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Theorem cvmliftlem15 34289
Description: Lemma for cvmlift 34290. Discharge the assumptions of cvmliftlem14 34288. 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 24482, 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 9287 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 34288. (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 4078 . . 3 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ⊆ II
2 cvmliftlem.g . . . . . . . . . . 11 (𝜑𝐺 ∈ (II Cn 𝐽))
32ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝐺 ∈ (II Cn 𝐽))
4 simprl 770 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝑗𝐽)
5 cnima 22769 . . . . . . . . . 10 ((𝐺 ∈ (II Cn 𝐽) ∧ 𝑗𝐽) → (𝐺𝑗) ∈ II)
63, 4, 5syl2anc 585 . . . . . . . . 9 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝐺𝑗) ∈ II)
7 simplr 768 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝑥 ∈ (0[,]1))
8 simprrl 780 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝐺𝑥) ∈ 𝑗)
9 iiuni 24397 . . . . . . . . . . . . . 14 (0[,]1) = II
10 cvmliftlem.x . . . . . . . . . . . . . 14 𝑋 = 𝐽
119, 10cnf 22750 . . . . . . . . . . . . 13 (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
122, 11syl 17 . . . . . . . . . . . 12 (𝜑𝐺:(0[,]1)⟶𝑋)
1312ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝐺:(0[,]1)⟶𝑋)
14 ffn 6718 . . . . . . . . . . 11 (𝐺:(0[,]1)⟶𝑋𝐺 Fn (0[,]1))
15 elpreima 7060 . . . . . . . . . . 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 712 . . . . . . . . 9 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝑥 ∈ (𝐺𝑗))
18 simprrr 781 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝑆𝑗) ≠ ∅)
19 ffun 6721 . . . . . . . . . . . . 13 (𝐺:(0[,]1)⟶𝑋 → Fun 𝐺)
20 funimacnv 6630 . . . . . . . . . . . . 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 4229 . . . . . . . . . . . 12 (𝑗 ∩ ran 𝐺) ⊆ 𝑗
2321, 22eqsstrdi 4037 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
2423ralrimivw 3151 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → ∀𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
25 r19.2z 4495 . . . . . . . . . 10 (((𝑆𝑗) ≠ ∅ ∧ ∀𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗) → ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
2618, 24, 25syl2anc 585 . . . . . . . . 9 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
27 eleq2 2823 . . . . . . . . . . 11 (𝑢 = (𝐺𝑗) → (𝑥𝑢𝑥 ∈ (𝐺𝑗)))
28 imaeq2 6056 . . . . . . . . . . . . 13 (𝑢 = (𝐺𝑗) → (𝐺𝑢) = (𝐺 “ (𝐺𝑗)))
2928sseq1d 4014 . . . . . . . . . . . 12 (𝑢 = (𝐺𝑗) → ((𝐺𝑢) ⊆ 𝑗 ↔ (𝐺 “ (𝐺𝑗)) ⊆ 𝑗))
3029rexbidv 3179 . . . . . . . . . . 11 (𝑢 = (𝐺𝑗) → (∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗 ↔ ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗))
3127, 30anbi12d 632 . . . . . . . . . 10 (𝑢 = (𝐺𝑗) → ((𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ (𝑥 ∈ (𝐺𝑗) ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)))
3231rspcev 3613 . . . . . . . . 9 (((𝐺𝑗) ∈ II ∧ (𝑥 ∈ (𝐺𝑗) ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)) → ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
336, 17, 26, 32syl12anc 836 . . . . . . . 8 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
34 cvmliftlem.f . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
3534adantr 482 . . . . . . . . 9 ((𝜑𝑥 ∈ (0[,]1)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
3612ffvelcdmda 7087 . . . . . . . . 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 34254 . . . . . . . . 9 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺𝑥) ∈ 𝑋) → ∃𝑗𝐽 ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))
3935, 36, 38syl2anc 585 . . . . . . . 8 ((𝜑𝑥 ∈ (0[,]1)) → ∃𝑗𝐽 ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))
4033, 39reximddv 3172 . . . . . . 7 ((𝜑𝑥 ∈ (0[,]1)) → ∃𝑗𝐽𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
41 r19.42v 3191 . . . . . . . . 9 (∃𝑗𝐽 (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ (𝑥𝑢 ∧ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
4241rexbii 3095 . . . . . . . 8 (∃𝑢 ∈ II ∃𝑗𝐽 (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
43 rexcom 3288 . . . . . . . 8 (∃𝑗𝐽𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ ∃𝑢 ∈ II ∃𝑗𝐽 (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
