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Theorem cvmliftlem15 31728
Description: Lemma for cvmlift 31729. Discharge the assumptions of cvmliftlem14 31727. 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 23044, 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 8411 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 31727. (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 3847 . . 3 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ⊆ II
2 cvmliftlem.g . . . . . . . . . . 11 (𝜑𝐺 ∈ (II Cn 𝐽))
32ad2antrr 717 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝐺 ∈ (II Cn 𝐽))
4 simprl 787 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝑗𝐽)
5 cnima 21349 . . . . . . . . . 10 ((𝐺 ∈ (II Cn 𝐽) ∧ 𝑗𝐽) → (𝐺𝑗) ∈ II)
63, 4, 5syl2anc 579 . . . . . . . . 9 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝐺𝑗) ∈ II)
7 simplr 785 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝑥 ∈ (0[,]1))
8 simprrl 799 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝐺𝑥) ∈ 𝑗)
9 iiuni 22963 . . . . . . . . . . . . . 14 (0[,]1) = II
10 cvmliftlem.x . . . . . . . . . . . . . 14 𝑋 = 𝐽
119, 10cnf 21330 . . . . . . . . . . . . 13 (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
122, 11syl 17 . . . . . . . . . . . 12 (𝜑𝐺:(0[,]1)⟶𝑋)
1312ad2antrr 717 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝐺:(0[,]1)⟶𝑋)
14 ffn 6223 . . . . . . . . . . 11 (𝐺:(0[,]1)⟶𝑋𝐺 Fn (0[,]1))
15 elpreima 6527 . . . . . . . . . . 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 704 . . . . . . . . 9 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝑥 ∈ (𝐺𝑗))
18 simprrr 800 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝑆𝑗) ≠ ∅)
19 ffun 6226 . . . . . . . . . . . . 13 (𝐺:(0[,]1)⟶𝑋 → Fun 𝐺)
20 funimacnv 6148 . . . . . . . . . . . . 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 3992 . . . . . . . . . . . 12 (𝑗 ∩ ran 𝐺) ⊆ 𝑗
2321, 22syl6eqss 3815 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
2423ralrimivw 3114 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → ∀𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
25 r19.2z 4219 . . . . . . . . . 10 (((𝑆𝑗) ≠ ∅ ∧ ∀𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗) → ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
2618, 24, 25syl2anc 579 . . . . . . . . 9 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
27 eleq2 2833 . . . . . . . . . . 11 (𝑢 = (𝐺𝑗) → (𝑥𝑢𝑥 ∈ (𝐺𝑗)))
28 imaeq2 5644 . . . . . . . . . . . . 13 (𝑢 = (𝐺𝑗) → (𝐺𝑢) = (𝐺 “ (𝐺𝑗)))
2928sseq1d 3792 . . . . . . . . . . . 12 (𝑢 = (𝐺𝑗) → ((𝐺𝑢) ⊆ 𝑗 ↔ (𝐺 “ (𝐺𝑗)) ⊆ 𝑗))
3029rexbidv 3199 . . . . . . . . . . 11 (𝑢 = (𝐺𝑗) → (∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗 ↔ ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗))
3127, 30anbi12d 624 . . . . . . . . . 10 (𝑢 = (𝐺𝑗) → ((𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ (𝑥 ∈ (𝐺𝑗) ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)))
3231rspcev 3461 . . . . . . . . 9 (((𝐺𝑗) ∈ II ∧ (𝑥 ∈ (𝐺𝑗) ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)) → ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
336, 17, 26, 32syl12anc 865 . . . . . . . 8 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
34 cvmliftlem.f . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
3534adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (0[,]1)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
3612ffvelrnda 6549 . . . . . . . . 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 31693 . . . . . . . . 9 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺𝑥) ∈ 𝑋) → ∃𝑗𝐽 ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))
3935, 36, 38syl2anc 579 . . . . . . . 8 ((𝜑𝑥 ∈ (0[,]1)) → ∃𝑗𝐽 ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))
4033, 39reximddv 3164 . . . . . . 7 ((𝜑𝑥 ∈ (0[,]1)) → ∃𝑗𝐽𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
41 r19.42v 3239 . . . . . . . . 9 (∃𝑗𝐽 (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ (𝑥𝑢 ∧ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
4241rexbii 3188 . . . . . . . 8 (∃𝑢 ∈ II ∃𝑗𝐽 (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
43 rexcom 3246 . . . . . . . 8 (∃𝑗𝐽𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ ∃𝑢 ∈ II ∃𝑗𝐽 (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
44 elunirab 4606 . . . . . . . 8 (𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ↔ ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
4542, 43, 443bitr4i 294 . . . . . . 7 (∃𝑗𝐽𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ 𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
