cvmliftlem15 - Mathbox for Mario Carneiro

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Theorem cvmliftlem15 36243

Description: Lemma for cvmlift 36244. Discharge the assumptions of cvmliftlem14 36242. 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 25320, 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 9306 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 36242. (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 4008	. . 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 23616	. . . . . . . . . 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 25235	. . . . . . . . . . . . . 14 ⊢ (0[,]1) = ∪ II
10		cvmliftlem.x	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ∪ 𝐽
11	9, 10	cnf 23597	. . . . . . . . . . . . 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 6734	. . . . . . . . . . 11 ⊢ (𝐺:(0[,]1)⟶𝑋 → 𝐺 Fn (0[,]1))
15		elpreima 7083	. . . . . . . . . . 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 6737	. . . . . . . . . . . . 13 ⊢ (𝐺:(0[,]1)⟶𝑋 → Fun 𝐺)
20		funimacnv 6646	. . . . . . . . . . . . 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 4136	. . . . . . . . . . . 12 ⊢ (𝑗 ∩ ran 𝐺) ⊆ 𝑗
23	21, 22	eqsstrdi 3957	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗)
24	23	ralrimivw 3162	. . . . . . . . . 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 2853	. . . . . . . . . . 11 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ (^◡𝐺 “ 𝑗)))
28		imaeq2 6061	. . . . . . . . . . . . 13 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → (𝐺 “ 𝑢) = (𝐺 “ (^◡𝐺 “ 𝑗)))
29	28	sseq1d 3944	. . . . . . . . . . . 12 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → ((𝐺 “ 𝑢) ⊆ 𝑗 ↔ (𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗))
30	29	rexbidv 3190	. . . . . . . . . . 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 3580	. . . . . . . . 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 7111	. . . . . . . . 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 36208	. . . . . . . . 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 3182	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → ∃𝑗 ∈ 𝐽 ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗))
41		r19.42v 3198	. . . . . . . . 9 ⊢ (∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗) ↔ (𝑥 ∈ 𝑢 ∧ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗))
42	41	rexbii 3113	. . . . . . . 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 3295	. . . . . . . 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 3919	. . . 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 3954	. . . 4 ⊢ (𝜑 → ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ (0[,]1))
52	48, 51	eqssd 3930	. . 3 ⊢ (𝜑 → (0[,]1) = ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗})
53		lebnumii 25320	. . 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 14145	. . . . 5 ⊢ (1...𝑛) ∈ Fin
56		imaeq2 6061	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝐺 “ 𝑢) = (𝐺 “ 𝑣))
57	56	sseq1d 3944	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → ((𝐺 “ 𝑢) ⊆ 𝑗 ↔ (𝐺 “ 𝑣) ⊆ 𝑗))
58	57	2rexbidv 3231	. . . . . . . 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 3458	. . . . . . . . . . . . 13 ⊢ 𝑗 ∈ V
61		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
62	60, 61	op1std 8038	. . . . . . . . . . . 12 ⊢ (𝑢 = ⟨𝑗, 𝑠⟩ → (1^st ‘𝑢) = 𝑗)
63	62	sseq2d 3945	. . . . . . . . . . 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 6068	. . . . . . . . . . . 12 ⊢ ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (𝐺 “ 𝑣))
66		sstr2 3920	. . . . . . . . . . . 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 3181	. . . . . . . . . 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 3163	. . . . . . 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 3103	. . . . 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 6911	. . . . . . 7 ⊢ (𝑢 = (𝑔‘𝑘) → (1^st ‘𝑢) = (1^st ‘(𝑔‘𝑘)))
75	74	sseq2d 3945	. . . . . 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 9306	. . . . 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 4594	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑎 → {𝑗} = {𝑎})
88		fveq2 6911	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑎 → (𝑆‘𝑗) = (𝑆‘𝑎))
89	87, 88	xpeq12d 5687	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → ({𝑗} × (𝑆‘𝑗)) = ({𝑎} × (𝑆‘𝑎)))
90	89	cbviunv 5000	. . . . . . . . 9 ⊢ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) = ∪ 𝑎 ∈ 𝐽 ({𝑎} × (𝑆‘𝑎))
91		feq3 6714	. . . . . . . . 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 2763	. . . . . . 7 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
96		2fveq3 6916	. . . . . . . . . . 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 2853	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑏 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐 ↔ (𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
99	98	cbvriotavw 7413	. . . . . . . . . . . . . . 15 ⊢ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)
100		fveq1 6910	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑦‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑤 − 1) / 𝑛)))
101	100	eleq1d 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
102	101	riotabidv 7405	. . . . . . . . . . . . . . 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 2810	. . . . . . . . . . . . . 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 5982	. . . . . . . . . . . . 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 5861	. . . . . . . . . . . 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 6913	. . . . . . . . . . 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 2810	. . . . . . . . 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 7453	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑚 → (𝑤 − 1) = (𝑚 − 1))
110	109	oveq1d 7461	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑚 → ((𝑤 − 1) / 𝑛) = ((𝑚 − 1) / 𝑛))
111		oveq1 7453	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑚 → (𝑤 / 𝑛) = (𝑚 / 𝑛))
112	110, 111	oveq12d 7464	. . . . . . . . . 10 ⊢ (𝑤 = 𝑚 → (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) = (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)))
113		2fveq3 6916	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑚 → (2^nd ‘(𝑔‘𝑤)) = (2^nd ‘(𝑔‘𝑚)))
114	110	fveq2d 6915	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑚 → (𝑥‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑚 − 1) / 𝑛)))
115	114	eleq1d 2849	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑚 → ((𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))
116	113, 115	riotaeqbidv 7406	. . . . . . . . . . . . 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 5982	. . . . . . . . . . . 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 5861	. . . . . . . . . . 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 6913	. . . . . . . . . 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 7541	. . . . . . . 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 14178	. . . . . . . 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 2763	. . . . . . 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 36242	. . . . . 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 1968	. . . 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 3167	. 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 2147 ≠ wne 2959 ∀wral 3080 ∃wrex 3090 ∃!wreu 3367 {crab 3416 Vcvv 3454 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 𝒫 cpw 4560 {csn 4588 ⟨cop 4838 ∪ cuni 4870 ∪ ciun 4954 ↦ cmpt 5190 I cid 5550 × cxp 5654 ^◡ccnv 5655 ran crn 5657 ↾ cres 5658 “ cima 5659 ∘ ccom 5660 Fun wfun 6555 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 ℩crio 7402 (class class class)co 7446 ∈ cmpo 7448 1^st c1st 8026 2^nd c2nd 8027 Fincfn 9004 0cc0 11230 1c1 11231 − cmin 11571 / cdiv 12003 ℕcn 12365 (,)cioo 13506 [,]cicc 13509 ...cfz 13669 seqcseq 14174 ↾_t crest 17623 topGenctg 17640 Cn ccn 23575 Homeochmeo 24105 IIcii 25229 CovMap ccvm 36200

