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Theorem cvmliftlem15 34587
Description: Lemma for cvmlift 34588. Discharge the assumptions of cvmliftlem14 34586. The set of all open subsets 𝑢 of the unit interval such that 𝐺𝑢 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 24712, 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 9289 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 34586. (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypotheses
Ref Expression
cvmliftlem.1 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
cvmliftlem.b 𝐵 = 𝐶
cvmliftlem.x 𝑋 = 𝐽
cvmliftlem.f (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftlem.g (𝜑𝐺 ∈ (II Cn 𝐽))
cvmliftlem.p (𝜑𝑃𝐵)
cvmliftlem.e (𝜑 → (𝐹𝑃) = (𝐺‘0))
Assertion
Ref Expression
cvmliftlem15 (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 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.
StepHypRef Expression
1 ssrab2 4076 . . 3 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ⊆ II
2 cvmliftlem.g . . . . . . . . . . 11 (𝜑𝐺 ∈ (II Cn 𝐽))
32ad2antrr 722 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝐺 ∈ (II Cn 𝐽))
4 simprl 767 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝑗𝐽)
5 cnima 22989 . . . . . . . . . 10 ((𝐺 ∈ (II Cn 𝐽) ∧ 𝑗𝐽) → (𝐺𝑗) ∈ II)
63, 4, 5syl2anc 582 . . . . . . . . 9 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝐺𝑗) ∈ II)
7 simplr 765 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝑥 ∈ (0[,]1))
8 simprrl 777 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝐺𝑥) ∈ 𝑗)
9 iiuni 24621 . . . . . . . . . . . . . 14 (0[,]1) = II
10 cvmliftlem.x . . . . . . . . . . . . . 14 𝑋 = 𝐽
119, 10cnf 22970 . . . . . . . . . . . . 13 (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
122, 11syl 17 . . . . . . . . . . . 12 (𝜑𝐺:(0[,]1)⟶𝑋)
1312ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝐺:(0[,]1)⟶𝑋)
14 ffn 6716 . . . . . . . . . . 11 (𝐺:(0[,]1)⟶𝑋𝐺 Fn (0[,]1))
15 elpreima 7058 . . . . . . . . . . 11 (𝐺 Fn (0[,]1) → (𝑥 ∈ (𝐺𝑗) ↔ (𝑥 ∈ (0[,]1) ∧ (𝐺𝑥) ∈ 𝑗)))
1613, 14, 153syl 18 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝑥 ∈ (𝐺𝑗) ↔ (𝑥 ∈ (0[,]1) ∧ (𝐺𝑥) ∈ 𝑗)))
177, 8, 16mpbir2and 709 . . . . . . . . 9 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝑥 ∈ (𝐺𝑗))
18 simprrr 778 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝑆𝑗) ≠ ∅)
19 ffun 6719 . . . . . . . . . . . . 13 (𝐺:(0[,]1)⟶𝑋 → Fun 𝐺)
20 funimacnv 6628 . . . . . . . . . . . . 13 (Fun 𝐺 → (𝐺 “ (𝐺𝑗)) = (𝑗 ∩ ran 𝐺))
2113, 19, 203syl 18 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝐺 “ (𝐺𝑗)) = (𝑗 ∩ ran 𝐺))
22 inss1 4227 . . . . . . . . . . . 12 (𝑗 ∩ ran 𝐺) ⊆ 𝑗
2321, 22eqsstrdi 4035 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
2423ralrimivw 3148 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → ∀𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
25 r19.2z 4493 . . . . . . . . . 10 (((𝑆𝑗) ≠ ∅ ∧ ∀𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗) → ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
2618, 24, 25syl2anc 582 . . . . . . . . 9 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
27 eleq2 2820 . . . . . . . . . . 11 (𝑢 = (𝐺𝑗) → (𝑥𝑢𝑥 ∈ (𝐺𝑗)))
28 imaeq2 6054 . . . . . . . . . . . . 13 (𝑢 = (𝐺𝑗) → (𝐺𝑢) = (𝐺 “ (𝐺𝑗)))
2928sseq1d 4012 . . . . . . . . . . . 12 (𝑢 = (𝐺𝑗) → ((𝐺𝑢) ⊆ 𝑗 ↔ (𝐺 “ (𝐺𝑗)) ⊆ 𝑗))
3029rexbidv 3176 . . . . . . . . . . 11 (𝑢 = (𝐺𝑗) → (∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗 ↔ ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗))
3127, 30anbi12d 629 . . . . . . . . . 10 (𝑢 = (𝐺𝑗) → ((𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ (𝑥 ∈ (𝐺𝑗) ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)))
3231rspcev 3611 . . . . . . . . 9 (((𝐺𝑗) ∈ II ∧ (𝑥 ∈ (𝐺𝑗) ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)) → ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
336, 17, 26, 32syl12anc 833 . . . . . . . 8 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
34 cvmliftlem.f . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
3534adantr 479 . . . . . . . . 9 ((𝜑𝑥 ∈ (0[,]1)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
3612ffvelcdmda 7085 . . . . . . . . 9 ((𝜑𝑥 ∈ (0[,]1)) → (𝐺𝑥) ∈ 𝑋)
37 cvmliftlem.1 . . . . . . . . . 10 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
3837, 10cvmcov 34552 . . . . . . . . 9 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺𝑥) ∈ 𝑋) → ∃𝑗𝐽 ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))
