cvmliftlem15 - Mathbox for Mario Carneiro

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Theorem cvmliftlem15 35885

Description: Lemma for cvmlift 35886. Discharge the assumptions of cvmliftlem14 35884. The set of all open subsets 𝑢 of the unit interval such that 𝐺 “ 𝑢 is contained in an even covering of some open set in 𝐽 is a cover of II by the definition of a covering map, so by the Lebesgue number lemma lebnumii 25200, there is a subdivision of the closed unit interval into 𝑁 equal parts such that each part is entirely contained within one such open set of 𝐽. Then using finite choice ac6sfi 9258 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 35884. (Contributed by Mario Carneiro, 14-Feb-2015.)

**Hypotheses**
Ref	Expression
cvmliftlem.1	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾_t 𝑢)Homeo(𝐽 ↾_t 𝑘))))})
cvmliftlem.b	⊢ 𝐵 = ∪ 𝐶
cvmliftlem.x	⊢ 𝑋 = ∪ 𝐽
cvmliftlem.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftlem.g	⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
cvmliftlem.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmliftlem.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))

**Assertion**
Ref	Expression
cvmliftlem15	⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))

Distinct variable groups: 𝑣,𝐵 𝑓,𝑘,𝑠,𝑢,𝑣,𝐹 𝑃,𝑓,𝑘,𝑢,𝑣 𝐶,𝑓,𝑘,𝑠,𝑢,𝑣 𝜑,𝑓,𝑠 𝑆,𝑓,𝑘,𝑠,𝑢,𝑣 𝑓,𝐺,𝑘,𝑠,𝑢,𝑣 𝑓,𝐽,𝑘,𝑠,𝑢,𝑣 Allowed substitution hints: 𝜑(𝑣, 𝑢, 𝑘) 𝐵(𝑢, 𝑓, 𝑘, 𝑠) 𝑃(𝑠) 𝑋(𝑣, 𝑢, 𝑓, 𝑘, 𝑠)

