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Theorem cvmcov 32624
 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 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
cvmcov.2 𝑋 = 𝐽
Assertion
Ref Expression
cvmcov ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃𝑋) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅))
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
StepHypRef Expression
1 cvmcov.1 . . . . 5 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
2 cvmcov.2 . . . . 5 𝑋 = 𝐽
31, 2iscvm 32620 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅)))
43simprbi 500 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅))
5 eleq1 2880 . . . . . 6 (𝑥 = 𝑃 → (𝑥𝑘𝑃𝑘))
65anbi1d 632 . . . . 5 (𝑥 = 𝑃 → ((𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) ↔ (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
76rexbidv 3259 . . . 4 (𝑥 = 𝑃 → (∃𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) ↔ ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
87rspcv 3569 . . 3 (𝑃𝑋 → (∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) → ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
94, 8mpan9 510 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃𝑋) → ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅))
10 nfv 1915 . . . 4 𝑘 𝑃𝑥
11 nfmpt1 5131 . . . . . . 7 𝑘(𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
121, 11nfcxfr 2956 . . . . . 6 𝑘𝑆
13 nfcv 2958 . . . . . 6 𝑘𝑥
1412, 13nffv 6659 . . . . 5 𝑘(𝑆𝑥)
15 nfcv 2958 . . . . 5 𝑘
1614, 15nfne 3090 . . . 4 𝑘(𝑆𝑥) ≠ ∅
1710, 16nfan 1900 . . 3 𝑘(𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅)
18 nfv 1915 . . 3 𝑥(𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)
19 eleq2w 2876 . . . 4 (𝑥 = 𝑘 → (𝑃𝑥𝑃𝑘))
20 fveq2 6649 . . . . 5 (𝑥 = 𝑘 → (𝑆𝑥) = (𝑆𝑘))
2120neeq1d 3049 . . . 4 (𝑥 = 𝑘 → ((𝑆𝑥) ≠ ∅ ↔ (𝑆𝑘) ≠ ∅))
2219, 21anbi12d 633 . . 3 (𝑥 = 𝑘 → ((𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅) ↔ (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
2317, 18, 22cbvrexw 3391 . 2 (∃𝑥𝐽 (𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅) ↔ ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅))
249, 23sylibr 237 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃𝑋) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  ∃wrex 3110  {crab 3113   ∖ cdif 3881   ∩ cin 3883  ∅c0 4246  𝒫 cpw 4500  {csn 4528  ∪ cuni 4803   ↦ cmpt 5113  ◡ccnv 5522   ↾ cres 5525   “ cima 5526  ‘cfv 6328  (class class class)co 7139   ↾t crest 16690  Topctop 21502   Cn ccn 21833  Homeochmeo 22362   CovMap ccvm 32616 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-cvm 32617 This theorem is referenced by:  cvmcov2  32636  cvmopnlem  32639  cvmfolem  32640  cvmliftmolem2  32643  cvmliftlem15  32659  cvmlift2lem10  32673  cvmlift3lem8  32687
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