3935, 36, 38syl2anc 582 . . . . . . . 8 ((𝜑𝑥 ∈ (0[,]1)) → ∃𝑗𝐽 ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))
4033, 39reximddv 3169 . . . . . . 7 ((𝜑𝑥 ∈ (0[,]1)) → ∃𝑗𝐽𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
41 r19.42v 3188 . . . . . . . . 9 (∃𝑗𝐽 (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ (𝑥𝑢 ∧ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
4241rexbii 3092 . . . . . . . 8 (∃𝑢 ∈ II ∃𝑗𝐽 (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
43 rexcom 3285 . . . . . . . 8 (∃𝑗𝐽𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ ∃𝑢 ∈ II ∃𝑗𝐽 (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
44 elunirab 4923 . . . . . . . 8 (𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ↔ ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
4542, 43, 443bitr4i 302 . . . . . . 7 (∃𝑗𝐽𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ 𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
4640, 45sylib 217 . . . . . 6 ((𝜑𝑥 ∈ (0[,]1)) → 𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
4746ex 411 . . . . 5 (𝜑 → (𝑥 ∈ (0[,]1) → 𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗}))
4847ssrdv 3987 . . . 4 (𝜑 → (0[,]1) ⊆ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
49 uniss 4915 . . . . . 6 ({𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ⊆ II → {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ⊆ II)
501, 49mp1i 13 . . . . 5 (𝜑 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ⊆ II)
5150, 9sseqtrrdi 4032 . . . 4 (𝜑 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ⊆ (0[,]1))
5248, 51eqssd 3998 . . 3 (𝜑 → (0[,]1) = {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
53 lebnumii 24712 . . 3 (({𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ⊆ II ∧ (0[,]1) = {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗}) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣)
541, 52, 53sylancr 585 . 2 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣)
55 fzfi 13941 . . . . 5 (1...𝑛) ∈ Fin
56 imaeq2 6054 . . . . . . . . . 10 (𝑢 = 𝑣 → (𝐺𝑢) = (𝐺𝑣))
5756sseq1d 4012 . . . . . . . . 9 (𝑢 = 𝑣 → ((𝐺𝑢) ⊆ 𝑗 ↔ (𝐺𝑣) ⊆ 𝑗))
58572rexbidv 3217 . . . . . . . 8 (𝑢 = 𝑣 → (∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗 ↔ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗))
5958rexrab 3691 . . . . . . 7 (∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 ↔ ∃𝑣 ∈ II (∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗 ∧ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣))
60 vex 3476 . . . . . . . . . . . . 13 𝑗 ∈ V
61 vex 3476 . . . . . . . . . . . . 13 𝑠 ∈ V
6260, 61op1std 7987 . . . . . . . . . . . 12 (𝑢 = ⟨𝑗, 𝑠⟩ → (1st𝑢) = 𝑗)
6362sseq2d 4013 . . . . . . . . . . 11 (𝑢 = ⟨𝑗, 𝑠⟩ → ((𝐺𝑣) ⊆ (1st𝑢) ↔ (𝐺𝑣) ⊆ 𝑗))
6463rexiunxp 5839 . . . . . . . . . 10 (∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺𝑣) ⊆ (1st𝑢) ↔ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗)
65 imass2 6100 . . . . . . . . . . . 12 ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (𝐺𝑣))
66 sstr2 3988 . . . . . . . . . . . 12 ((𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (𝐺𝑣) → ((𝐺𝑣) ⊆ (1st𝑢) → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢)))
6765, 66syl 17 . . . . . . . . . . 11 ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ((𝐺𝑣) ⊆ (1st𝑢) → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢)))
6867reximdv 3168 . . . . . . . . . 10 ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺𝑣) ⊆ (1st𝑢) → ∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢)))
6964, 68biimtrrid 242 . . . . . . . . 9 ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗 → ∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢)))
7069impcom 406 . . . . . . . 8 ((∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗 ∧ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣) → ∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢))
7170rexlimivw 3149 . . . . . . 7 (∃𝑣 ∈ II (∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗 ∧ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣) → ∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢))
7259, 71sylbi 216 . . . . . 6 (∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢))
7372ralimi 3081 . . . . 5 (∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∀𝑘 ∈ (1...𝑛)∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢))
