cvmliftlem15 - Mathbox for Mario Carneiro

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Theorem cvmliftlem15 35325

Description: Lemma for cvmlift 35326. Discharge the assumptions of cvmliftlem14 35324. 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 24921, 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 9297 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 35324. (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 4060	. . 3 ⊢ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ II
2		cvmliftlem.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
3	2	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝐺 ∈ (II Cn 𝐽))
4		simprl 770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝑗 ∈ 𝐽)
5		cnima 23208	. . . . . . . . . 10 ⊢ ((𝐺 ∈ (II Cn 𝐽) ∧ 𝑗 ∈ 𝐽) → (^◡𝐺 “ 𝑗) ∈ II)
6	3, 4, 5	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (^◡𝐺 “ 𝑗) ∈ II)
7		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝑥 ∈ (0[,]1))
8		simprrl 780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝐺‘𝑥) ∈ 𝑗)
9		iiuni 24830	. . . . . . . . . . . . . 14 ⊢ (0[,]1) = ∪ II
10		cvmliftlem.x	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ∪ 𝐽
11	9, 10	cnf 23189	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
12	2, 11	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:(0[,]1)⟶𝑋)
13	12	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝐺:(0[,]1)⟶𝑋)
14		ffn 6711	. . . . . . . . . . 11 ⊢ (𝐺:(0[,]1)⟶𝑋 → 𝐺 Fn (0[,]1))
15		elpreima 7053	. . . . . . . . . . 11 ⊢ (𝐺 Fn (0[,]1) → (𝑥 ∈ (^◡𝐺 “ 𝑗) ↔ (𝑥 ∈ (0[,]1) ∧ (𝐺‘𝑥) ∈ 𝑗)))
16	13, 14, 15	3syl 18	. . . . . . . . . 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 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → 𝑥 ∈ (^◡𝐺 “ 𝑗))
18		simprrr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝑆‘𝑗) ≠ ∅)
19		ffun 6714	. . . . . . . . . . . . 13 ⊢ (𝐺:(0[,]1)⟶𝑋 → Fun 𝐺)
20		funimacnv 6622	. . . . . . . . . . . . 13 ⊢ (Fun 𝐺 → (𝐺 “ (^◡𝐺 “ 𝑗)) = (𝑗 ∩ ran 𝐺))
21	13, 19, 20	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝐺 “ (^◡𝐺 “ 𝑗)) = (𝑗 ∩ ran 𝐺))
22		inss1 4217	. . . . . . . . . . . 12 ⊢ (𝑗 ∩ ran 𝐺) ⊆ 𝑗
23	21, 22	eqsstrdi 4008	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → (𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗)
24	23	ralrimivw 3137	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → ∀𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗)
25		r19.2z 4475	. . . . . . . . . 10 ⊢ (((𝑆‘𝑗) ≠ ∅ ∧ ∀𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗) → ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗)
26	18, 24, 25	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (0[,]1)) ∧ (𝑗 ∈ 𝐽 ∧ ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))) → ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗)
27		eleq2 2824	. . . . . . . . . . 11 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ (^◡𝐺 “ 𝑗)))
28		imaeq2 6048	. . . . . . . . . . . . 13 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → (𝐺 “ 𝑢) = (𝐺 “ (^◡𝐺 “ 𝑗)))
29	28	sseq1d 3995	. . . . . . . . . . . 12 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → ((𝐺 “ 𝑢) ⊆ 𝑗 ↔ (𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗))
30	29	rexbidv 3165	. . . . . . . . . . 11 ⊢ (𝑢 = (^◡𝐺 “ 𝑗) → (∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗 ↔ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ (^◡𝐺 “ 𝑗)) ⊆ 𝑗))
31	27, 30	anbi12d 632	. . . . . . . . . 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 3606	. . . . . . . . 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 836	. . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
36	12	ffvelcdmda 7079	. . . . . . . . 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 35290	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺‘𝑥) ∈ 𝑋) → ∃𝑗 ∈ 𝐽 ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))
39	35, 36, 38	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → ∃𝑗 ∈ 𝐽 ((𝐺‘𝑥) ∈ 𝑗 ∧ (𝑆‘𝑗) ≠ ∅))
40	33, 39	reximddv 3157	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → ∃𝑗 ∈ 𝐽 ∃𝑢 ∈ II (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗))
41		r19.42v 3177	. . . . . . . . 9 ⊢ (∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗) ↔ (𝑥 ∈ 𝑢 ∧ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗))