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-inf2 9672 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307 ax-pre-sup 11308 ax-addf 11309

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-of 7720 df-om 7905 df-1st 8028 df-2nd 8029 df-supp 8208 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-2o 8508 df-er 8748 df-ec 8750 df-map 8880 df-ixp 8957 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-fsupp 9385 df-fi 9433 df-sup 9464 df-inf 9465 df-oi 9534 df-card 10050 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-div 12004 df-nn 12366 df-2 12435 df-3 12436 df-4 12437 df-5 12438 df-6 12439 df-7 12440 df-8 12441 df-9 12442 df-n0 12637 df-z 12724 df-dec 12845 df-uz 12996 df-q 13106 df-rp 13151 df-xneg 13271 df-xadd 13272 df-xmul 13273 df-ioo 13510 df-ico 13512 df-icc 13513 df-fz 13670 df-fzo 13819 df-fl 13962 df-seq 14175 df-exp 14235 df-hash 14505 df-cj 15298 df-re 15299 df-im 15300 df-sqrt 15434 df-abs 15435 df-clim 15687 df-sum 15886 df-struct 17357 df-sets 17374 df-slot 17392 df-ndx 17404 df-base 17420 df-ress 17441 df-plusg 17473 df-mulr 17474 df-starv 17475 df-sca 17476 df-vsca 17477 df-ip 17478 df-tset 17479 df-ple 17480 df-ds 17482 df-unif 17483 df-hom 17484 df-cco 17485 df-rest 17625 df-topn 17626 df-0g 17644 df-gsum 17645 df-topgen 17646 df-pt 17647 df-prds 17650 df-xrs 17706 df-qtop 17711 df-imas 17712 df-xps 17714 df-mre 17788 df-mrc 17789 df-acs 17791 df-mgm 18848 df-sgrp 18940 df-mnd 18956 df-submnd 19011 df-mulg 19310 df-cntz 19564 df-cmn 20029 df-psmet 21703 df-xmet 21704 df-met 21705 df-bl 21706 df-mopn 21707 df-cnfld 21712 df-top 23245 df-topon 23262 df-topsp 23284 df-bases 23297 df-cld 23370 df-ntr 23371 df-cls 23372 df-nei 23449 df-cn 23578 df-cnp 23579 df-cmp 23738 df-conn 23763 df-lly 23818 df-nlly 23819 df-tx 23914 df-hmeo 24107 df-xms 24672 df-ms 24673 df-tms 24674 df-ii 25231 df-cncf 25232 df-htpy 25324 df-phtpy 25325 df-phtpc 25346 df-pconn 36166 df-sconn 36167 df-cvm 36201

This theorem is used by: cvmlift 36244

W3C validator