Theorem cvmcov 35461

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 35457	. . . 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 2825	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 ∈ 𝑘 ↔ 𝑃 ∈ 𝑘))
6	5	anbi1d 632	. . . . 5 ⊢ (𝑥 = 𝑃 → ((𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) ↔ (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
7	6	rexbidv 3162	. . . 4 ⊢ (𝑥 = 𝑃 → (∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) ↔ ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
8	7	rspcv 3561	. . 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 1916	. . . 4 ⊢ Ⅎ𝑘 𝑃 ∈ 𝑥
11		nfmpt1 5185	. . . . . . 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 2897	. . . . . 6 ⊢ Ⅎ𝑘𝑆
13		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑘𝑥
14	12, 13	nffv 6844	. . . . 5 ⊢ Ⅎ𝑘(𝑆‘𝑥)
15		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑘∅
16	14, 15	nfne 3034	. . . 4 ⊢ Ⅎ𝑘(𝑆‘𝑥) ≠ ∅
17	10, 16	nfan 1901	. . 3 ⊢ Ⅎ𝑘(𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅)
18		nfv 1916	. . 3 ⊢ Ⅎ𝑥(𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)
19		eleq2w 2821	. . . 4 ⊢ (𝑥 = 𝑘 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑘))
20		fveq2 6834	. . . . 5 ⊢ (𝑥 = 𝑘 → (𝑆‘𝑥) = (𝑆‘𝑘))
21	20	neeq1d 2992	. . . 4 ⊢ (𝑥 = 𝑘 → ((𝑆‘𝑥) ≠ ∅ ↔ (𝑆‘𝑘) ≠ ∅))
22	19, 21	anbi12d 633	. . 3 ⊢ (𝑥 = 𝑘 → ((𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅) ↔ (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
23	17, 18, 22	cbvrexw 3281	. 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 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390   ∖ cdif 3887   ∩ cin 3889  ∅c0 4274  𝒫 cpw 4542  {csn 4568  ∪ cuni 4851   ↦ cmpt 5167  ^◡ccnv 5623   ↾ cres 5626   “ cima 5627  ‘cfv 6492  (class class class)co 7360   ↾_t crest 17374  Topctop 22868   Cn ccn 23199  Homeochmeo 23728   CovMap ccvm 35453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-cvm 35454

This theorem is referenced by:  cvmcov2  35473  cvmopnlem  35476  cvmfolem  35477  cvmliftmolem2  35480  cvmliftlem15  35496  cvmlift2lem10  35510  cvmlift3lem8  35524