Theorem cvmliftlem1 35270

Description: Lemma for cvmlift 35284. In cvmliftlem15 35283, 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 1^st ‘(𝑇‘𝑀) is a neighborhood of (𝐺 “ [(𝑀 − 1) / 𝑁, 𝑀 / 𝑁]), and 2^nd ‘(𝑇‘𝑀) is an even covering of 1^st ‘(𝑇‘𝑀), which is to say a disjoint union of open sets in 𝐶 whose image is 1^st ‘(𝑇‘𝑀). (Contributed by Mario Carneiro, 14-Feb-2015.)

Hypotheses

Ref	Expression
cvmliftlem.1	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})
cvmliftlem.b	⊢ 𝐵 = ∪ 𝐶
cvmliftlem.x	⊢ 𝑋 = ∪ 𝐽
cvmliftlem.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftlem.g	⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
cvmliftlem.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmliftlem.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))
cvmliftlem.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
cvmliftlem.t	⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
cvmliftlem.a	⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))
cvmliftlem.l	⊢ 𝐿 = (topGen‘ran (,))
cvmliftlem1.m	⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁))

Assertion

Ref	Expression
cvmliftlem1	⊢ ((𝜑 ∧ 𝜓) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))

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

Step	Hyp	Ref	Expression
1		relxp 5707	. . . . . 6 ⊢ Rel ({𝑗} × (𝑆‘𝑗))
2	1	rgenw 3063	. . . . 5 ⊢ ∀𝑗 ∈ 𝐽 Rel ({𝑗} × (𝑆‘𝑗))
3		reliun 5829	. . . . 5 ⊢ (Rel ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ↔ ∀𝑗 ∈ 𝐽 Rel ({𝑗} × (𝑆‘𝑗)))
4	2, 3	mpbir 231	. . . 4 ⊢ Rel ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))
5		cvmliftlem.t	. . . . . 6 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
7		cvmliftlem1.m	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁))
8	6, 7	ffvelcdmd 7105	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑇‘𝑀) ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
9		1st2nd 8063	. . . 4 ⊢ ((Rel ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ∧ (𝑇‘𝑀) ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) → (𝑇‘𝑀) = ⟨(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))⟩)
10	4, 8, 9	sylancr 587	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑇‘𝑀) = ⟨(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))⟩)
11	10, 8	eqeltrrd 2840	. 2 ⊢ ((𝜑 ∧ 𝜓) → ⟨(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))⟩ ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
12		fveq2 6907	. . . 4 ⊢ (𝑗 = (1st ‘(𝑇‘𝑀)) → (𝑆‘𝑗) = (𝑆‘(1st ‘(𝑇‘𝑀))))
13	12	opeliunxp2 5852	. . 3 ⊢ (⟨(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))⟩ ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ↔ ((1st ‘(𝑇‘𝑀)) ∈ 𝐽 ∧ (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀)))))
14	13	simprbi 496	. 2 ⊢ (⟨(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))⟩ ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))
15	11, 14	syl 17	1 ⊢ ((𝜑 ∧ 𝜓) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  𝒫 cpw 4605  {csn 4631  ⟨cop 4637  ∪ cuni 4912  ∪ ciun 4996   ↦ cmpt 5231   × cxp 5687  ^◡ccnv 5688  ran crn 5690   ↾ cres 5691   “ cima 5692  Rel wrel 5694  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  1^st c1st 8011  2^nd c2nd 8012  0cc0 11153  1c1 11154   − cmin 11490   / cdiv 11918  ℕcn 12264  (,)cioo 13384  [,]cicc 13387  ...cfz 13544   ↾_t crest 17467  topGenctg 17484   Cn ccn 23248  Homeochmeo 23777  IIcii 24915   CovMap ccvm 35240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-1st 8013  df-2nd 8014

This theorem is referenced by:  cvmliftlem6  35275  cvmliftlem8  35277  cvmliftlem9  35278