Theorem poimirlem26 38565

Description: Lemma for poimir 38572 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 14117	. . . . . 6 ⊢ (0..^𝐾) ∈ Fin
2		fzfi 14115	. . . . . 6 ⊢ (1...𝑁) ∈ Fin
3		mapfi 9337	. . . . . 6 ⊢ (((0..^𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((0..^𝐾) ↑m (1...𝑁)) ∈ Fin)
4	1, 2, 3	mp2an 705	. . . . 5 ⊢ ((0..^𝐾) ↑m (1...𝑁)) ∈ Fin
5		mapfi 9337	. . . . . . 7 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ↑m (1...𝑁)) ∈ Fin)
6	2, 2, 5	mp2an 705	. . . . . 6 ⊢ ((1...𝑁) ↑m (1...𝑁)) ∈ Fin
7		f1of 6824	. . . . . . . 8 ⊢ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁))
8	7	ss2abi 3991	. . . . . . 7 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}
9		ovex 7453	. . . . . . . 8 ⊢ (1...𝑁) ∈ V
10	9, 9	mapval 8858	. . . . . . 7 ⊢ ((1...𝑁) ↑m (1...𝑁)) = {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}
11	8, 10	sseqtrri 3957	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑m (1...𝑁))
12		ssfi 9188	. . . . . 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 705	. . . . 5 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin
14	4, 13	pm3.2i 476	. . . 4 ⊢ (((0..^𝐾) ↑m (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
15		xpfi 9311	. . . 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 14	. . 3 ⊢ (𝜑 → (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin)
17		2z 12728	. . . 4 ⊢ 2 ∈ ℤ
18	17	a1i 11	. . 3 ⊢ (𝜑 → 2 ∈ ℤ)
19		snfi 9071	. . . . . . 7 ⊢ {𝑥} ∈ Fin
20		fzfi 14115	. . . . . . . 8 ⊢ (0...𝑁) ∈ Fin
21		rabfi 9262	. . . . . . . 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 9311	. . . . . . 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 705	. . . . . 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 14500	. . . . . 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 12721	. . . 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 739	. . . . . . . 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 1946	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0})))
32		nfcsb1v 3870	. . . . . . . . . . 11 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
33	32	nfeq2 2939	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
34	31, 33	nfim 1928	. . . . . . . . 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 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (1...𝑗) = (1...𝑘))
36	35	imaeq2d 6051	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((2nd ‘𝑡) “ (1...𝑗)) = ((2nd ‘𝑡) “ (1...𝑘)))
37	36	xpeq1d 5679	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (((2nd ‘𝑡) “ (1...𝑗)) × {1}) = (((2nd ‘𝑡) “ (1...𝑘)) × {1}))
38		oveq1 7427	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1))
39	38	oveq1d 7435	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → ((𝑗 + 1)...𝑁) = ((𝑘 + 1)...𝑁))
40	39	imaeq2d 6051	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)))
41	40	xpeq1d 5679	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))
42	37, 41	uneq12d 4168	. . . . . . . . . . . 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 7436	. . . . . . . . . . 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 2771	. . . . . . . . . 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 3860	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ⦋𝑡 / 𝑠⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
46	45	eqeq2d 2771	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝐵 = ⦋𝑡 / 𝑠⦌𝐶 ↔ 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
47	44, 46	imbi12d 347	. . . . . . . . 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 1946	. . . . . . . . . . 11 ⊢ Ⅎ𝑠 𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0})))
49		nfcsb1v 3870	. . . . . . . . . . . 12 ⊢ Ⅎ𝑠⦋𝑡 / 𝑠⦌𝐶
50	49	nfeq2 2939	. . . . . . . . . . 11 ⊢ Ⅎ𝑠 𝐵 = ⦋𝑡 / 𝑠⦌𝐶
51	48, 50	nfim 1928	. . . . . . . . . 10 ⊢ Ⅎ𝑠(𝑝 = ((1st ‘𝑡) ∘f + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑡 / 𝑠⦌𝐶)
52		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → (1st ‘𝑠) = (1st ‘𝑡))
53		fveq2 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑡 → (2nd ‘𝑠) = (2nd ‘𝑡))
54	53	imaeq1d 6050	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd ‘𝑡) “ (1...𝑗)))
55	54	xpeq1d 5679	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = (((2nd ‘𝑡) “ (1...𝑗)) × {1}))
56	53	imaeq1d 6050	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)))
57	56	xpeq1d 5679	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))
58	55, 57	uneq12d 4168	. . . . . . . . . . . . 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 7438	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 3860	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → 𝐶 = ⦋𝑡 / 𝑠⦌𝐶)
62	61	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (𝐵 = 𝐶 ↔ 𝐵 = ⦋𝑡 / 𝑠⦌𝐶))
63	60, 62	imbi12d 347	. . . . . . . . . 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 2276	. . . . . . . . 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 2276	. . . . . . . 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 765	. . . . . . . 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 8033	. . . . . . . . . 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 8869	. . . . . . . . . 10 ⊢ ((1st ‘𝑥) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘𝑥):(1...𝑁)⟶(0..^𝐾))
71	69, 70	syl 18	. . . . . . . . 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 740	. . . . . . . 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 8034	. . . . . . . . . 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 6898	. . . . . . . . . . 11 ⊢ (2nd ‘𝑥) ∈ V
75		f1oeq1 6812	. . . . . . . . . . 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 3632	. . . . . . . . . 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 221	. . . . . . . . 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 740	. . . . . . . 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 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑁
80		nfcv 2922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑥
81	80, 32	nfcsbw 3872	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
82	79, 81	nfne 3058	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
83		nfcv 2922	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡𝐶
84	83, 49, 61	cbvcsbw 3856	. . . . . . . . . . . . . 14 ⊢ ⦋𝑥 / 𝑠⦌𝐶 = ⦋𝑥 / 𝑡⦌⦋𝑡 / 𝑠⦌𝐶
85	45	csbeq2dv 3853	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ⦋𝑥 / 𝑡⦌⦋𝑡 / 𝑠⦌𝐶 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
