Theorem cvmcov 35231

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 35227	. . . 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 2832	. . . . . 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 3185	. . . 4 ⊢ (𝑥 = 𝑃 → (∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) ↔ ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
8	7	rspcv 3631	. . 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 1913	. . . 4 ⊢ Ⅎ𝑘 𝑃 ∈ 𝑥
11		nfmpt1 5274	. . . . . . 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 2906	. . . . . 6 ⊢ Ⅎ𝑘𝑆
13		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑘𝑥
14	12, 13	nffv 6930	. . . . 5 ⊢ Ⅎ𝑘(𝑆‘𝑥)
15		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑘∅
16	14, 15	nfne 3049	. . . 4 ⊢ Ⅎ𝑘(𝑆‘𝑥) ≠ ∅
17	10, 16	nfan 1898	. . 3 ⊢ Ⅎ𝑘(𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅)
18		nfv 1913	. . 3 ⊢ Ⅎ𝑥(𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)
19		eleq2w 2828	. . . 4 ⊢ (𝑥 = 𝑘 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑘))
20		fveq2 6920	. . . . 5 ⊢ (𝑥 = 𝑘 → (𝑆‘𝑥) = (𝑆‘𝑘))
21	20	neeq1d 3006	. . . 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 3313	. 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 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  {crab 3443   ∖ cdif 3973   ∩ cin 3975  ∅c0 4352  𝒫 cpw 4622  {csn 4648  ∪ cuni 4931   ↦ cmpt 5249  ^◡ccnv 5699   ↾ cres 5702   “ cima 5703  ‘cfv 6573  (class class class)co 7448   ↾_t crest 17480  Topctop 22920   Cn ccn 23253  Homeochmeo 23782   CovMap ccvm 35223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-cvm 35224

This theorem is referenced by:  cvmcov2  35243  cvmopnlem  35246  cvmfolem  35247  cvmliftmolem2  35250  cvmliftlem15  35266  cvmlift2lem10  35280  cvmlift3lem8  35294