab 5159   ↦ cmpt 5178   × cxp 5621  dom cdm 5623   ↾ cres 5625   “ cima 5626  Fun wfun 6485  ⟶wf 6487  –1-1→wf1 6488  –onto→wfo 6489  –1-1-onto→wf1o 6490  ‘cfv 6491  ℩crio 7314  (class class class)co 7358   ∘_f cof 7620  1^st c1st 7931  2^nd c2nd 7932   ↑_m cmap 8765   ≈ cen 8882   ≼ cdom 8883  Fincfn 8885  ℂcc 11026  0cc0 11028  1c1 11029   + caddc 11031   · cmul 11033   < clt 11168   ≤ cle 11169   − cmin 11366  ℕcn 12147  2c2 12202  ℕ₀cn0 12403  ℤcz 12490  ℤ_≥cuz 12753  ...cfz 13425  ..^cfzo 13572  ♯chash 14255  Σcsu 15611   ∥ cdvds 16181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-inf2 9552  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-disj 5065  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-isom 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-map 8767  df-pm 8768  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-sup 9347  df-oi 9417  df-dju 9815  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12148  df-2 12210  df-3 12211  df-n0 12404  df-z 12491  df-uz 12754  df-rp 12908  df-fz 13426  df-fzo 13573  df-seq 13927  df-exp 13987  df-hash 14256  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-clim 15413  df-sum 15612  df-dvds 16182

This theorem is referenced by:  poimirlem28  37818