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Theorem cvmliftlem15 31015
Description: Lemma for cvmlift 31016. Discharge the assumptions of cvmliftlem14 31014. 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 22684, there is a subdivision of the 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 8155 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 31014. (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 3671 . . 3 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ⊆ II
2 cvmliftlem.g . . . . . . . . . . 11 (𝜑𝐺 ∈ (II Cn 𝐽))
32ad2antrr 761 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝐺 ∈ (II Cn 𝐽))
4 simprl 793 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝑗𝐽)
5 cnima 20988 . . . . . . . . . 10 ((𝐺 ∈ (II Cn 𝐽) ∧ 𝑗𝐽) → (𝐺𝑗) ∈ II)
63, 4, 5syl2anc 692 . . . . . . . . 9 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝐺𝑗) ∈ II)
7 simplr 791 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝑥 ∈ (0[,]1))
8 simprrl 803 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝐺𝑥) ∈ 𝑗)
9 iiuni 22603 . . . . . . . . . . . . . 14 (0[,]1) = II
10 cvmliftlem.x . . . . . . . . . . . . . 14 𝑋 = 𝐽
119, 10cnf 20969 . . . . . . . . . . . . 13 (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
122, 11syl 17 . . . . . . . . . . . 12 (𝜑𝐺:(0[,]1)⟶𝑋)
1312ad2antrr 761 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝐺:(0[,]1)⟶𝑋)
14 ffn 6007 . . . . . . . . . . 11 (𝐺:(0[,]1)⟶𝑋𝐺 Fn (0[,]1))
15 elpreima 6298 . . . . . . . . . . 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 956 . . . . . . . . 9 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → 𝑥 ∈ (𝐺𝑗))
18 simprrr 804 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝑆𝑗) ≠ ∅)
19 ffun 6010 . . . . . . . . . . . . 13 (𝐺:(0[,]1)⟶𝑋 → Fun 𝐺)
20 funimacnv 5933 . . . . . . . . . . . . 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 3816 . . . . . . . . . . . 12 (𝑗 ∩ ran 𝐺) ⊆ 𝑗
2321, 22syl6eqss 3639 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → (𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
2423ralrimivw 2962 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → ∀𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
25 r19.2z 4037 . . . . . . . . . 10 (((𝑆𝑗) ≠ ∅ ∧ ∀𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗) → ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
2618, 24, 25syl2anc 692 . . . . . . . . 9 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)
27 eleq2 2687 . . . . . . . . . . 11 (𝑢 = (𝐺𝑗) → (𝑥𝑢𝑥 ∈ (𝐺𝑗)))
28 imaeq2 5426 . . . . . . . . . . . . 13 (𝑢 = (𝐺𝑗) → (𝐺𝑢) = (𝐺 “ (𝐺𝑗)))
2928sseq1d 3616 . . . . . . . . . . . 12 (𝑢 = (𝐺𝑗) → ((𝐺𝑢) ⊆ 𝑗 ↔ (𝐺 “ (𝐺𝑗)) ⊆ 𝑗))
3029rexbidv 3046 . . . . . . . . . . 11 (𝑢 = (𝐺𝑗) → (∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗 ↔ ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗))
3127, 30anbi12d 746 . . . . . . . . . 10 (𝑢 = (𝐺𝑗) → ((𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ (𝑥 ∈ (𝐺𝑗) ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)))
3231rspcev 3298 . . . . . . . . 9 (((𝐺𝑗) ∈ II ∧ (𝑥 ∈ (𝐺𝑗) ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺 “ (𝐺𝑗)) ⊆ 𝑗)) → ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
336, 17, 26, 32syl12anc 1321 . . . . . . . 8 (((𝜑𝑥 ∈ (0[,]1)) ∧ (𝑗𝐽 ∧ ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))) → ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
34 cvmliftlem.f . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
3534adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (0[,]1)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
3612ffvelrnda 6320 . . . . . . . . 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 30980 . . . . . . . . 9 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺𝑥) ∈ 𝑋) → ∃𝑗𝐽 ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))