Proof of Theorem cvmliftlem15 Dummy variables 𝑏 𝑦 𝑧 𝑎 𝑐 𝑔 𝑗 𝑚 𝑛 𝑡 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4031	. . 3 ⊢ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ II
2		cvmliftlem.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
3	2	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝐺 ∈ (II Cn 𝐽))
4		simprl 783	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝑗 ∈ 𝐽)
5		cnima 23496	. . . . . . . . . 10 ⊢ ((𝐺 ∈ (II Cn 𝐽) ∧ 𝑗 ∈ 𝐽) → (^◡𝐺 “ 𝑗) ∈ II)
6	3, 4, 5	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (^◡𝐺 “ 𝑗) ∈ II)
7		simplr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝑥 ∈ (0[,]1))
8		simprrl 793	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝐺‘𝑥) ∈ 𝑗)
9		iiuni 25115	. . . . . . . . . . . . . 14 ⊢ (0[,]1) = ∪ II
10		cvmliftlem.x	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ∪ 𝐽
11	9, 10	cnf 23477	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
12	2, 11	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:(0[,]1)⟶𝑋)
13	12	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝐺:(0[,]1)⟶𝑋)
14		ffn 6706	. . . . . . . . . . 11 ⊢ (𝐺:(0[,]1)⟶𝑋 → 𝐺 Fn (0[,]1))
15		elpreima 7054	. . . . . . . . . . 11 ⊢ (𝐺 Fn (0[,]1) → (𝑥 ∈ (^◡𝐺 “ 𝑗) ↔ (𝑥 ∈ (0[,]1) ∧ (𝐺‘𝑥) ∈ 𝑗)))
16	13, 14, 15	3syl 19	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝑥 ∈ (^◡𝐺 “ 𝑗) ↔ (𝑥 ∈ (0[,]1) ∧ (𝐺‘𝑥) ∈ 𝑗)))
17	7, 8, 16	mpbir2and 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝑥 ∈ (^◡𝐺 “ 𝑗))
18		simprrr 794	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝑆‘𝑗) ≠ ∅)
19		ffun 6709	. . . . . . . . . . . . 13 ⊢ (𝐺:(0[,]1)⟶𝑋 → Fun 𝐺)
20		funimacnv 6618	. . . . . . . . . . . . 13 ⊢ (Fun 𝐺 → (𝐺 “ (^◡𝐺 “ 𝑗)) = (𝑗 ∩ ran 𝐺))
21	13, 19, 20	3syl 19	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝐺 “ (^◡𝐺 “ 𝑗)) = (𝑗 ∩ ran 𝐺))
22		inss1 4185	. . . . . . . . . . . 12 ⊢ (𝑗 ∩ ran 𝐺) ⊆ 𝑗
23	21, 22	eqsstrdi 3978	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗)
24	23	ralrimivw 3160	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → ∀𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗)
25		r19.2z 4458	. . . . . . . . . 10 ⊢ (((𝑆‘𝑗) ≠ ∅ ∧ ∀𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗) → ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗)
26	18, 24, 25	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗)
27		eleq2 2851	. . . . . . . . . . 11 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ (^◡𝐺 “ 𝑗)))
28		imaeq2 6056	. . . . . . . . . . . . 13 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → (𝐺 “ 𝑢) = (𝐺 “ (^◡𝐺 “ 𝑗)))
29	28	sseq1d 3965	. . . . . . . . . . . 12 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → ((𝐺 “ 𝑢) ⊆ 𝑗 ↔ (𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗))
30	29	rexbidv 3188	. . . . . . . . . . 11 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → (∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗 ↔ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗))
31	27, 30	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → ((𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗) ↔ (𝑥 ∈ (^◡𝐺 “ 𝑗) ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗)))
32	31	rspcev 3579	. . . . . . . . 9 ⊢ (((^◡𝐺 “ 𝑗) ∈ II ∧ (𝑥 ∈ (^◡𝐺 “ 𝑗) ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗)) → ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗))
33	6, 17, 26, 32	syl12anc 850	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗))
34		cvmliftlem.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
35	34	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
36	12	ffvelcdmda 7081	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → (𝐺‘𝑥) ∈ 𝑋)
37		cvmliftlem.1	. . . . . . . . . 10 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾_t 𝑢)Homeo(𝐽 ↾_t 𝑘))))})
38	37, 10	cvmcov 35850	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺‘𝑥) ∈ 𝑋) → ∃𝑗 ∈ 𝐽 ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))
39	35, 36, 38	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → ∃𝑗 ∈ 𝐽 ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))
40	33, 39	reximddv 3180	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → ∃𝑗 ∈ 𝐽 ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗))
41		r19.42v 3196	. . . . . . . . 9 ⊢ (∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗) ↔ (𝑥 ∈ 𝑢 ∧ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗))
42	41	rexbii 3111	. . . . . . . 8 ⊢ (∃𝑢 ∈ II ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗) ↔ ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗))
43		rexcom 3293	. . . . . . . 8 ⊢ (∃𝑗 ∈ 𝐽 ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗) ↔ ∃𝑢 ∈ II ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗))
44		elunirab 4885	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ↔ ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗))
45	42, 43, 44	3bitr4i 306	. . . . . . 7 ⊢ (∃𝑗 ∈ 𝐽 ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗) ↔ 𝑥 ∈ ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗})
46	40, 45	sylib 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → 𝑥 ∈ ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗})
47	46	ex 418	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,]1) → 𝑥 ∈ ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗}))
48	47	ssrdv 3940	. . . 4 ⊢ (𝜑 → (0[,]1) ⊆ ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗})
49		uniss 4878	. . . . . 6 ⊢ ({𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ II → ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ ∪ II)
50	1, 49	mp1i 14	. . . . 5 ⊢ (𝜑 → ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ ∪ II)