86	84, 85	eqtrid 2807	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ⦋𝑥 / 𝑠⦌𝐶 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
87	86	neeq2d 3015	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
88	82, 87	rspc 3564	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑁) → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
89	88	impcom 413	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
90	89	adantll 727	. . . . . . . . 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 8040	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
92	91	csbeq1d 3850	. . . . . . . . . 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 744	. . . . . . . . 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 3024	. . . . . . . 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 38564	. . . . . . 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 1946	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 = ⦋𝑥 / 𝑠⦌𝐶
97	81	nfeq2 2939	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗 𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
98	86	eqeq2d 2771	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
99	96, 97, 98	cbvrexw 3305	. . . . . . . . . . . . 13 ⊢ (∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ ∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
100	92	eqeq2d 2771	. . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . 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 287	. . . . . . . . . . . 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 3185	. . . . . . . . . . 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 537	. . . . . . . . . . 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 519	. . . . . . . . . 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 3419	. . . . . . . . 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 6889	. . . . . . . 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 727	. . . . . . 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 5130	. . . . . 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 418	. . . . 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 16441	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∥ 0)
112	17, 111	ax-mp 5	. . . . . . 7 ⊢ 2 ∥ 0
113		hash0 14511	. . . . . . 7 ⊢ (♯‘∅) = 0
114	112, 113	breqtrri 5131	. . . . . 6 ⊢ 2 ∥ (♯‘∅)
115		simpr 490	. . . . . . . . . 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 155	. . . . . . . . 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 3158	. . . . . . . 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 4337	. . . . . . . 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 237	. . . . . . 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 6889	. . . . . 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 5142	. . . . 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 182	. . . 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 14579	. . . . . 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 705	. . . . 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 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
126		hashsng 14513	. . . . . . 7 ⊢ (𝑥 ∈ V → (♯‘{𝑥}) = 1)
127	125, 126	ax-mp 5	. . . . . 6 ⊢ (♯‘{𝑥}) = 1
128	127	oveq1i 7430	. . . . 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 14500	. . . . . . . 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 12618	. . . . . 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 11314	. . . . 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 2787	. . . 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 5146	. . 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 16478	. 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 705	. . . . . 6 ⊢ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin
137		xpfi 9311	. . . . . 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 705	. . . . 5 ⊢ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin
139		rabfi 9262	. . . . 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 12351	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
142		npcan1 11741	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
143	141, 142	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
144		nnm1nn0 12647	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
145	29, 144	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
146	145	nn0zd 12718	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
147		uzid 12980	. . . . . . . . . . . 12 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)))
148		peano2uz 13028	. . . . . . . . . . . 12 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
149	146, 147, 148	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
150	143, 149	eqeltrrd 2861	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
151		fzss2 13698	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
152		ssralv 3977	. . . . . . . . . 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 19	. . . . . . . . 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 486	. . . . . . . 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 4085	. . . . . . . . . . . 12 ⊢ (∀𝑗 ∈ (0...𝑁)(𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ ∀𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
156		nfv 1946	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝜑
157		nfcsb1v 3870	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
158	157	nfeq2 2939	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
159	156, 158	nfan 1931	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
160		nfv 1946	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗 𝑖 ∈ (0...(𝑁 − 1))
161	159, 160	nfan 1931	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1)))
162		nnel 3071	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑗 ∉ {(2nd ‘𝑡)} ↔ 𝑗 ∈ {(2nd ‘𝑡)})
163		velsn 4599	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ {(2nd ‘𝑡)} ↔ 𝑗 = (2nd ‘𝑡))
164	162, 163	bitri 278	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑗 ∉ {(2nd ‘𝑡)} ↔ 𝑗 = (2nd ‘𝑡))
165		csbeq1a 3860	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑡) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
166	165	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (2nd ‘𝑡) → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
167	166	biimparc 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ 𝑗 = (2nd ‘𝑡)) → 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