74 fveq2 6890 . . . . . . 7 (𝑢 = (𝑔𝑘) → (1st𝑢) = (1st ‘(𝑔𝑘)))
7574sseq2d 4013 . . . . . 6 (𝑢 = (𝑔𝑘) → ((𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢) ↔ (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘))))
7675ac6sfi 9289 . . . . 5 (((1...𝑛) ∈ Fin ∧ ∀𝑘 ∈ (1...𝑛)∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢)) → ∃𝑔(𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘))))
7755, 73, 76sylancr 585 . . . 4 (∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∃𝑔(𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘))))
78 cvmliftlem.b . . . . . . 7 𝐵 = 𝐶
7934ad2antrr 722 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘)))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
802ad2antrr 722 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘)))) → 𝐺 ∈ (II Cn 𝐽))
81 cvmliftlem.p . . . . . . . 8 (𝜑𝑃𝐵)
8281ad2antrr 722 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘)))) → 𝑃𝐵)
83 cvmliftlem.e . . . . . . . 8 (𝜑 → (𝐹𝑃) = (𝐺‘0))
8483ad2antrr 722 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘)))) → (𝐹𝑃) = (𝐺‘0))
85 simplr 765 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘)))) → 𝑛 ∈ ℕ)
86 simprl 767 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘)))) → 𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
87 sneq 4637 . . . . . . . . . . 11 (𝑗 = 𝑎 → {𝑗} = {𝑎})
88 fveq2 6890 . . . . . . . . . . 11 (𝑗 = 𝑎 → (𝑆𝑗) = (𝑆𝑎))
8987, 88xpeq12d 5706 . . . . . . . . . 10 (𝑗 = 𝑎 → ({𝑗} × (𝑆𝑗)) = ({𝑎} × (𝑆𝑎)))
9089cbviunv 5042 . . . . . . . . 9 𝑗𝐽 ({𝑗} × (𝑆𝑗)) = 𝑎𝐽 ({𝑎} × (𝑆𝑎))
91 feq3 6699 . . . . . . . . 9 ( 𝑗𝐽 ({𝑗} × (𝑆𝑗)) = 𝑎𝐽 ({𝑎} × (𝑆𝑎)) → (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ↔ 𝑔:(1...𝑛)⟶ 𝑎𝐽 ({𝑎} × (𝑆𝑎))))
9290, 91ax-mp 5 . . . . . . . 8 (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ↔ 𝑔:(1...𝑛)⟶ 𝑎𝐽 ({𝑎} × (𝑆𝑎)))
9386, 92sylib 217 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘)))) → 𝑔:(1...𝑛)⟶ 𝑎𝐽 ({𝑎} × (𝑆𝑎)))
94 simprr 769 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘)))) → ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘)))
95 eqid 2730 . . . . . . 7 (topGen‘ran (,)) = (topGen‘ran (,))
96 2fveq3 6895 . . . . . . . . . . 11 (𝑡 = 𝑧 → ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑡)) = ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑧)))
9796cbvmptv 5260 . . . . . . . . . 10 (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑡))) = (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑧)))
98 eleq2 2820 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑏 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐 ↔ (𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
9998cbvriotavw 7377 . . . . . . . . . . . . . . 15 (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)
100 fveq1 6889 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → (𝑦‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑤 − 1) / 𝑛)))
101100eleq1d 2816 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
102101riotabidv 7369 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏) = (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
10399, 102eqtrid 2782 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
104103reseq2d 5980 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)))
105104cnveqd 5874 . . . . . . . . . . . 12 (𝑦 = 𝑥(𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)))
106105fveq1d 6892 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑧)) = ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧)))
107106mpteq2dv 5249 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑧))) = (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧))))
10897, 107eqtrid 2782 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑡))) = (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧))))
109 oveq1 7418 . . . . . . . . . . . 12 (𝑤 = 𝑚 → (𝑤 − 1) = (𝑚 − 1))
110109oveq1d 7426 . . . . . . . . . . 11 (𝑤 = 𝑚 → ((𝑤 − 1) / 𝑛) = ((𝑚 − 1) / 𝑛))
111 oveq1 7418 . . . . . . . . . . 11 (𝑤 = 𝑚 → (𝑤 / 𝑛) = (𝑚 / 𝑛))
112110, 111oveq12d 7429 . . . . . . . . . 10 (𝑤 = 𝑚 → (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) = (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)))
113 2fveq3 6895 . . . . . . . . . . . . . 14 (𝑤 = 𝑚 → (2nd ‘(𝑔𝑤)) = (2nd ‘(𝑔𝑚)))
114110fveq2d 6894 . . . . . . . . . . . . . . 15 (𝑤 = 𝑚 → (𝑥‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑚 − 1) / 𝑛)))
