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Theorem cvmliftlem1 31599
 Description: Lemma for cvmlift 31613. In cvmliftlem15 31612, 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 5266 . . . . . 6 Rel ({𝑗} × (𝑆𝑗))
21rgenw 3072 . . . . 5 𝑗𝐽 Rel ({𝑗} × (𝑆𝑗))
3 reliun 5378 . . . . 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 466 . . . . 5 ((𝜑𝜓) → 𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
7 cvmliftlem1.m . . . . 5 ((𝜑𝜓) → 𝑀 ∈ (1...𝑁))
86, 7ffvelrnd 6503 . . . 4 ((𝜑𝜓) → (𝑇𝑀) ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
9 1st2nd 7362 . . . 4 ((Rel 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ (𝑇𝑀) ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗))) → (𝑇𝑀) = ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩)
104, 8, 9sylancr 567 . . 3 ((𝜑𝜓) → (𝑇𝑀) = ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩)
1110, 8eqeltrrd 2850 . 2 ((𝜑𝜓) → ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩ ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
12 fveq2 6332 . . . 4 (𝑗 = (1st ‘(𝑇𝑀)) → (𝑆𝑗) = (𝑆‘(1st ‘(𝑇𝑀))))
1312opeliunxp2 5399 . . 3 (⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩ ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ↔ ((1st ‘(𝑇𝑀)) ∈ 𝐽 ∧ (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀)))))
1413simprbi 478 . 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 382   = wceq 1630   ∈ wcel 2144  ∀wral 3060  {crab 3064   ∖ cdif 3718   ∩ cin 3720   ⊆ wss 3721  ∅c0 4061  𝒫 cpw 4295  {csn 4314  ⟨cop 4320  ∪ cuni 4572  ∪ ciun 4652   ↦ cmpt 4861   × cxp 5247  ◡ccnv 5248  ran crn 5250   ↾ cres 5251   “ cima 5252  Rel wrel 5254  ⟶wf 6027  ‘cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  0cc0 10137  1c1 10138   − cmin 10467   / cdiv 10885  ℕcn 11221  (,)cioo 12379  [,]cicc 12382  ...cfz 12532   ↾t crest 16288  topGenctg 16305   Cn ccn 21248  Homeochmeo 21776  IIcii 22897   CovMap ccvm 31569 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-1st 7314  df-2nd 7315 This theorem is referenced by:  cvmliftlem6  31604  cvmliftlem8  31606  cvmliftlem9  31607
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