51	50, 9	sseqtrrdi 3975	. . . 4 ⊢ (𝜑 → ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ (0[,]1))
52	48, 51	eqssd 3951	. . 3 ⊢ (𝜑 → (0[,]1) = ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗})
53		lebnumii 25200	. . 3 ⊢ (({𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ II ∧ (0[,]1) = ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗}) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣)
54	1, 52, 53	sylancr 599	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣)
55		fzfi 14040	. . . . 5 ⊢ (1...𝑛) ∈ Fin
56		imaeq2 6056	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝐺 “ 𝑢) = (𝐺 “ 𝑣))
57	56	sseq1d 3965	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → ((𝐺 “ 𝑢) ⊆ 𝑗 ↔ (𝐺 “ 𝑣) ⊆ 𝑗))
58	57	2rexbidv 3229	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → (∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗 ↔ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗))
59	58	rexrab 3657	. . . . . . 7 ⊢ (∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 ↔ ∃𝑣 ∈ II (∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗 ∧ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣))
60		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑗 ∈ V
61		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
62	60, 61	op1std 8000	. . . . . . . . . . . 12 ⊢ (𝑢 = ⟨𝑗, 𝑠⟩ → (1^st ‘𝑢) = 𝑗)
63	62	sseq2d 3966	. . . . . . . . . . 11 ⊢ (𝑢 = ⟨𝑗, 𝑠⟩ → ((𝐺 “ 𝑣) ⊆ (1^st ‘𝑢) ↔ (𝐺 “ 𝑣) ⊆ 𝑗))
64	63	rexiunxp 5824	. . . . . . . . . 10 ⊢ (∃𝑢 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ 𝑣) ⊆ (1^st ‘𝑢) ↔ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗)
65		imass2 6102	. . . . . . . . . . . 12 ⊢ ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (𝐺 “ 𝑣))
66		sstr2 3941	. . . . . . . . . . . 12 ⊢ ((𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (𝐺 “ 𝑣) → ((𝐺 “ 𝑣) ⊆ (1^st ‘𝑢) → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘𝑢)))
67	65, 66	syl 18	. . . . . . . . . . 11 ⊢ ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ((𝐺 “ 𝑣) ⊆ (1^st ‘𝑢) → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘𝑢)))
68	67	reximdv 3179	. . . . . . . . . 10 ⊢ ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (∃𝑢 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ 𝑣) ⊆ (1^st ‘𝑢) → ∃𝑢 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘𝑢)))
69	64, 68	biimtrrid 246	. . . . . . . . 9 ⊢ ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗 → ∃𝑢 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘𝑢)))
70	69	impcom 413	. . . . . . . 8 ⊢ ((∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗 ∧ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣) → ∃𝑢 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘𝑢))
71	70	rexlimivw 3161	. . . . . . 7 ⊢ (∃𝑣 ∈ II (∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗 ∧ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣) → ∃𝑢 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘𝑢))
72	59, 71	sylbi 220	. . . . . 6 ⊢ (∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∃𝑢 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘𝑢))
73	72	ralimi 3101	. . . . 5 ⊢ (∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∀𝑘 ∈ (1...𝑛)∃𝑢 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘𝑢))
74		fveq2 6882	. . . . . . 7 ⊢ (𝑢 = (𝑔‘𝑘) → (1^st ‘𝑢) = (1^st ‘(𝑔‘𝑘)))
75	74	sseq2d 3966	. . . . . 6 ⊢ (𝑢 = (𝑔‘𝑘) → ((𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘𝑢) ↔ (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘))))
76	75	ac6sfi 9258	. . . . 5 ⊢ (((1...𝑛) ∈ Fin ∧ ∀𝑘 ∈ (1...𝑛)∃𝑢 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘𝑢)) → ∃𝑔(𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘))))
77	55, 73, 76	sylancr 599	. . . 4 ⊢ (∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∃𝑔(𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘))))
78		cvmliftlem.b	. . . . . . 7 ⊢ 𝐵 = ∪ 𝐶
79	34	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘)))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
80	2	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘)))) → 𝐺 ∈ (II Cn 𝐽))
81		cvmliftlem.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
82	81	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘)))) → 𝑃 ∈ 𝐵)
83		cvmliftlem.e	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))
84	83	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘)))) → (𝐹‘𝑃) = (𝐺‘0))
85		simplr 781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘)))) → 𝑛 ∈ ℕ)
86		simprl 783	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘)))) → 𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
87		sneq 4597	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑎 → {𝑗} = {𝑎})
88		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑎 → (𝑆‘𝑗) = (𝑆‘𝑎))
89	87, 88	xpeq12d 5690	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → ({𝑗} × (𝑆‘𝑗)) = ({𝑎} × (𝑆‘𝑎)))
90	89	cbviunv 5001	. . . . . . . . 9 ⊢ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) = ∪ 𝑎 ∈ 𝐽 ({𝑎} × (𝑆‘𝑎))
91		feq3 6686	. . . . . . . . 9 ⊢ (∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) = ∪ 𝑎 ∈ 𝐽 ({𝑎} × (𝑆‘𝑎)) → (𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ↔ 𝑔:(1...𝑛)⟶∪ 𝑎 ∈ 𝐽 ({𝑎} × (𝑆‘𝑎))))
92	90, 91	ax-mp 5	. . . . . . . 8 ⊢ (𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ↔ 𝑔:(1...𝑛)⟶∪ 𝑎 ∈ 𝐽 ({𝑎} × (𝑆‘𝑎)))
93	86, 92	sylib 221	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘)))) → 𝑔:(1...𝑛)⟶∪ 𝑎 ∈ 𝐽 ({𝑎} × (𝑆‘𝑎)))
94		simprr 785	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘)))) → ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘)))
95		eqid 2762	. . . . . . 7 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