42	41	rexbii 3084	. . . . . . . 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 3275	. . . . . . . 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 4903	. . . . . . . 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 303	. . . . . . 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 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → 𝑥 ∈ ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗})
47	46	ex 412	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,]1) → 𝑥 ∈ ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗}))
48	47	ssrdv 3969	. . . 4 ⊢ (𝜑 → (0[,]1) ⊆ ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗})
49		uniss 4896	. . . . . 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 13	. . . . 5 ⊢ (𝜑 → ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ ∪ II)
51	50, 9	sseqtrrdi 4005	. . . 4 ⊢ (𝜑 → ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗} ⊆ (0[,]1))
52	48, 51	eqssd 3981	. . 3 ⊢ (𝜑 → (0[,]1) = ∪ {𝑢 ∈ II ∣ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗})
53		lebnumii 24921	. . 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 587	. 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 13995	. . . . 5 ⊢ (1...𝑛) ∈ Fin
56		imaeq2 6048	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝐺 “ 𝑢) = (𝐺 “ 𝑣))
57	56	sseq1d 3995	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → ((𝐺 “ 𝑢) ⊆ 𝑗 ↔ (𝐺 “ 𝑣) ⊆ 𝑗))
58	57	2rexbidv 3210	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → (∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑢) ⊆ 𝑗 ↔ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗))
59	58	rexrab 3684	. . . . . . 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 3468	. . . . . . . . . . . . 13 ⊢ 𝑗 ∈ V
61		vex 3468	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
62	60, 61	op1std 8003	. . . . . . . . . . . 12 ⊢ (𝑢 = ⟨𝑗, 𝑠⟩ → (1^st ‘𝑢) = 𝑗)
63	62	sseq2d 3996	. . . . . . . . . . 11 ⊢ (𝑢 = ⟨𝑗, 𝑠⟩ → ((𝐺 “ 𝑣) ⊆ (1^st ‘𝑢) ↔ (𝐺 “ 𝑣) ⊆ 𝑗))
64	63	rexiunxp 5825	. . . . . . . . . 10 ⊢ (∃𝑢 ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))(𝐺 “ 𝑣) ⊆ (1^st ‘𝑢) ↔ ∃𝑗 ∈ 𝐽 ∃𝑠 ∈ (𝑆‘𝑗)(𝐺 “ 𝑣) ⊆ 𝑗)
65		imass2 6094	. . . . . . . . . . . 12 ⊢ ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (𝐺 “ 𝑣))
66		sstr2 3970	. . . . . . . . . . . 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 17	. . . . . . . . . . 11 ⊢ ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ((𝐺 “ 𝑣) ⊆ (1^st ‘𝑢) → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1^st ‘𝑢)))
68	67	reximdv 3156	. . . . . . . . . 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 243	. . . . . . . . 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 407	. . . . . . . 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 3138	. . . . . . 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 217	. . . . . 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 3074	. . . . 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 6881	. . . . . . 7 ⊢ (𝑢 = (𝑔‘𝑘) → (1^st ‘𝑢) = (1^st ‘(𝑔‘𝑘)))
75	74	sseq2d 3996	. . . . . 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 9297	. . . . 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 587	. . . 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 726	. . . . . . 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 726	. . . . . . 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 726	. . . . . . 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 726	. . . . . . 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 768	. . . . . . 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 770	. . . . . . . 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 4616	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑎 → {𝑗} = {𝑎})
88		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑎 → (𝑆‘𝑗) = (𝑆‘𝑎))
89	87, 88	xpeq12d 5690	. . . . . . . . . 10 ⊢ (𝑗 = 𝑎 → ({𝑗} × (𝑆‘𝑗)) = ({𝑎} × (𝑆‘𝑎)))
90	89	cbviunv 5021	. . . . . . . . 9 ⊢ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) = ∪ 𝑎 ∈ 𝐽 ({𝑎} × (𝑆‘𝑎))
91		feq3 6693	. . . . . . . . 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 218	. . . . . . 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 772	. . . . . . 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 2736	. . . . . . 7 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
