cvmliftlem15 - Mathbox for Mario Carneiro

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Theorem cvmliftlem15 36149

Description: Lemma for cvmlift 36150. Discharge the assumptions of cvmliftlem14 36148. 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 25292, 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 9278 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 36148. (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 4005	. . 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 23588	. . . . . . . . . 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 25207	. . . . . . . . . . . . . 14 ⊢ (0[,]1) = ∪ II
10		cvmliftlem.x	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ∪ 𝐽
11	9, 10	cnf 23569	. . . . . . . . . . . . 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 6709	. . . . . . . . . . 11 ⊢ (𝐺:(0[,]1)⟶𝑋 → 𝐺 Fn (0[,]1))
15		elpreima 7057	. . . . . . . . . . 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 6712	. . . . . . . . . . . . 13 ⊢ (𝐺:(0[,]1)⟶𝑋 → Fun 𝐺)
20		funimacnv 6621	. . . . . . . . . . . . 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 4133	. . . . . . . . . . . 12 ⊢ (𝑗 ∩ ran 𝐺) ⊆ 𝑗
23	21, 22	eqsstrdi 3952	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗)
24	23	ralrimivw 3158	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → ∀𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗)
25		r19.2z 4454	. . . . . . . . . 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 2849	. . . . . . . . . . 11 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ (^◡𝐺 “ 𝑗)))
28		imaeq2 6047	. . . . . . . . . . . . 13 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → (𝐺 “ 𝑢) = (𝐺 “ (^◡𝐺 “ 𝑗)))
29	28	sseq1d 3939	. . . . . . . . . . . 12 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → ((𝐺 “ 𝑢) ⊆ 𝑗 ↔ (𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗))
30	29	rexbidv 3186	. . . . . . . . . . 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 3576	. . . . . . . . 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 7084	. . . . . . . . 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 36114	. . . . . . . . 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 3178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → ∃𝑗 ∈ 𝐽 ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗))
41		r19.42v 3194	. . . . . . . . 9 ⊢ (∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗) ↔ (𝑥 ∈ 𝑢 ∧ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗))
42	41	rexbii 3109	. . . . . . . 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 3291	. . . . . . . 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 4881	. . . . . . . 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 3914	. . . 4 ⊢ (𝜑 → (0[,]1) ⊆ ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗})
49		uniss 4874	. . . . . 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 3949	. . . 4 ⊢ (𝜑 → ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ (0[,]1))
52	48, 51	eqssd 3925	. . 3 ⊢ (𝜑 → (0[,]1) = ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗})
53		lebnumii 25292	. . 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 14118	. . . . 5 ⊢ (1...𝑛) ∈ Fin
56		imaeq2 6047	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝐺 “ 𝑢) = (𝐺 “ 𝑣))
57	56	sseq1d 3939	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → ((𝐺 “ 𝑢) ⊆ 𝑗 ↔ (𝐺 “ 𝑣) ⊆ 𝑗))
58	57	2rexbidv 3227	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → (∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗 ↔ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗))
59	58	rexrab 3653	. . . . . . 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 3454	. . . . . . . . . . . . 13 ⊢ 𝑗 ∈ V
61		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
62	60, 61	op1std 8011	. . . . . . . . . . . 12 ⊢ (𝑢 = ⟨𝑗, 𝑠⟩ → (1^st ‘𝑢) = 𝑗)
63	62	sseq2d 3940	. . . . . . . . . . 11 ⊢ (𝑢 = ⟨𝑗, 𝑠⟩ → ((𝐺 “ 𝑣) ⊆ (1^st ‘𝑢) ↔ (𝐺 “ 𝑣) ⊆ 𝑗))
64	63	rexiunxp 5816	. . . . . . . . . 10 ⊢ (∃𝑢 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ 𝑣) ⊆ (1^st ‘𝑢) ↔ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗)
65		imass2 6054	. . . . . . . . . . . 12 ⊢ ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (𝐺 “ 𝑣))
66		sstr2 3915	. . . . . . . . . . . 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 3177	. . . . . . . . . 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 3159	. . . . . . 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 3099	. . . . 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 6885	. . . . . . 7 ⊢ (𝑢 = (𝑔‘𝑘) → (1^st ‘𝑢) = (1^st ‘(𝑔‘𝑘)))
75	74	sseq2d 3940	. . . . . 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 9278	. . . . 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 4590	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑎 → {𝑗} = {𝑎})
88		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑎 → (𝑆‘𝑗) = (𝑆‘𝑎))
89	87, 88	xpeq12d 5681	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → ({𝑗} × (𝑆‘𝑗)) = ({𝑎} × (𝑆‘𝑎)))
90	89	cbviunv 4996	. . . . . . . . 9 ⊢ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) = ∪ 𝑎 ∈ 𝐽 ({𝑎} × (𝑆‘𝑎))
91		feq3 6689	. . . . . . . . 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 2760	. . . . . . 7 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
96		2fveq3 6890	. . . . . . . . . . 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 5208	. . . . . . . . . 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 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑏 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐 ↔ (𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
99	98	cbvriotavw 7387	. . . . . . . . . . . . . . 15 ⊢ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)
100		fveq1 6884	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑦‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑤 − 1) / 𝑛)))
101	100	eleq1d 2845	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
102	101	riotabidv 7379	. . . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . . 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 5969	. . . . . . . . . . . . 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 5852	. . . . . . . . . . . 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 6887	. . . . . . . . . . 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 5198	. . . . . . . . . 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 2807	. . . . . . . . 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 7427	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑚 → (𝑤 − 1) = (𝑚 − 1))
110	109	oveq1d 7435	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑚 → ((𝑤 − 1) / 𝑛) = ((𝑚 − 1) / 𝑛))
111		oveq1 7427	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑚 → (𝑤 / 𝑛) = (𝑚 / 𝑛))
112	110, 111	oveq12d 7438	. . . . . . . . . 10 ⊢ (𝑤 = 𝑚 → (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) = (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)))
113		2fveq3 6890	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑚 → (2^nd ‘(𝑔‘𝑤)) = (2^nd ‘(𝑔‘𝑚)))
114	110	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑚 → (𝑥‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑚 − 1) / 𝑛)))
115	114	eleq1d 2845	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑚 → ((𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))
116	113, 115	riotaeqbidv 7380	. . . . . . . . . . . . 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 5969	. . . . . . . . . . . 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 5852	. . . . . . . . . . 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 6887	. . . . . . . . . 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 5191	. . . . . . . . 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 7515	. . . . . . . 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 14151	. . . . . . . 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 2760	. . . . . . 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 36148	. . . . . 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 1965	. . . 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 3163	. 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 1569 ∃wex 1811 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ∃!wreu 3363 {crab 3412 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 ∩ cin 4089 ∪ cun 4149 ∅c0 4278 𝒫 cpw 4556 {csn 4584 ⟨cop 4834 ∪ cuni 4866 ∪ ciun 4950 ↦ cmpt 5185 I cid 5544 × cxp 5648 ^◡ccnv 5649 ran crn 5651 ↾ cres 5652 “ cima 5653 ∘ ccom 5654 Fun wfun 6532 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 ℩crio 7376 (class class class)co 7420 ∈ cmpo 7422 1^st c1st 7999 2^nd c2nd 8000 Fincfn 8976 0cc0 11203 1c1 11204 − cmin 11544 / cdiv 11976 ℕcn 12338 (,)cioo 13479 [,]cicc 13482 ...cfz 13642 seqcseq 14147 ↾_t crest 17596 topGenctg 17613 Cn ccn 23547 Homeochmeo 24077 IIcii 25201 CovMap ccvm 36106