3935, 36, 38syl2anc 692 . . . . . . . 8 ((𝜑𝑥 ∈ (0[,]1)) → ∃𝑗𝐽 ((𝐺𝑥) ∈ 𝑗 ∧ (𝑆𝑗) ≠ ∅))
4033, 39reximddv 3013 . . . . . . 7 ((𝜑𝑥 ∈ (0[,]1)) → ∃𝑗𝐽𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
41 r19.42v 3085 . . . . . . . . 9 (∃𝑗𝐽 (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ (𝑥𝑢 ∧ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
4241rexbii 3035 . . . . . . . 8 (∃𝑢 ∈ II ∃𝑗𝐽 (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
43 rexcom 3092 . . . . . . . 8 (∃𝑗𝐽𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ ∃𝑢 ∈ II ∃𝑗𝐽 (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
44 elunirab 4419 . . . . . . . 8 (𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ↔ ∃𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗))
4542, 43, 443bitr4i 292 . . . . . . 7 (∃𝑗𝐽𝑢 ∈ II (𝑥𝑢 ∧ ∃𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗) ↔ 𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
4640, 45sylib 208 . . . . . 6 ((𝜑𝑥 ∈ (0[,]1)) → 𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
4746ex 450 . . . . 5 (𝜑 → (𝑥 ∈ (0[,]1) → 𝑥 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗}))
4847ssrdv 3593 . . . 4 (𝜑 → (0[,]1) ⊆ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
49 uniss 4429 . . . . . 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 3636 . . . 4 (𝜑 {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} ⊆ (0[,]1))
5248, 51eqssd 3604 . . 3 (𝜑 → (0[,]1) = {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗})
53 lebnumii 22684 . . 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 694 . 2 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑣 ∈ {𝑢 ∈ II ∣ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗} (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣)
55 fzfi 12718 . . . . 5 (1...𝑛) ∈ Fin
56 imaeq2 5426 . . . . . . . . . 10 (𝑢 = 𝑣 → (𝐺𝑢) = (𝐺𝑣))
5756sseq1d 3616 . . . . . . . . 9 (𝑢 = 𝑣 → ((𝐺𝑢) ⊆ 𝑗 ↔ (𝐺𝑣) ⊆ 𝑗))
58572rexbidv 3051 . . . . . . . 8 (𝑢 = 𝑣 → (∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑢) ⊆ 𝑗 ↔ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗))
5958rexrab 3356 . . . . . . 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 3192 . . . . . . . . . . . . 13 𝑗 ∈ V
61 vex 3192 . . . . . . . . . . . . 13 𝑠 ∈ V
6260, 61op1std 7130 . . . . . . . . . . . 12 (𝑢 = ⟨𝑗, 𝑠⟩ → (1st𝑢) = 𝑗)
6362sseq2d 3617 . . . . . . . . . . 11 (𝑢 = ⟨𝑗, 𝑠⟩ → ((𝐺𝑣) ⊆ (1st𝑢) ↔ (𝐺𝑣) ⊆ 𝑗))
6463rexiunxp 5227 . . . . . . . . . 10 (∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺𝑣) ⊆ (1st𝑢) ↔ ∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗)
65 imass2 5465 . . . . . . . . . . . 12 ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (𝐺𝑣))
66 sstr2 3594 . . . . . . . . . . . 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 3011 . . . . . . . . . 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 233 . . . . . . . . 9 ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣 → (∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗 → ∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢)))
7069impcom 446 . . . . . . . 8 ((∃𝑗𝐽𝑠 ∈ (𝑆𝑗)(𝐺𝑣) ⊆ 𝑗 ∧ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑣) → ∃𝑢 𝑗𝐽 ({𝑗} × (𝑆𝑗))(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢))
7170rexlimivw 3023 . . . . . . 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 207 . . . . . 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 2947 . . . . 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 6153 . . . . . . 7 (𝑢 = (𝑔𝑘) → (1st𝑢) = (1st ‘(𝑔𝑘)))
7574sseq2d 3617 . . . . . 6 (𝑢 = (𝑔𝑘) → ((𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st𝑢) ↔ (𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘))))