168	29	nnred 12350	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑁 ∈ ℝ)
169	168	ltm1d 12249	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
170	145	nn0red 12668	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
171	170, 168	ltnled 11457	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
172	169, 171	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
173		elfzle2 13661	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ (0...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
174	172, 173	nsyl 141	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
175		eleq1 2848	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 𝑁 → (𝑖 ∈ (0...(𝑁 − 1)) ↔ 𝑁 ∈ (0...(𝑁 − 1))))
176	175	notbid 321	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑁 → (¬ 𝑖 ∈ (0...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (0...(𝑁 − 1))))
177	174, 176	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑖 = 𝑁 → ¬ 𝑖 ∈ (0...(𝑁 − 1))))
178	177	con2d 135	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑖 ∈ (0...(𝑁 − 1)) → ¬ 𝑖 = 𝑁))
179	178	imp 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) → ¬ 𝑖 = 𝑁)
180		eqeq2 2772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑖 = 𝑁 ↔ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
181	180	notbid 321	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (¬ 𝑖 = 𝑁 ↔ ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
182	179, 181	syl5ibcom 248	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
183	167, 182	syl5 35	. . . . . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . . . . 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 665	. . . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (¬ 𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
187		idd 25	. . . . . . . . . . . . . . 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 189	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 3263	. . . . . . . . . . . 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 246	. . . . . . . . . . 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 153	. . . . . . . . . 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 3089	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
194		dfrex2 3089	. . . . . . . . . 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 299	. . . . . . . . 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 3174	. . . . . . . 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 48	. . . . . . 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 459	. . . . . 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 486	. . . . 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 4000	. . . 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 14557	. . . 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 599	. . 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 8034	. . . . . . . 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 2956	. . . . . . . . . . . 12 ⊢ (𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
205	204	ralbii 3108	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁) ¬ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
206		ralnex 3088	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ (0...𝑁) ¬ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
207	205, 206	bitri 278	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
208	29	nnnn0d 12667	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
209		nn0uz 13003	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
210	208, 209	eleqtrdi 2870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
211	143, 210	eqeltrd 2860	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘0))
212		fzsplit2 13683	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
213	211, 150, 212	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
214	143	oveq1d 7435	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
215	29	nnzd 12719	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
216		fzsn 13700	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
217	215, 216	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
218	214, 217	eqtrd 2795	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
219	218	uneq2d 4167	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((0...(𝑁 − 1)) ∪ {𝑁}))
220	213, 219	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁}))
221	220	raleqdv 3319	. . . . . . . . . . . . . 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 4195	. . . . . . . . . . . . . . 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 4072	. . . . . . . . . . . . . . . . . 18 ⊢ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ⊆ (0...𝑁)
224		ssrexv 3978	. . . . . . . . . . . . . . . . . 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 3099	. . . . . . . . . . . . . . . 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 542	. . . . . . . . . . . . . . 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 293	. . . . . . . . . . . . . 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 519	. . . . . . . . . . . . 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 728	. . . . . . . . . . . 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 13759	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁))
232	208, 231	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
233	232	ad2antrr 739	. . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑁 → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
235	234	rexbidv 3186	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑁 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
236	235	rspcva 3574	. . . . . . . . . . . . . . 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 1946	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗(𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁))
238		nfcv 2922	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗(0...(𝑁 − 1))
239		nfre1 3287	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
240	238, 239	nfralw 3309	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
241	237, 240	nfan 1931	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
242		eleq1 2848	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑁 ∈ (0...(𝑁 − 1)) ↔ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
243	242	notbid 321	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (¬ 𝑁 ∈ (0...(𝑁 − 1)) ↔ ¬ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
