Theorem cvmliftlem1 36019

Description: Lemma for cvmlift 36033. In cvmliftlem15 36032, 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 5669	. . . . . 6 ⊢ Rel ({𝑗} × (𝑆‘𝑗))
2	1	rgenw 3081	. . . . 5 ⊢ ∀𝑗 ∈ 𝐽 Rel ({𝑗} × (𝑆‘𝑗))
3		reliun 5794	. . . . 5 ⊢ (Rel ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) ↔ ∀𝑗 ∈ 𝐽 Rel ({𝑗} × (𝑆‘𝑗)))
4	2, 3	mpbir 234	. . . 4 ⊢ Rel ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))
5		cvmliftlem.t	. . . . . 6 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
6	5	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
7		cvmliftlem1.m	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁))
8	6, 7	ffvelcdmd 7077	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑇‘𝑀) ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
9		1st2nd 8039	. . . 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 599	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑇‘𝑀) = ⟨(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))⟩)
11	10, 8	eqeltrrd 2862	. 2 ⊢ ((𝜑 ∧ 𝜓) → ⟨(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))⟩ ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
12		fveq2 6877	. . . 4 ⊢ (𝑗 = (1st ‘(𝑇‘𝑀)) → (𝑆‘𝑗) = (𝑆‘(1st ‘(𝑇‘𝑀))))
13	12	opeliunxp2 5815	. . 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 503	. 2 ⊢ (⟨(1st ‘(𝑇‘𝑀)), (2nd ‘(𝑇‘𝑀))⟩ ∈ ∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))
15	11, 14	syl 18	1 ⊢ ((𝜑 ∧ 𝜓) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ⟨cop 4590  ∪ cuni 4867  ∪ ciun 4951   ↦ cmpt 5186   × cxp 5649  ^◡ccnv 5650  ran crn 5652   ↾ cres 5653   “ cima 5654  Rel wrel 5656  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  1^st c1st 7988  2^nd c2nd 7989  0cc0 11181  1c1 11182   − cmin 11522   / cdiv 11954  ℕcn 12316  (,)cioo 13457  [,]cicc 13460  ...cfz 13620   ↾_t crest 17571  topGenctg 17588   Cn ccn 23522  Homeochmeo 24052  IIcii 25176   CovMap ccvm 35989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-1st 7990  df-2nd 7991

This theorem is used by:  cvmliftlem6  36024  cvmliftlem8  36026  cvmliftlem9  36027