96		2fveq3 6886	. . . . . . . . . . 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 5230	. . . . . . . . . 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 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑏 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐 ↔ (𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
99	98	cbvriotavw 7377	. . . . . . . . . . . . . . 15 ⊢ (℩𝑐 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (℩𝑏 ∈ (2^nd ‘(𝑔‘𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)
100		fveq1 6880	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑦‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑤 − 1) / 𝑛)))
101	100	eleq1d 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
102	101	riotabidv 7369	. . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . 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 5971	. . . . . . . . . . . . 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 5860	. . . . . . . . . . . 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 6883	. . . . . . . . . . 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 5220	. . . . . . . . . 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 2783	. . . . . . . . 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 7417	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑚 → (𝑤 − 1) = (𝑚 − 1))
110	109	oveq1d 7425	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑚 → ((𝑤 − 1) / 𝑛) = ((𝑚 − 1) / 𝑛))
111		oveq1 7417	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑚 → (𝑤 / 𝑛) = (𝑚 / 𝑛))
112	110, 111	oveq12d 7428	. . . . . . . . . 10 ⊢ (𝑤 = 𝑚 → (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) = (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)))
113		2fveq3 6886	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑚 → (2^nd ‘(𝑔‘𝑤)) = (2^nd ‘(𝑔‘𝑚)))
114	110	fveq2d 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑚 → (𝑥‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑚 − 1) / 𝑛)))
115	114	eleq1d 2820	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑚 → ((𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))
116	113, 115	riotaeqbidv 7370	. . . . . . . . . . . . 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 5971	. . . . . . . . . . . 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 5860	. . . . . . . . . . 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 6883	. . . . . . . . . 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 5212	. . . . . . . . 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 7507	. . . . . . . 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 14028	. . . . . . . 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 2736	. . . . . . 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 35324	. . . . . 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 412	. . . . 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 1933	. . . 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 34	. . 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 3142	. 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 15	1 ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 ∃!wreu 3362 {crab 3420 Vcvv 3464 ∖ cdif 3928 ∪ cun 3929 ∩ cin 3930 ⊆ wss 3931 ∅c0 4313 𝒫 cpw 4580 {csn 4606 ⟨cop 4612 ∪ cuni 4888 ∪ ciun 4972 ↦ cmpt 5206 I cid 5552 × cxp 5657 ^◡ccnv 5658 ran crn 5660 ↾ cres 5661 “ cima 5662 ∘ ccom 5663 Fun wfun 6530 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 ℩crio 7366 (class class class)co 7410 ∈ cmpo 7412 1^st c1st 7991 2^nd c2nd 7992 Fincfn 8964 0cc0 11134 1c1 11135 − cmin 11471 / cdiv 11899 ℕcn 12245 (,)cioo 13367 [,]cicc 13370 ...cfz 13529 seqcseq 14024 ↾_t crest 17439 topGenctg 17456 Cn ccn 23167 Homeochmeo 23696 IIcii 24824 CovMap ccvm 35282

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 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 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-ec 8726 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-seq 14025 df-exp 14085 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509 df-sum 15708 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-cn 23170 df-cnp 23171 df-cmp 23330 df-conn 23355 df-lly 23409 df-nlly 23410 df-tx 23505 df-hmeo 23698 df-xms 24264 df-ms 24265 df-tms 24266 df-ii 24826 df-cncf 24827 df-htpy 24925 df-phtpy 24926 df-phtpc 24947 df-pconn 35248 df-sconn 35249 df-cvm 35283

This theorem is referenced by: cvmlift 35326

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