44 elunirab 4925 . . . . . . . 8 (𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ↔ ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
4542, 43, 443bitr4i 303 . . . . . . 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 414 . . . . 5 (𝜑 → (𝑥 ∈ (0[,]1) → 𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗}))
4847ssrdv 3989 . . . 4 (𝜑 → (0[,]1) ⊆ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
49 uniss 4917 . . . . . 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 4034 . . . 4 (𝜑 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ⊆ (0[,]1))
5248, 51eqssd 4000 . . 3 (𝜑 → (0[,]1) = {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
53 lebnumii 24482 . . 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 588 . 2 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣)
55 fzfi 13937 . . . . 5 (1...𝑛) ∈ Fin
56 imaeq2 6056 . . . . . . . . . 10 (𝑢 = 𝑣 → (𝐺𝑢) = (𝐺𝑣))
5756sseq1d 4014 . . . . . . . . 9 (𝑢 = 𝑣 → ((𝐺𝑢) ⊆ 𝑗 ↔ (𝐺𝑣) ⊆ 𝑗))
58572rexbidv 3220 . . . . . . . 8 (𝑢 = 𝑣 → (∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗 ↔ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗))
5958rexrab 3693 . . . . . . 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 3479 . . . . . . . . . . . . 13 𝑗 ∈ V
61 vex 3479 . . . . . . . . . . . . 13 𝑠 ∈ V
6260, 61op1std 7985 . . . . . . . . . . . 12 (𝑢 = ⟨𝑗, 𝑠⟩ → (1st𝑢) = 𝑗)
6362sseq2d 4015 . . . . . . . . . . 11 (𝑢 = ⟨𝑗, 𝑠⟩ → ((𝐺𝑣) ⊆ (1st𝑢) ↔ (𝐺𝑣) ⊆ 𝑗))
6463rexiunxp 5841 . . . . . . . . . 10 (∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺𝑣) ⊆ (1st𝑢) ↔ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗)
65 imass2 6102 . . . . . . . . . . . 12 ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (𝐺𝑣))
66 sstr2 3990 . . . . . . . . . . . 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 3171 . . . . . . . . . 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 409 . . . . . . . 8 ((∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗 ∧ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣) → ∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢))
7170rexlimivw 3152 . . . . . . 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 3084 . . . . 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 6892 . . . . . . 7 (𝑢 = (𝑔𝑘) → (1st𝑢) = (1st ‘(𝑔𝑘)))
7574sseq2d 4015 . . . . . 6 (𝑢 = (𝑔𝑘) → ((𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢) ↔ (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘))))
7675ac6sfi 9287 . . . . 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 588 . . . 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 725 . . . . . . 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 725 . . . . . . 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 725 . . . . . . 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 725 . . . . . . 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 768 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘)))) → 𝑛 ∈ ℕ)
86 simprl 770 . . . . . . . 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 4639 . . . . . . . . . . 11 (𝑗 = 𝑎 → {𝑗} = {𝑎})
88 fveq2 6892 . . . . . . . . . . 11 (𝑗 = 𝑎 → (𝑆𝑗) = (𝑆𝑎))
8987, 88xpeq12d 5708 . . . . . . . . . 10 (𝑗 = 𝑎 → ({𝑗} × (𝑆𝑗)) = ({𝑎} × (𝑆𝑎)))
9089cbviunv 5044 . . . . . . . . 9 𝑗𝐽 ({𝑗} × (𝑆𝑗)) = 𝑎𝐽 ({𝑎} × (𝑆𝑎))
91 feq3 6701 . . . . . . . . 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 772 . . . . . . 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 2733 . . . . . . 7 (topGen‘ran (,)) = (topGen‘ran (,))
96 2fveq3 6897 . . . . . . . . . . 11 (𝑡 = 𝑧 → ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑡)) = ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑧)))
9796cbvmptv 5262 . . . . . . . . . 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 2823 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑏 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐 ↔ (𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
9998cbvriotavw 7375 . . . . . . . . . . . . . . 15 (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)
100 fveq1 6891 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → (𝑦‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑤 − 1) / 𝑛)))
101100eleq1d 2819 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
102101riotabidv 7367 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏) = (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