7675ac6sfi 8155 . . . . 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 694 . . . 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 761 . . . . . . 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 761 . . . . . . 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 761 . . . . . . 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 761 . . . . . . 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 791 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ (𝑔:(1...𝑛)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ ∀𝑘 ∈ (1...𝑛)(𝐺 “ (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛))) ⊆ (1st ‘(𝑔𝑘)))) → 𝑛 ∈ ℕ)
86 simprl 793 . . . . . . . 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 4163 . . . . . . . . . . 11 (𝑗 = 𝑎 → {𝑗} = {𝑎})
88 fveq2 6153 . . . . . . . . . . 11 (𝑗 = 𝑎 → (𝑆𝑗) = (𝑆𝑎))
8987, 88xpeq12d 5105 . . . . . . . . . 10 (𝑗 = 𝑎 → ({𝑗} × (𝑆𝑗)) = ({𝑎} × (𝑆𝑎)))
9089cbviunv 4530 . . . . . . . . 9 𝑗𝐽 ({𝑗} × (𝑆𝑗)) = 𝑎𝐽 ({𝑎} × (𝑆𝑎))
91 feq3 5990 . . . . . . . . 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 208 . . . . . . 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 795 . . . . . . 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 2621 . . . . . . 7 (topGen‘ran (,)) = (topGen‘ran (,))
96 fveq2 6153 . . . . . . . . . . . 12 (𝑡 = 𝑧 → (𝐺𝑡) = (𝐺𝑧))
9796fveq2d 6157 . . . . . . . . . . 11 (𝑡 = 𝑧 → ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑡)) = ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑧)))
9897cbvmptv 4715 . . . . . . . . . 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)))
99 eleq2 2687 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑏 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐 ↔ (𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
10099cbvriotav 6582 . . . . . . . . . . . . . . 15 (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)
101 fveq1 6152 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → (𝑦‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑤 − 1) / 𝑛)))
102101eleq1d 2683 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → ((𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
103102riotabidv 6573 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑏) = (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
104100, 103syl5eq 2667 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐) = (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))
105104reseq2d 5361 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)))
106105cnveqd 5263 . . . . . . . . . . . 12 (𝑦 = 𝑥(𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)))
107106fveq1d 6155 . . . . . . . . . . 11 (𝑦 = 𝑥 → ((𝐹 ↾ (𝑐 ∈ (2nd ‘(𝑔𝑤))(𝑦‘((𝑤 − 1) / 𝑛)) ∈ 𝑐))‘(𝐺𝑧)) = ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧)))
108107mpteq2dv 4710 . . . . . . . . . 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))))
10998, 108syl5eq 2667 . . . . . . . . 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))))
110 oveq1 6617 . . . . . . . . . . . 12 (𝑤 = 𝑚 → (𝑤 − 1) = (𝑚 − 1))
111110oveq1d 6625 . . . . . . . . . . 11 (𝑤 = 𝑚 → ((𝑤 − 1) / 𝑛) = ((𝑚 − 1) / 𝑛))
112 oveq1 6617 . . . . . . . . . . 11 (𝑤 = 𝑚 → (𝑤 / 𝑛) = (𝑚 / 𝑛))
113111, 112oveq12d 6628 . . . . . . . . . 10 (𝑤 = 𝑚 → (((𝑤 − 1) / 𝑛)[,](𝑤 / 𝑛)) = (((𝑚 − 1) / 𝑛)[,](𝑚 / 𝑛)))
114 fveq2 6153 . . . . . . . . . . . . . . 15 (𝑤 = 𝑚 → (𝑔𝑤) = (𝑔𝑚))
115114fveq2d 6157 . . . . . . . . . . . . . 14 (𝑤 = 𝑚 → (2nd ‘(𝑔𝑤)) = (2nd ‘(𝑔𝑚)))
116111fveq2d 6157 . . . . . . . . . . . . . . 15 (𝑤 = 𝑚 → (𝑥‘((𝑤 − 1) / 𝑛)) = (𝑥‘((𝑚 − 1) / 𝑛)))
117116eleq1d 2683 . . . . . . . . . . . . . 14 (𝑤 = 𝑚 → ((𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏 ↔ (𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))
118115, 117riotaeqbidv 6574 . . . . . . . . . . . . 13 (𝑤 = 𝑚 → (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏) = (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))
119118reseq2d 5361 . . . . . . . . . . . 12 (𝑤 = 𝑚 → (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏)))
