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Theorem cvmliftlem1 30975
 Description: Lemma for cvmlift 30989. In cvmliftlem15 30988, we picked an 𝑁 large enough so that the sections (𝐺 “ [(𝑘 − 1) / 𝑁, 𝑘 / 𝑁]) are all contained in an even covering, and the function 𝑇 enumerates these even coverings. So 1st ‘(𝑇‘𝑀) is a neighborhood of (𝐺 “ [(𝑀 − 1) / 𝑁, 𝑀 / 𝑁]), and 2nd ‘(𝑇‘𝑀) is an even covering of 1st ‘(𝑇‘𝑀), which is to say a disjoint union of open sets in 𝐶 whose image is 1st ‘(𝑇‘𝑀). (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))
cvmliftlem.n (𝜑𝑁 ∈ ℕ)
cvmliftlem.t (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
cvmliftlem.a (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))
cvmliftlem.l 𝐿 = (topGen‘ran (,))
cvmliftlem1.m ((𝜑𝜓) → 𝑀 ∈ (1...𝑁))
Assertion
Ref Expression
cvmliftlem1 ((𝜑𝜓) → (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))))
Distinct variable groups:   𝑣,𝐵   𝑗,𝑘,𝑠,𝑢,𝑣,𝐹   𝑗,𝑀,𝑘,𝑠,𝑢,𝑣   𝑃,𝑘,𝑢,𝑣   𝐶,𝑗,𝑘,𝑠,𝑢,𝑣   𝜑,𝑗,𝑠   𝑘,𝑁,𝑢,𝑣   𝑆,𝑗,𝑘,𝑠,𝑢,𝑣   𝑗,𝑋   𝑗,𝐺,𝑘,𝑠,𝑢,𝑣   𝑇,𝑗,𝑘,𝑠,𝑢,𝑣   𝑗,𝐽,𝑘,𝑠,𝑢,𝑣
Allowed substitution hints:   𝜑(𝑣,𝑢,𝑘)   𝜓(𝑣,𝑢,𝑗,𝑘,𝑠)   𝐵(𝑢,𝑗,𝑘,𝑠)   𝑃(𝑗,𝑠)   𝐿(𝑣,𝑢,𝑗,𝑘,𝑠)   𝑁(𝑗,𝑠)   𝑋(𝑣,𝑢,𝑘,𝑠)

Proof of Theorem cvmliftlem1
StepHypRef Expression
1 relxp 5188 . . . . . 6 Rel ({𝑗} × (𝑆𝑗))
21rgenw 2919 . . . . 5 𝑗𝐽 Rel ({𝑗} × (𝑆𝑗))
3 reliun 5200 . . . . 5 (Rel 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ↔ ∀𝑗𝐽 Rel ({𝑗} × (𝑆𝑗)))
42, 3mpbir 221 . . . 4 Rel 𝑗𝐽 ({𝑗} × (𝑆𝑗))
5 cvmliftlem.t . . . . . 6 (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
65adantr 481 . . . . 5 ((𝜑𝜓) → 𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
7 cvmliftlem1.m . . . . 5 ((𝜑𝜓) → 𝑀 ∈ (1...𝑁))
86, 7ffvelrnd 6316 . . . 4 ((𝜑𝜓) → (𝑇𝑀) ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
9 1st2nd 7159 . . . 4 ((Rel 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ (𝑇𝑀) ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗))) → (𝑇𝑀) = ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩)
104, 8, 9sylancr 694 . . 3 ((𝜑𝜓) → (𝑇𝑀) = ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩)
1110, 8eqeltrrd 2699 . 2 ((𝜑𝜓) → ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩ ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
12 fveq2 6148 . . . 4 (𝑗 = (1st ‘(𝑇𝑀)) → (𝑆𝑗) = (𝑆‘(1st ‘(𝑇𝑀))))
1312opeliunxp2 5220 . . 3 (⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩ ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ↔ ((1st ‘(𝑇𝑀)) ∈ 𝐽 ∧ (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀)))))
1413simprbi 480 . 2 (⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩ ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) → (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))))
1511, 14syl 17 1 ((𝜑𝜓) → (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  {crab 2911   ∖ cdif 3552   ∩ cin 3554   ⊆ wss 3555  ∅c0 3891  𝒫 cpw 4130  {csn 4148  ⟨cop 4154  ∪ cuni 4402  ∪ ciun 4485   ↦ cmpt 4673   × cxp 5072  ◡ccnv 5073  ran crn 5075   ↾ cres 5076   “ cima 5077  Rel wrel 5079  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  0cc0 9880  1c1 9881   − cmin 10210   / cdiv 10628  ℕcn 10964  (,)cioo 12117  [,]cicc 12120  ...cfz 12268   ↾t crest 16002  topGenctg 16019   Cn ccn 20938  Homeochmeo 21466  IIcii 22586   CovMap ccvm 30945 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-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  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-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-1st 7113  df-2nd 7114 This theorem is referenced by:  cvmliftlem6  30980  cvmliftlem8  30982  cvmliftlem9  30983
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