244	174, 243	syl5ibcom 248	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ¬ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
245	244	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 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 4747	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ↔ (𝑗 ∈ (0...𝑁) ∧ 𝑗 ≠ (2nd ‘𝑡)))
247		diffi 9190	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0...𝑁) ∈ Fin → ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin)
248	20, 247	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin
249		ssrab2 4005	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ⊆ ((0...𝑁) ∖ {(2nd ‘𝑡)})
250		ssdomg 9027	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14559	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((0...𝑁) ∈ Fin ∧ (2nd ‘𝑡) ∈ (0...𝑁)) → (♯‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = ((♯‘(0...𝑁)) − 1))
253	20, 252	mpan 703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((2nd ‘𝑡) ∈ (0...𝑁) → (♯‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = ((♯‘(0...𝑁)) − 1))
254		1cnd 11302	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → 1 ∈ ℂ)
255	141, 254, 254	addsubd 11690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑁 + 1) − 1) = ((𝑁 − 1) + 1))
256		hashfz0 14577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑁 ∈ ℕ0 → (♯‘(0...𝑁)) = (𝑁 + 1))
257	208, 256	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (♯‘(0...𝑁)) = (𝑁 + 1))
258	257	oveq1d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((♯‘(0...𝑁)) − 1) = ((𝑁 + 1) − 1))
259		hashfz0 14577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑁 − 1) ∈ ℕ0 → (♯‘(0...(𝑁 − 1))) = ((𝑁 − 1) + 1))
260	145, 259	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (♯‘(0...(𝑁 − 1))) = ((𝑁 − 1) + 1))
261	255, 258, 260	3eqtr4d 2805	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((♯‘(0...𝑁)) − 1) = (♯‘(0...(𝑁 − 1))))
262	253, 261	sylan9eqr 2817	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) → (♯‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = (♯‘(0...(𝑁 − 1))))
263		fzfi 14115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0...(𝑁 − 1)) ∈ Fin
264		hashen 14491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((♯‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = (♯‘(0...(𝑁 − 1))) ↔ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≈ (0...(𝑁 − 1)))
266	262, 265	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) → ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≈ (0...(𝑁 − 1)))
267		rabfi 9262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑖 ∈ (0...(𝑁 − 1)) ↔ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
270	269	biimpac 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
271		rabid 3432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 508	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 561	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 502	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} → 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}))
279	274	pm4.71rd 572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5186	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1946	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ Ⅎ𝑘(𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
282		nfrab1 3431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ Ⅎ𝑗{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}
283	282	nfcri 2914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ Ⅎ𝑗 𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}
284		nfcsb1v 3870	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
285	284	nfeq2 2939	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ Ⅎ𝑗 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
286	283, 285	nfan 1931	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ Ⅎ𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
287		eleq1 2848	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3860	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑗 = 𝑘 → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
289	288	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑗 = 𝑘 → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
290	287, 289	anbi12d 644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5108	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5497	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 300	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 605	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3184	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2922	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ Ⅎ𝑝{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}
301		nfv 1946	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2922	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ Ⅎ𝑗𝑝
303	282, 284	nfmpt 5202	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ Ⅎ𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
304		nfcv 2922	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ Ⅎ𝑗𝑖
305	302, 303, 304	nfbr 5151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3303	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3096	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2922	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ Ⅎ𝑗𝑘
313		nfcv 2922	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ Ⅎ𝑗((0...𝑁) ∖ {(2nd ‘𝑡)})
314	284	nfel1 2938	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))
315	288	eleq1d 2845	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3641	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9039	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9116	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9254	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1494	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . 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 2052	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑗 → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
333	332	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑗 → ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
334	333	eleq1d 2845	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 = 𝑗 → (⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ↔ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