96		2fveq3 6887	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑧 → (^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)) = (^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑧)))
97	96	cbvmptv 5213	. . . . . . . . . 10 ⊢ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡))) = (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑧)))
98		eleq2 2851	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑏 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐 ↔ (𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
99	98	cbvriotavw 7384	. . . . . . . . . . . . . . 15 ⊢ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)
100		fveq1 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑦‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑤 − 1) / 𝑛)))
101	100	eleq1d 2847	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
102	101	riotabidv 7376	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏) = (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
103	99, 102	eqtrid 2809	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
104	103	reseq2d 5976	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐)) = (𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)))
105	104	cnveqd 5859	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐)) = ^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)))
106	105	fveq1d 6884	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑧)) = (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)))
107	106	mpteq2dv 5203	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑧))) = (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧))))
108	97, 107	eqtrid 2809	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡))) = (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧))))
109		oveq1 7424	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑚 → (𝑤 − 1) = (𝑚 − 1))
110	109	oveq1d 7432	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑚 → ((𝑤 − 1) / 𝑛) = ((𝑚 − 1) / 𝑛))
111		oveq1 7424	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑚 → (𝑤 / 𝑛) = (𝑚 / 𝑛))
112	110, 111	oveq12d 7435	. . . . . . . . . 10 ⊢ (𝑤 = 𝑚 → (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) = (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)))
113		2fveq3 6887	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑚 → (2^nd ‘(𝑔‘𝑤)) = (2^nd ‘(𝑔‘𝑚)))
114	110	fveq2d 6886	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑚 → (𝑥‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑚 − 1) / 𝑛)))
115	114	eleq1d 2847	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑚 → ((𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))
116	113, 115	riotaeqbidv 7377	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑚 → (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏) = (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))
117	116	reseq2d 5976	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑚 → (𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) = (𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏)))
118	117	cnveqd 5859	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑚 → ^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) = ^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏)))
119	118	fveq1d 6884	. . . . . . . . . 10 ⊢ (𝑤 = 𝑚 → (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)) = (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)))
120	112, 119	mpteq12dv 5196	. . . . . . . . 9 ⊢ (𝑤 = 𝑚 → (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧))) = (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧))))
121	108, 120	cbvmpov 7512	. . . . . . . 8 ⊢ (𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)))) = (𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧))))
122		seqeq2 14073	. . . . . . . 8 ⊢ ((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)))) = (𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)))) → seq0((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})))
123	121, 122	ax-mp 5	. . . . . . 7 ⊢ seq0((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
124		eqid 2762	. . . . . . 7 ⊢ ∪ 𝑘 ∈ (1...𝑛)(seq0((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))‘𝑘) = ∪ 𝑘 ∈ (1...𝑛)(seq0((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ (^◡(𝐹 ↾ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺‘𝑡)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))‘𝑘)
125	37, 78, 10, 79, 80, 82, 84, 85, 93, 94, 95, 123, 124	cvmliftlem14 35884	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘)))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
126	125	ex 418	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)))
127	126	exlimdv 1966	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∃𝑔(𝑔:(1...𝑛)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘(𝑔‘𝑘))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)))
128	77, 127	syl5 35	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)))
129	128	rexlimdva 3165	. 2 ⊢ (𝜑 → (∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)))
130	54, 129	mpd 16	1 ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 ∃!wreu 3365 {crab 3414 Vcvv 3453 ∖ cdif 3899 ∪ cun 3900 ∩ cin 3901 ⊆ wss 3902 ∅c0 4282 𝒫 cpw 4560 {csn 4587 ⟨cop 4593 ∪ cuni 4870 ∪ ciun 4954 ↦ cmpt 5190 I cid 5553 × cxp 5657 ^◡ccnv 5658 ran crn 5660 ↾ cres 5661 “ cima 5662 ∘ ccom 5663 Fun wfun 6531 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 ℩crio 7373 (class class class)co 7417 ∈ cmpo 7419 1^st c1st 7988 2^nd c2nd 7989 Fincfn 8956 0cc0 11128 1c1 11129 − cmin 11469 / cdiv 11899 ℕcn 12261 (,)cioo 13402 [,]cicc 13405 ...cfz 13565 seqcseq 14069 ↾_t crest 17511 topGenctg 17528 Cn ccn 23455 Homeochmeo 23985 IIcii 25109 CovMap ccvm 35842

