Theorem cvmcov 33225

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 33221	. . . 4 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
4	3	simprbi 497	. . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))
5		eleq1 2826	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 ∈ 𝑘 ↔ 𝑃 ∈ 𝑘))
6	5	anbi1d 630	. . . . 5 ⊢ (𝑥 = 𝑃 → ((𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) ↔ (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
7	6	rexbidv 3226	. . . 4 ⊢ (𝑥 = 𝑃 → (∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) ↔ ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
8	7	rspcv 3557	. . 3 ⊢ (𝑃 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) → ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
9	4, 8	mpan9 507	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))
10		nfv 1917	. . . 4 ⊢ Ⅎ𝑘 𝑃 ∈ 𝑥
11		nfmpt1 5182	. . . . . . 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 2905	. . . . . 6 ⊢ Ⅎ𝑘𝑆
13		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑘𝑥
14	12, 13	nffv 6784	. . . . 5 ⊢ Ⅎ𝑘(𝑆‘𝑥)
15		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑘∅
16	14, 15	nfne 3045	. . . 4 ⊢ Ⅎ𝑘(𝑆‘𝑥) ≠ ∅
17	10, 16	nfan 1902	. . 3 ⊢ Ⅎ𝑘(𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅)
18		nfv 1917	. . 3 ⊢ Ⅎ𝑥(𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)
19		eleq2w 2822	. . . 4 ⊢ (𝑥 = 𝑘 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑘))
20		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝑘 → (𝑆‘𝑥) = (𝑆‘𝑘))
21	20	neeq1d 3003	. . . 4 ⊢ (𝑥 = 𝑘 → ((𝑆‘𝑥) ≠ ∅ ↔ (𝑆‘𝑘) ≠ ∅))
22	19, 21	anbi12d 631	. . 3 ⊢ (𝑥 = 𝑘 → ((𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅) ↔ (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
23	17, 18, 22	cbvrexw 3374	. 2 ⊢ (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅) ↔ ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))
24	9, 23	sylibr 233	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068   ∖ cdif 3884   ∩ cin 3886  ∅c0 4256  𝒫 cpw 4533  {csn 4561  ∪ cuni 4839   ↦ cmpt 5157  ^◡ccnv 5588   ↾ cres 5591   “ cima 5592  ‘cfv 6433  (class class class)co 7275   ↾_t crest 17131  Topctop 22042   Cn ccn 22375  Homeochmeo 22904   CovMap ccvm 33217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-cvm 33218

This theorem is referenced by:  cvmcov2  33237  cvmopnlem  33240  cvmfolem  33241  cvmliftmolem2  33244  cvmliftlem15  33260  cvmlift2lem10  33274  cvmlift3lem8  33288