10399, 102eqtrid 2785 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
104103reseq2d 5982 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)))
105104cnveqd 5876 . . . . . . . . . . . 12 (𝑦 = 𝑥(𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)))
106105fveq1d 6894 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑧)) = ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧)))
107106mpteq2dv 5251 . . . . . . . . . 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 2785 . . . . . . . . 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 7416 . . . . . . . . . . . 12 (𝑤 = 𝑚 → (𝑤 − 1) = (𝑚 − 1))
110109oveq1d 7424 . . . . . . . . . . 11 (𝑤 = 𝑚 → ((𝑤 − 1) / 𝑛) = ((𝑚 − 1) / 𝑛))
111 oveq1 7416 . . . . . . . . . . 11 (𝑤 = 𝑚 → (𝑤 / 𝑛) = (𝑚 / 𝑛))
112110, 111oveq12d 7427 . . . . . . . . . 10 (𝑤 = 𝑚 → (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) = (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)))
113 2fveq3 6897 . . . . . . . . . . . . . 14 (𝑤 = 𝑚 → (2nd ‘(𝑔𝑤)) = (2nd ‘(𝑔𝑚)))
114110fveq2d 6896 . . . . . . . . . . . . . . 15 (𝑤 = 𝑚 → (𝑥‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑚 − 1) / 𝑛)))
115114eleq1d 2819 . . . . . . . . . . . . . 14 (𝑤 = 𝑚 → ((𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))
116113, 115riotaeqbidv 7368 . . . . . . . . . . . . 13 (𝑤 = 𝑚 → (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏) = (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))
117116reseq2d 5982 . . . . . . . . . . . 12 (𝑤 = 𝑚 → (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏)))
118117cnveqd 5876 . . . . . . . . . . 11 (𝑤 = 𝑚(𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏)))
119118fveq1d 6894 . . . . . . . . . 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 5240 . . . . . . . . 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 7504 . . . . . . . 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 13970 . . . . . . . 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 2733 . . . . . . 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 34288 . . . . . 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 414 . . . . 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 1937 . . . 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 3156 . 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 397   = wceq 1542  wex 1782  wcel 2107  wne 2941  wral 3062  wrex 3071  ∃!wreu 3375  {crab 3433  Vcvv 3475  cdif 3946  cun 3947  cin 3948  wss 3949  c0 4323  𝒫 cpw 4603  {csn 4629  cop 4635   cuni 4909   ciun 4998  cmpt 5232   I cid 5574   × cxp 5675  ccnv 5676  ran crn 5678  cres 5679  cima 5680  ccom 5681  Fun wfun 6538   Fn wfn 6539  wf 6540  cfv 6544  crio 7364  (class class class)co 7409  cmpo 7411  1st c1st 7973  2nd c2nd 7974  Fincfn 8939  0cc0 11110  1c1 11111  cmin 11444   / cdiv 11871  cn 12212  (,)cioo 13324  [,]cicc 13327  ...cfz 13484  seqcseq 13966  t crest 17366  topGenctg 17383   Cn ccn 22728  Homeochmeo 23257  IIcii 24391   CovMap ccvm 34246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-addf 11189  ax-mulf 11190
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-ec 8705  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ioo 13328  df-ico 13330  df-icc 13331  df-fz 13485  df-fzo 13628  df-fl 13757  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-sum 15633  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-starv 17212  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-hom 17221  df-cco 17222  df-rest 17368  df-topn 17369  df-0g 17387  df-gsum 17388  df-topgen 17389  df-pt 17390  df-prds 17393  df-xrs 17448  df-qtop 17453  df-imas 17454  df-xps 17456  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-mulg 18951  df-cntz 19181  df-cmn 19650  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-cnfld 20945  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-cn 22731  df-cnp 22732  df-cmp 22891  df-conn 22916  df-lly 22970  df-nlly 22971  df-tx 23066  df-hmeo 23259  df-xms 23826  df-ms 23827  df-tms 23828  df-ii 24393  df-htpy 24486  df-phtpy 24487  df-phtpc 24508  df-pconn 34212  df-sconn 34213  df-cvm 34247
This theorem is referenced by:  cvmlift  34290
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