120119cnveqd 5263 . . . . . . . . . . 11 (𝑤 = 𝑚(𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏)) = (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏)))
121120fveq1d 6155 . . . . . . . . . 10 (𝑤 = 𝑚 → ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑤))(𝑥‘((𝑤 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧)) = ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑔𝑚))(𝑥‘((𝑚 − 1) / 𝑛)) ∈ 𝑏))‘(𝐺𝑧)))
122113, 121mpteq12dv 4698 . . . . . . . . 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))))
123109, 122cbvmpt2v 6695 . . . . . . . 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))))
124 seqeq2 12752 . . . . . . . 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⟩}⟩})))
125123, 124ax-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⟩}⟩}))
126 eqid 2621 . . . . . . 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)
12737, 78, 10, 79, 80, 82, 84, 85, 93, 94, 95, 125, 126cvmliftlem14 31014 . . . . . 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))
128127ex 450 . . . . 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)))
129128exlimdv 1858 . . . 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)))
13077, 129syl5 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)))
131130rexlimdva 3025 . 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)))
13254, 131mpd 15 1 (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  ∃!wreu 2909  {crab 2911  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153  cop 4159   cuni 4407   ciun 4490  cmpt 4678   I cid 4989   × cxp 5077  ccnv 5078  ran crn 5080  cres 5081  cima 5082  ccom 5083  Fun wfun 5846   Fn wfn 5847  wf 5848  cfv 5852  crio 6570  (class class class)co 6610  cmpt2 6612  1st c1st 7118  2nd c2nd 7119  Fincfn 7906  0cc0 9887  1c1 9888  cmin 10217   / cdiv 10635  cn 10971  (,)cioo 12124  [,]cicc 12127  ...cfz 12275  seqcseq 12748  t crest 16009  topGenctg 16026   Cn ccn 20947  Homeochmeo 21475  IIcii 22597   CovMap ccvm 30972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965  ax-addf 9966  ax-mulf 9967
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-ec 7696  df-map 7811  df-ixp 7860  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-fsupp 8227  df-fi 8268  df-sup 8299  df-inf 8300  df-oi 8366  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-4 11032  df-5 11033  df-6 11034  df-7 11035  df-8 11036  df-9 11037  df-n0 11244  df-z 11329  df-dec 11445  df-uz 11639  df-q 11740  df-rp 11784  df-xneg 11897  df-xadd 11898  df-xmul 11899  df-ioo 12128  df-ico 12130  df-icc 12131  df-fz 12276  df-fzo 12414  df-fl 12540  df-seq 12749  df-exp 12808  df-hash 13065  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-clim 14160  df-sum 14358  df-struct 15790  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-mulr 15883  df-starv 15884  df-sca 15885  df-vsca 15886  df-ip 15887  df-tset 15888  df-ple 15889  df-ds 15892  df-unif 15893  df-hom 15894  df-cco 15895  df-rest 16011  df-topn 16012  df-0g 16030  df-gsum 16031  df-topgen 16032  df-pt 16033  df-prds 16036  df-xrs 16090  df-qtop 16095  df-imas 16096  df-xps 16098  df-mre 16174  df-mrc 16175  df-acs 16177  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-submnd 17264  df-mulg 17469  df-cntz 17678  df-cmn 18123  df-psmet 19666  df-xmet 19667  df-met 19668  df-bl 19669  df-mopn 19670  df-cnfld 19675  df-top 20627  df-topon 20644  df-topsp 20657  df-bases 20670  df-cld 20742  df-ntr 20743  df-cls 20744  df-nei 20821  df-cn 20950  df-cnp 20951  df-cmp 21109  df-conn 21134  df-lly 21188  df-nlly 21189  df-tx 21284  df-hmeo 21477  df-xms 22044  df-ms 22045  df-tms 22046  df-ii 22599  df-htpy 22688  df-phtpy 22689  df-phtpc 22710  df-pconn 30938  df-sconn 30939  df-cvm 30973
This theorem is referenced by:  cvmlift  31016
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