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Theorem cvmcov 31371
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 31367 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅)))
43simprbi 479 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅))
5 eleq1 2718 . . . . . 6 (𝑥 = 𝑃 → (𝑥𝑘𝑃𝑘))
65anbi1d 741 . . . . 5 (𝑥 = 𝑃 → ((𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) ↔ (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
76rexbidv 3081 . . . 4 (𝑥 = 𝑃 → (∃𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) ↔ ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
87rspcv 3336 . . 3 (𝑃𝑋 → (∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) → ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
94, 8mpan9 485 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃𝑋) → ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅))
10 nfv 1883 . . . 4 𝑘 𝑃𝑥
11 nfmpt1 4780 . . . . . . 7 𝑘(𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
121, 11nfcxfr 2791 . . . . . 6 𝑘𝑆
13 nfcv 2793 . . . . . 6 𝑘𝑥
1412, 13nffv 6236 . . . . 5 𝑘(𝑆𝑥)
15 nfcv 2793 . . . . 5 𝑘
1614, 15nfne 2923 . . . 4 𝑘(𝑆𝑥) ≠ ∅
1710, 16nfan 1868 . . 3 𝑘(𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅)
18 nfv 1883 . . 3 𝑥(𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)
19 eleq2 2719 . . . 4 (𝑥 = 𝑘 → (𝑃𝑥𝑃𝑘))
20 fveq2 6229 . . . . 5 (𝑥 = 𝑘 → (𝑆𝑥) = (𝑆𝑘))
2120neeq1d 2882 . . . 4 (𝑥 = 𝑘 → ((𝑆𝑥) ≠ ∅ ↔ (𝑆𝑘) ≠ ∅))
2219, 21anbi12d 747 . . 3 (𝑥 = 𝑘 → ((𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅) ↔ (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
2317, 18, 22cbvrex 3198 . 2 (∃𝑥𝐽 (𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅) ↔ ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅))
249, 23sylibr 224 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃𝑋) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  cdif 3604  cin 3606  c0 3948  𝒫 cpw 4191  {csn 4210   cuni 4468  cmpt 4762  ccnv 5142  cres 5145  cima 5146  cfv 5926  (class class class)co 6690  t crest 16128  Topctop 20746   Cn ccn 21076  Homeochmeo 21604   CovMap ccvm 31363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-cvm 31364
This theorem is referenced by:  cvmcov2  31383  cvmopnlem  31386  cvmfolem  31387  cvmliftmolem2  31390  cvmliftlem15  31406  cvmlift2lem10  31420  cvmlift3lem8  31434
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