335	334, 317	vtoclga 3536	. . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . 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 607	. . . . . . . . . . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . . . . . . . . . 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 2973	. . . . . . . . . . . . . . . . . . . . 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 48	. . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . . . . 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 3269	. . . . . . . . . . . . . . 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 35	. . . . . . . . . . . . . 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 708	. . . . . . . . . . . . 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 572	. . . . . . . . . . . 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 4635	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
354	353	ad2antrr 739	. . . . . . . . . . . 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 313	. . . . . . . . . . 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 321	. . . . . . . . . 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 286	. . . . . . . . 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 590	. . . . . . . 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 605	. . . . . . 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 3418	. . . . . 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 1946	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝑡 = ⟨𝑥, 𝑘⟩
362		nfv 1946	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 𝑥 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
363		nfrab1 3431	. . . . . . . . . . . . . 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 2914	. . . . . . . . . . . . 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 1931	. . . . . . . . . . . 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 1931	. . . . . . . . . . 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 1946	. . . . . . . . . . 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 4833	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → ⟨𝑥, 𝑘⟩ = ⟨𝑥, 𝑦⟩)
369	368	eqeq2d 2771	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑦 → (𝑡 = ⟨𝑥, 𝑘⟩ ↔ 𝑡 = ⟨𝑥, 𝑦⟩))
370		eleq1 2848	. . . . . . . . . . . . . . 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 3432	. . . . . . . . . . . . . . 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 290	. . . . . . . . . . . . . 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 642	. . . . . . . . . . . . 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 1111	. . . . . . . . . . . . 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 292	. . . . . . . . . . . 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 644	. . . . . . . . . . 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 2371	. . . . . . . . . 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 1880	. . . . . . . . 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 5813	. . . . . . . . 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 5500	. . . . . . . . 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 306	. . . . . . . 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 2757	. . . . . . 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 3454	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
384	125, 383	op2ndd 8012	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑡) = 𝑦)
385	384	sneqd 4595	. . . . . . . . . . . 12 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → {(2nd ‘𝑡)} = {𝑦})
386	385	difeq2d 4063	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → ((0...𝑁) ∖ {(2nd ‘𝑡)}) = ((0...𝑁) ∖ {𝑦}))
387	125, 383	op1std 8011	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑡) = 𝑥)
388	387	csbeq1d 3850	. . . . . . . . . . . 12 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋𝑥 / 𝑠⦌𝐶)
389	388	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑥 / 𝑠⦌𝐶))
390	386, 389	rexeqbidv 3335	. . . . . . . . . 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 3185	. . . . . . . . 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 3015	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))
393	392	ralbidv 3185	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))
394	391, 393	anbi12d 644	. . . . . . . 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 5698	. . . . . . 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 2786	. . . . . 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 4263	. . . . . 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 2820	. . . . 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 6889	. . . 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 5808	. . . . . . . . . 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 2956	. . . . . . . . . . . . 13 ⊢ (𝑥 ≠ 𝑡 ↔ ¬ 𝑥 = 𝑡)
403		disjsn2 4672	. . . . . . . . . . . . 13 ⊢ (𝑥 ≠ 𝑡 → ({𝑥} ∩ {𝑡}) = ∅)
404	402, 403	sylbir 238	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = 𝑡 → ({𝑥} ∩ {𝑡}) = ∅)
405	404	xpeq1d 5679	. . . . . . . . . . 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 5749	. . . . . . . . . . 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 2811	. . . . . . . . . 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 2807	. . . . . . . . 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 876	. . . . . . . 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 3081	. . . . . . 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 4593	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → {𝑥} = {𝑡})
412		csbeq1 3849	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → ⦋𝑥 / 𝑠⦌𝐶 = ⦋𝑡 / 𝑠⦌𝐶)
413	412	eqeq2d 2771	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑡 / 𝑠⦌𝐶))
414	413	rexbidv 3186	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶))
415	414	ralbidv 3185	. . . . . . . . . . 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 3015	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶))
417	416	ralbidv 3185	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶))
418	415, 417	anbi12d 644	. . . . . . . . . 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 3419	. . . . . . . . 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 5681	. . . . . . . 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 5084	. . . . . . 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 234	. . . . . 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 15989	. . . 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 2797	. . 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 8021	. . . . . . . . . . . 12 ⊢ 1st :V–onto→V