4640, 45sylib 209 . . . . . 6 ((𝜑𝑥 ∈ (0[,]1)) → 𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
4746ex 401 . . . . 5 (𝜑 → (𝑥 ∈ (0[,]1) → 𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗}))
4847ssrdv 3767 . . . 4 (𝜑 → (0[,]1) ⊆ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
49 uniss 4617 . . . . . 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, 9syl6sseqr 3812 . . . 4 (𝜑 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ⊆ (0[,]1))
5248, 51eqssd 3778 . . 3 (𝜑 → (0[,]1) = {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
53 lebnumii 23044 . . 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 581 . 2 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣)
55 fzfi 12979 . . . . 5 (1...𝑛) ∈ Fin
56 imaeq2 5644 . . . . . . . . . 10 (𝑢 = 𝑣 → (𝐺𝑢) = (𝐺𝑣))
5756sseq1d 3792 . . . . . . . . 9 (𝑢 = 𝑣 → ((𝐺𝑢) ⊆ 𝑗 ↔ (𝐺𝑣) ⊆ 𝑗))
58572rexbidv 3204 . . . . . . . 8 (𝑢 = 𝑣 → (∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗 ↔ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗))
5958rexrab 3527 . . . . . . 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 3353 . . . . . . . . . . . . 13 𝑗 ∈ V
61 vex 3353 . . . . . . . . . . . . 13 𝑠 ∈ V
6260, 61op1std 7376 . . . . . . . . . . . 12 (𝑢 = ⟨𝑗, 𝑠⟩ → (1st𝑢) = 𝑗)
6362sseq2d 3793 . . . . . . . . . . 11 (𝑢 = ⟨𝑗, 𝑠⟩ → ((𝐺𝑣) ⊆ (1st𝑢) ↔ (𝐺𝑣) ⊆ 𝑗))
6463rexiunxp 5431 . . . . . . . . . 10 (∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺𝑣) ⊆ (1st𝑢) ↔ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗)
65 imass2 5683 . . . . . . . . . . . 12 ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (𝐺𝑣))
66 sstr2 3768 . . . . . . . . . . . 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 3162 . . . . . . . . . 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, 68syl5bir 234 . . . . . . . . 9 ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗 → ∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢)))
7069impcom 396 . . . . . . . 8 ((∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗 ∧ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣) → ∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢))
7170rexlimivw 3176 . . . . . . 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 208 . . . . . 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 3099 . . . . 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 6375 . . . . . . 7 (𝑢 = (𝑔𝑘) → (1st𝑢) = (1st ‘(𝑔𝑘)))
7574sseq2d 3793 . . . . . 6 (𝑢 = (𝑔𝑘) → ((𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢) ↔ (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘))))
7675ac6sfi 8411 . . . . 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 581 . . . 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 717 . . . . . . 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 717 . . . . . . 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 717 . . . . . . 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 717 . . . . . . 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 785 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘)))) → 𝑛 ∈ ℕ)
86 simprl 787 . . . . . . . 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 4344 . . . . . . . . . . 11 (𝑗 = 𝑎 → {𝑗} = {𝑎})
88 fveq2 6375 . . . . . . . . . . 11 (𝑗 = 𝑎 → (𝑆𝑗) = (𝑆𝑎))
8987, 88xpeq12d 5308 . . . . . . . . . 10 (𝑗 = 𝑎 → ({𝑗} × (𝑆𝑗)) = ({𝑎} × (𝑆𝑎)))
9089cbviunv 4715 . . . . . . . . 9 𝑗𝐽 ({𝑗} × (𝑆𝑗)) = 𝑎𝐽 ({𝑎} × (𝑆𝑎))
91 feq3 6206 . . . . . . . . 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 209 . . . . . . 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 789 . . . . . . 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 2765 . . . . . . 7 (topGen‘ran (,)) = (topGen‘ran (,))
96 2fveq3 6380 . . . . . . . . . . 11 (𝑡 = 𝑧 → ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑡)) = ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑧)))
9796cbvmptv 4909 . . . . . . . . . 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 2833 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑏 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐 ↔ (𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
9998cbvriotav 6814 . . . . . . . . . . . . . . 15 (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)
100 fveq1 6374 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → (𝑦‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑤 − 1) / 𝑛)))
101100eleq1d 2829 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