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 9645 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281 ax-addf 11282

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 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 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-supp 8180 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-er 8720 df-ec 8722 df-map 8852 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9357 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9507 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-q 13079 df-rp 13124 df-xneg 13244 df-xadd 13245 df-xmul 13246 df-ioo 13483 df-ico 13485 df-icc 13486 df-fz 13643 df-fzo 13792 df-fl 13935 df-seq 14148 df-exp 14208 df-hash 14478 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-clim 15660 df-sum 15859 df-struct 17330 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-mulr 17447 df-starv 17448 df-sca 17449 df-vsca 17450 df-ip 17451 df-tset 17452 df-ple 17453 df-ds 17455 df-unif 17456 df-hom 17457 df-cco 17458 df-rest 17598 df-topn 17599 df-0g 17617 df-gsum 17618 df-topgen 17619 df-pt 17620 df-prds 17623 df-xrs 17679 df-qtop 17684 df-imas 17685 df-xps 17687 df-mre 17761 df-mrc 17762 df-acs 17764 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-submnd 18984 df-mulg 19283 df-cntz 19536 df-cmn 20001 df-psmet 21675 df-xmet 21676 df-met 21677 df-bl 21678 df-mopn 21679 df-cnfld 21684 df-top 23217 df-topon 23234 df-topsp 23256 df-bases 23269 df-cld 23342 df-ntr 23343 df-cls 23344 df-nei 23421 df-cn 23550 df-cnp 23551 df-cmp 23710 df-conn 23735 df-lly 23790 df-nlly 23791 df-tx 23886 df-hmeo 24079 df-xms 24644 df-ms 24645 df-tms 24646 df-ii 25203 df-cncf 25204 df-htpy 25296 df-phtpy 25297 df-phtpc 25318 df-pconn 36072 df-sconn 36073 df-cvm 36107

This theorem is used by: cvmlift 36150

W3C validator