427		fofun 6797	. . . . . . . . . . . 12 ⊢ (1st :V–onto→V → Fun 1st )
428	426, 427	ax-mp 5	. . . . . . . . . . 11 ⊢ Fun 1st
429		ssv 3932	. . . . . . . . . . . 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 6796	. . . . . . . . . . . . . 14 ⊢ (1st :V–onto→V → 1st :V⟶V)
431	426, 430	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 1st :V⟶V
432	431	fdmi 6721	. . . . . . . . . . . 12 ⊢ dom 1st = V
433	429, 432	sseqtrri 3957	. . . . . . . . . . 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 6806	. . . . . . . . . . 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 705	. . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → (2nd ‘𝑡) = (2nd ‘𝑥))
437	436	csbeq1d 3850	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
438		fveq2 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑥 → (1st ‘𝑡) = (1st ‘𝑥))
439	438	csbeq1d 3850	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
440	439	csbeq2dv 3853	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
441	437, 440	eqtrd 2795	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑥 → ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
442	441	eqeq2d 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑥 → (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
443	439	eqeq2d 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
444	443	rexbidv 3186	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑥 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
445	444	ralbidv 3185	. . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . 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 8033	. . . . . . . . . . . . . . . . . . . 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 627	. . . . . . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . . . 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 3860	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 = (1st ‘𝑥) → 𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
452	451	eqcoms 2768	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1st ‘𝑥) = 𝑠 → 𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
453	452	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1st ‘𝑥) = 𝑠 → ⦋(1st ‘𝑥) / 𝑠⦌𝐶 = 𝐶)
454	453	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝑥) = 𝑠 → (𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶))
455	454	rexbidv 3186	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑥) = 𝑠 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
456	455	ralbidv 3185	. . . . . . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . . . . . 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 248	. . . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . 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 7453	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0...𝑁) ∈ V
464	463	enref 9012	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0...𝑁) ≈ (0...𝑁)
465		phpreu 38528	. . . . . . . . . . . . . . . . . . . . . . 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 705	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
467	466	biimpi 219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 → ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
468		eqeq1 2764	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑁 → (𝑖 = 𝐶 ↔ 𝑁 = 𝐶))
469	468	reubidv 3381	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑁 → (∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶))
470	469	rspcva 3574	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶)
471	232, 467, 470	syl2an 608	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶)
472		riotacl 7394	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶 → (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁))
473	471, 472	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁))
474	473	adantlr 728	. . . . . . . . . . . . . . . . . 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 5687	. . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . 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 7395	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶 → [(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶)
478	471, 477	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → [(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶)
479		riotaex 7381	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ V
480		sbceq2g 4376	. . . . . . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶)
483	482	expcom 419	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 → (𝜑 → 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶))
484	483	imdistanri 580	. . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . 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 3454	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑠 ∈ V
487	486, 479	op2ndd 8012	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (2nd ‘𝑥) = (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶))
488	487	csbeq1d 3850	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ⦋(2nd ‘𝑥) / 𝑗⦌𝐶 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶)
489		nfcv 2922	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑗𝑠
490		nfriota1 7384	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑗(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)
491	489, 490	nfop 4848	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑗⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩
492	491	nfeq2 2939	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑗 𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩
493	486, 479	op1std 8011	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (1st ‘𝑥) = 𝑠)
494	493	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → 𝑠 = (1st ‘𝑥))
495	494, 451	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → 𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
496	492, 495	csbeq2d 3852	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ⦋(2nd ‘𝑥) / 𝑗⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
497	488, 496	eqtr3d 2797	. . . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (𝑖 = 𝐶 ↔ 𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
500	492, 499	rexbid 3276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