102101riotabidv 6805 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏) = (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
10399, 102syl5eq 2811 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
104103reseq2d 5565 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)))
105104cnveqd 5466 . . . . . . . . . . . 12 (𝑦 = 𝑥(𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)))
106105fveq1d 6377 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑧)) = ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧)))
107106mpteq2dv 4904 . . . . . . . . . 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, 107syl5eq 2811 . . . . . . . . 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 6849 . . . . . . . . . . . 12 (𝑤 = 𝑚 → (𝑤 − 1) = (𝑚 − 1))
110109oveq1d 6857 . . . . . . . . . . 11 (𝑤 = 𝑚 → ((𝑤 − 1) / 𝑛) = ((𝑚 − 1) / 𝑛))
111 oveq1 6849 . . . . . . . . . . 11 (𝑤 = 𝑚 → (𝑤 / 𝑛) = (𝑚 / 𝑛))
112110, 111oveq12d 6860 . . . . . . . . . 10 (𝑤 = 𝑚 → (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) = (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)))
113 2fveq3 6380 . . . . . . . . . . . . . 14 (𝑤 = 𝑚 → (2nd ‘(𝑔𝑤)) = (2nd ‘(𝑔𝑚)))
114110fveq2d 6379 . . . . . . . . . . . . . . 15 (𝑤 = 𝑚 → (𝑥‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑚 − 1) / 𝑛)))
115114eleq1d 2829 . . . . . . . . . . . . . 14 (𝑤 = 𝑚 → ((𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))
116113, 115riotaeqbidv 6806 . . . . . . . . . . . . 13 (𝑤 = 𝑚 → (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏) = (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))
117116reseq2d 5565 . . . . . . . . . . . 12 (𝑤 = 𝑚 → (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏)))
118117cnveqd 5466 . . . . . . . . . . 11 (𝑤 = 𝑚(𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏)))
119118fveq1d 6377 . . . . . . . . . 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 4892 . . . . . . . . 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, 120cbvmpt2v 6933 . . . . . . . 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 13012 . . . . . . . 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 2765 . . . . . . 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 31727 . . . . . 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 401 . . . . 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 2028 . . . 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 3178 . 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 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  wrex 3056  ∃!wreu 3057  {crab 3059  Vcvv 3350  cdif 3729  cun 3730  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315  {csn 4334  cop 4340   cuni 4594   ciun 4676  cmpt 4888   I cid 5184   × cxp 5275  ccnv 5276  ran crn 5278  cres 5279  cima 5280  ccom 5281  Fun wfun 6062   Fn wfn 6063  wf 6064  cfv 6068  crio 6802  (class class class)co 6842  cmpt2 6844  1st c1st 7364  2nd c2nd 7365  Fincfn 8160  0cc0 10189  1c1 10190  cmin 10520   / cdiv 10938  cn 11274  (,)cioo 12377  [,]cicc 12380  ...cfz 12533  seqcseq 13008  t crest 16347  topGenctg 16364   Cn ccn 21308  Homeochmeo 21836  IIcii 22957   CovMap ccvm 31685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268  ax-mulf 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-ec 7949  df-map 8062  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-fi 8524  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12146  df-xadd 12147  df-xmul 12148  df-ioo 12381  df-ico 12383  df-icc 12384  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-sum 14702  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-starv 16229  df-sca 16230  df-vsca 16231  df-ip 16232  df-tset 16233  df-ple 16234  df-ds 16236  df-unif 16237  df-hom 16238  df-cco 16239  df-rest 16349  df-topn 16350  df-0g 16368  df-gsum 16369  df-topgen 16370  df-pt 16371  df-prds 16374  df-xrs 16428  df-qtop 16433  df-imas 16434  df-xps 16436  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-submnd 17602  df-mulg 17808  df-cntz 18013  df-cmn 18461  df-psmet 20011  df-xmet 20012  df-met 20013  df-bl 20014  df-mopn 20015  df-cnfld 20020  df-top 20978  df-topon 20995  df-topsp 21017  df-bases 21030  df-cld 21103  df-ntr 21104  df-cls 21105  df-nei 21182  df-cn 21311  df-cnp 21312  df-cmp 21470  df-conn 21495  df-lly 21549  df-nlly 21550  df-tx 21645  df-hmeo 21838  df-xms 22404  df-ms 22405  df-tms 22406  df-ii 22959  df-htpy 23048  df-phtpy 23049  df-phtpc 23070  df-pconn 31651  df-sconn 31652  df-cvm 31686
This theorem is referenced by:  cvmlift  31729
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