Theorem cvmcov 35248

Description: Property of a covering map. In order to make the covering property more manageable, we define here the set 𝑆(𝑘) of all even coverings of an open set 𝑘 in the range. Then the covering property states that every point has a neighborhood which has an even covering. (Contributed by Mario Carneiro, 13-Feb-2015.)

Hypotheses

Ref	Expression
cvmcov.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))))})
cvmcov.2	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cvmcov	⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅))

Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝑥,𝐶   𝑘,𝐹,𝑠,𝑢,𝑣,𝑥   𝑃,𝑘,𝑥   𝑘,𝐽,𝑠,𝑢,𝑣,𝑥   𝑥,𝑆   𝑥,𝑋

Allowed substitution hints:   𝑃(𝑣,𝑢,𝑠)   𝑆(𝑣,𝑢,𝑘,𝑠)   𝑋(𝑣,𝑢,𝑘,𝑠)

Proof of Theorem cvmcov

Step	Hyp	Ref	Expression
1		cvmcov.1	. . . . 5 ⊢ 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))))})
2		cvmcov.2	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	iscvm 35244	. . . 4 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
4	3	simprbi 496	. . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))
5		eleq1 2827	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 ∈ 𝑘 ↔ 𝑃 ∈ 𝑘))
6	5	anbi1d 631	. . . . 5 ⊢ (𝑥 = 𝑃 → ((𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) ↔ (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
7	6	rexbidv 3177	. . . 4 ⊢ (𝑥 = 𝑃 → (∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) ↔ ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
8	7	rspcv 3618	. . 3 ⊢ (𝑃 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) → ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
9	4, 8	mpan9 506	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))
10		nfv 1912	. . . 4 ⊢ Ⅎ𝑘 𝑃 ∈ 𝑥
11		nfmpt1 5256	. . . . . . 7 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})
12	1, 11	nfcxfr 2901	. . . . . 6 ⊢ Ⅎ𝑘𝑆
13		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑘𝑥
14	12, 13	nffv 6917	. . . . 5 ⊢ Ⅎ𝑘(𝑆‘𝑥)
15		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑘∅
16	14, 15	nfne 3041	. . . 4 ⊢ Ⅎ𝑘(𝑆‘𝑥) ≠ ∅
17	10, 16	nfan 1897	. . 3 ⊢ Ⅎ𝑘(𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅)
18		nfv 1912	. . 3 ⊢ Ⅎ𝑥(𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)
19		eleq2w 2823	. . . 4 ⊢ (𝑥 = 𝑘 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑘))
20		fveq2 6907	. . . . 5 ⊢ (𝑥 = 𝑘 → (𝑆‘𝑥) = (𝑆‘𝑘))
21	20	neeq1d 2998	. . . 4 ⊢ (𝑥 = 𝑘 → ((𝑆‘𝑥) ≠ ∅ ↔ (𝑆‘𝑘) ≠ ∅))
22	19, 21	anbi12d 632	. . 3 ⊢ (𝑥 = 𝑘 → ((𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅) ↔ (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
23	17, 18, 22	cbvrexw 3305	. 2 ⊢ (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅) ↔ ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))
24	9, 23	sylibr 234	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  {crab 3433   ∖ cdif 3960   ∩ cin 3962  ∅c0 4339  𝒫 cpw 4605  {csn 4631  ∪ cuni 4912   ↦ cmpt 5231  ^◡ccnv 5688   ↾ cres 5691   “ cima 5692  ‘cfv 6563  (class class class)co 7431   ↾_t crest 17467  Topctop 22915   Cn ccn 23248  Homeochmeo 23777   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-pow 5371  ax-pr 5438

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-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  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-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-cvm 35241

This theorem is referenced by:  cvmcov2  35260  cvmopnlem  35263  cvmfolem  35264  cvmliftmolem2  35267  cvmliftlem15  35283  cvmlift2lem10  35297  cvmlift3lem8  35311