501	500	ralbidv 3185	. . . . . . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . . . . . 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 541	. . . . . . . . . . . . . . . . . . 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 283	. . . . . . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . 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 286	. . . . . . . . . . . . 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 2826	. . . . . . . . . . . 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 6947	. . . . . . . . . . . . . 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 705	. . . . . . . . . . . . 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 2922	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠(2nd ‘𝑡)
514		nfcsb1v 3870	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠⦋(1st ‘𝑡) / 𝑠⦌𝐶
515	513, 514	nfcsbw 3872	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑠⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
516	515	nfeq2 2939	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑠 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
517		nfcv 2922	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑠(0...𝑁)
518	514	nfeq2 2939	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
519	517, 518	nfrexw 3310	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑠∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
520	517, 519	nfralw 3309	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑠∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
521	516, 520	nfan 1931	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑠(𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
522		nfcv 2922	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑠((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))
523	521, 522	nfrabw 3447	. . . . . . . . . . . . . . 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 1946	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑠(1st ‘𝑥) = 𝑦
525	523, 524	nfrexw 3310	. . . . . . . . . . . . . 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 1946	. . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑠 → ((1st ‘𝑥) = 𝑦 ↔ (1st ‘𝑥) = 𝑠))
528	527	rexbidv 3186	. . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . 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 2783	. . . . . . . . . . . 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 3413	. . . . . . . . . . . 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 2820	. . . . . . . . . . 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 6794	. . . . . . . . . . 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 18	. . . . . . . . . 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 236	. . . . . . . . 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 6796	. . . . . . . . 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 18	. . . . . . . 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 6904	. . . . . . . . . . . 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 6904	. . . . . . . . . . . 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 2774	. . . . . . . . . . 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 487	. . . . . . . . . 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 3644	. . . . . . . . . . . . . . 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 8034	. . . . . . . . . . . . . . . 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 627	. . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . 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 4001	. . . . . . . . . . . . . . . 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 3904	. . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑦 → (2nd ‘𝑡) = (2nd ‘𝑦))
551	550	csbeq1d 3850	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑦 → ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
552		fveq2 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑦 → (1st ‘𝑡) = (1st ‘𝑦))
553	552	csbeq1d 3850	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑦 → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(1st ‘𝑦) / 𝑠⦌𝐶)
554	553	csbeq2dv 3853	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑦 → ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)
555	551, 554	eqtrd 2795	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑦 → ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)
556	555	eqeq2d 2771	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑦 → (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
557	556	elrab 3644	. . . . . . . . . . . . . . . 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 8034	. . . . . . . . . . . . . . . . 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 627	. . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 625	. . . . . . . . . . . . 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 669	. . . . . . . . . . . . . . 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 635	. . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . . . . 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 280	. . . . . . . . . . . . . . . 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 636	. . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . 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 306	. . . . . . . . . . . . 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 221	. . . . . . . . . . . 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 38528	. . . . . . . . . . . . . . . . . . . . 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 705	. . . . . . . . . . . . . . . . . . . 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 3368	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃!𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 → ∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
577	576	ralimi 3099	. . . . . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑁 → (𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
580	579	rmobidv 3380	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑁 → (∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∃*𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
581	580	rspcva 3574	. . . . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → ∃*𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
583		nfv 1946	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘 𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶
584	583	rmo3 3835	. . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . 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 3870	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑗⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶
587	586	nfeq2 2939	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶
588		nfs1v 2192	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗[𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶
589	587, 588	nfan 1931	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗(𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
590		nfv 1946	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗(2nd ‘𝑥) = 𝑘
591	589, 590	nfim 1928	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = 𝑘)
592		nfv 1946	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦))
593		csbeq1a 3860	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑥) → ⦋(1st ‘𝑥) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
594	593	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (2nd ‘𝑥) → (𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
595	594	anbi1d 643	. . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (2nd ‘𝑥) → (𝑗 = 𝑘 ↔ (2nd ‘𝑥) = 𝑘))
597	595, 596	imbi12d 347	. . . . . . . . . . . . . . . . . 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 3742	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
599		vex 3454	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑘 ∈ V
600		sbceq2g 4376	. . . . . . . . . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ([𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
603		csbeq1 3849	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = (2nd ‘𝑦) → ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
604	603	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = (2nd ‘𝑦) → (𝑁 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
605	602, 604	bitrid 286	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = (2nd ‘𝑦) → ([𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
606	605	anbi2d 642	. . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (2nd ‘𝑦) → ((2nd ‘𝑥) = 𝑘 ↔ (2nd ‘𝑥) = (2nd ‘𝑦)))
608	606, 607	imbi12d 347	. . . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . 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 3849	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑥) = (1st ‘𝑦) → ⦋(1st ‘𝑥) / 𝑠⦌𝐶 = ⦋(1st ‘𝑦) / 𝑠⦌𝐶)
613	612	csbeq2dv 3853	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑥) = (1st ‘𝑦) → ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)
614	613	eqeq2d 2771	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑥) = (1st ‘𝑦) → (𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
615	614	anbi2d 642	. . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . 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 248	. . . . . . . . . . . . . 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 87	. . . . . . . . . . . . 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 460	. . . . . . . . . . . 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 605	. . . . . . . . . . 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 3640	. . . . . . . . . . . . 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 3640	. . . . . . . . . . . . 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 8042	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 421	. . . . . . . . . . . . 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 608	. . . . . . . . . . . 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 487	. . . . . . . . . . 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 44	. . . . . . . . . 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 243	. . . . . . . . 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 3205	. . . . . . . 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 7258	. . . . . . . 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 595	. . . . . . 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 6545	. . . . . . 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 595	. . . . . 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 9262	. . . . . . . . 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 3472	. . . . . . 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 8999	. . . . . 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 18	. . . . 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 9262	. . . . . . 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 14491	. . . . . 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 705	. . . . 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 237	. . . 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 7436	. . 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 2803	. 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 5130	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})))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569  ∃wex 1811  [wsb 2098   ∈ wcel 2144  {cab 2738   ≠ wne 2955   ∉ wnel 3061  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363  ∃*wrmo 3364  {crab 3412  Vcvv 3450  [wsbc 3738  ⦋csb 3846   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  {csn 4583  ⟨cop 4589  ∪ ciun 4950  Disj wdisj 5069   class class class wbr 5102  {copab 5166   ↦ cmpt 5185   × cxp 5648  dom cdm 5650   ↾ cres 5652   “ cima 5653  Fun wfun 6532  ⟶wf 6534  –1-1→wf1 6535  –onto→wfo 6536  –1-1-onto→wf1o 6537  ‘cfv 6538  ℩crio 7376  (class class class)co 7420   ∘_f cof 7691  1^st c1st 7999  2^nd c2nd 8000   ↑_m cmap 8847   ≈ cen 8970   ≼ cdom 8971  Fincfn 8973  ℂcc 11198  0cc0 11200  1c1 11201   + caddc 11203   · cmul 11205   < clt 11343   ≤ cle 11344   − cmin 11541  ℕcn 12335  2c2 12397  ℕ₀cn0 12606  ℤcz 12693  ℤ_≥cuz 12965  ...cfz 13639  ..^cfzo 13788  ♯chash 14474  Σcsu 15853   ∥ cdvds 16422

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9642  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-disj 5070  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-oadd 8480  df-er 8717  df-map 8849  df-pm 8850  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-oi 9504  df-dju 9982  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-n0 12607  df-z 12694  df-uz 12966  df-rp 13121  df-fz 13640  df-fzo 13789  df-seq 14145  df-exp 14205  df-hash 14475  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-clim 15655  df-sum 15854  df-dvds 16423

This theorem is used by:  poimirlem28  38567