115114eleq1d 2816 . . . . . . . . . . . . . 14 (𝑤 = 𝑚 → ((𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))
116113, 115riotaeqbidv 7370 . . . . . . . . . . . . 13 (𝑤 = 𝑚 → (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏) = (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))
117116reseq2d 5980 . . . . . . . . . . . 12 (𝑤 = 𝑚 → (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏)))
118117cnveqd 5874 . . . . . . . . . . 11 (𝑤 = 𝑚(𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏)))
119118fveq1d 6892 . . . . . . . . . 10 (𝑤 = 𝑚 → ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧)) = ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧)))
120112, 119mpteq12dv 5238 . . . . . . . . 9 (𝑤 = 𝑚 → (𝑧 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧))) = (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧))))
121108, 120cbvmpov 7506 . . . . . . . 8 (𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑡)))) = (𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧))))
122 seqeq2 13974 . . . . . . . 8 ((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑡)))) = (𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧)))) → seq0((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑡)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})))
123121, 122ax-mp 5 . . . . . . 7 seq0((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑡)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
124 eqid 2730 . . . . . . 7 𝑘 ∈ (1...𝑛)(seq0((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑡)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))‘𝑘) = 𝑘 ∈ (1...𝑛)(seq0((𝑦 ∈ V, 𝑤 ∈ ℕ ↦ (𝑡 ∈ (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) ↦ ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑡)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))‘𝑘)
12537, 78, 10, 79, 80, 82, 84, 85, 93, 94, 95, 123, 124cvmliftlem14 34586 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘)))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
126125ex 411 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ((𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)))
127126exlimdv 1934 . . . 4 ((𝜑𝑛 ∈ ℕ) → (∃𝑔(𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)))
12877, 127syl5 34 . . 3 ((𝜑𝑛 ∈ ℕ) → (∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)))
129128rexlimdva 3153 . 2 (𝜑 → (∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)))
13054, 129mpd 15 1 (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wex 1779  wcel 2104  wne 2938  wral 3059  wrex 3068  ∃!wreu 3372  {crab 3430  Vcvv 3472  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4321  𝒫 cpw 4601  {csn 4627  cop 4633   cuni 4907   ciun 4996  cmpt 5230   I cid 5572   × cxp 5673  ccnv 5674  ran crn 5676  cres 5677  cima 5678  ccom 5679  Fun wfun 6536   Fn wfn 6537  wf 6538  cfv 6542  crio 7366  (class class class)co 7411  cmpo 7413  1st c1st 7975  2nd c2nd 7976  Fincfn 8941  0cc0 11112  1c1 11113  cmin 11448   / cdiv 11875  cn 12216  (,)cioo 13328  [,]cicc 13331  ...cfz 13488  seqcseq 13970  t crest 17370  topGenctg 17387   Cn ccn 22948  Homeochmeo 23477  IIcii 24615   CovMap ccvm 34544
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-ec 8707  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-ioo 13332  df-ico 13334  df-icc 13335  df-fz 13489  df-fzo 13632  df-fl 13761  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-sum 15637  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-starv 17216  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-hom 17225  df-cco 17226  df-rest 17372  df-topn 17373  df-0g 17391  df-gsum 17392  df-topgen 17393  df-pt 17394  df-prds 17397  df-xrs 17452  df-qtop 17457  df-imas 17458  df-xps 17460  df-mre 17534  df-mrc 17535  df-acs 17537  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18706  df-mulg 18987  df-cntz 19222  df-cmn 19691  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-cnfld 21145  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-cn 22951  df-cnp 22952  df-cmp 23111  df-conn 23136  df-lly 23190  df-nlly 23191  df-tx 23286  df-hmeo 23479  df-xms 24046  df-ms 24047  df-tms 24048  df-ii 24617  df-htpy 24716  df-phtpy 24717  df-phtpc 24738  df-pconn 34510  df-sconn 34511  df-cvm 34545
This theorem is referenced by:  cvmlift  34588
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