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206 ax-addf 11207

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-ec 8702 df-map 8832 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-fi 9385 df-sup 9416 df-inf 9417 df-oi 9486 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-q 13002 df-rp 13047 df-xneg 13167 df-xadd 13168 df-xmul 13169 df-ioo 13406 df-ico 13408 df-icc 13409 df-fz 13566 df-fzo 13714 df-fl 13857 df-seq 14070 df-exp 14130 df-hash 14399 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-clim 15579 df-sum 15778 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-starv 17363 df-sca 17364 df-vsca 17365 df-ip 17366 df-tset 17367 df-ple 17368 df-ds 17370 df-unif 17371 df-hom 17372 df-cco 17373 df-rest 17513 df-topn 17514 df-0g 17532 df-gsum 17533 df-topgen 17534 df-pt 17535 df-prds 17538 df-xrs 17594 df-qtop 17599 df-imas 17600 df-xps 17602 df-mre 17676 df-mrc 17677 df-acs 17679 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-submnd 18898 df-mulg 19197 df-cntz 19450 df-cmn 19915 df-psmet 21583 df-xmet 21584 df-met 21585 df-bl 21586 df-mopn 21587 df-cnfld 21592 df-top 23125 df-topon 23142 df-topsp 23164 df-bases 23177 df-cld 23250 df-ntr 23251 df-cls 23252 df-nei 23329 df-cn 23458 df-cnp 23459 df-cmp 23618 df-conn 23643 df-lly 23698 df-nlly 23699 df-tx 23794 df-hmeo 23987 df-xms 24552 df-ms 24553 df-tms 24554 df-ii 25111 df-cncf 25112 df-htpy 25204 df-phtpy 25205 df-phtpc 25226 df-pconn 35808 df-sconn 35809 df-cvm 35843

This theorem is used by: cvmlift 35886

W3C validator