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Theorem cvmcov 30953
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 30949 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅)))
43simprbi 480 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅))
5 eleq1 2686 . . . . . 6 (𝑥 = 𝑃 → (𝑥𝑘𝑃𝑘))
65anbi1d 740 . . . . 5 (𝑥 = 𝑃 → ((𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) ↔ (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
76rexbidv 3045 . . . 4 (𝑥 = 𝑃 → (∃𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) ↔ ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
87rspcv 3291 . . 3 (𝑃𝑋 → (∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) → ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
94, 8mpan9 486 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃𝑋) → ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅))
10 nfv 1840 . . . 4 𝑘 𝑃𝑥
11 nfmpt1 4707 . . . . . . 7 𝑘(𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
121, 11nfcxfr 2759 . . . . . 6 𝑘𝑆
13 nfcv 2761 . . . . . 6 𝑘𝑥
1412, 13nffv 6155 . . . . 5 𝑘(𝑆𝑥)
15 nfcv 2761 . . . . 5 𝑘
1614, 15nfne 2890 . . . 4 𝑘(𝑆𝑥) ≠ ∅
1710, 16nfan 1825 . . 3 𝑘(𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅)
18 nfv 1840 . . 3 𝑥(𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)
19 eleq2 2687 . . . 4 (𝑥 = 𝑘 → (𝑃𝑥𝑃𝑘))
20 fveq2 6148 . . . . 5 (𝑥 = 𝑘 → (𝑆𝑥) = (𝑆𝑘))
2120neeq1d 2849 . . . 4 (𝑥 = 𝑘 → ((𝑆𝑥) ≠ ∅ ↔ (𝑆𝑘) ≠ ∅))
2219, 21anbi12d 746 . . 3 (𝑥 = 𝑘 → ((𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅) ↔ (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
2317, 18, 22cbvrex 3156 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  cdif 3552  cin 3554  c0 3891  𝒫 cpw 4130  {csn 4148   cuni 4402  cmpt 4673  ccnv 5073  cres 5076  cima 5077  cfv 5847  (class class class)co 6604  t crest 16002  Topctop 20617   Cn ccn 20938  Homeochmeo 21466   CovMap ccvm 30945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-cvm 30946
This theorem is referenced by:  cvmcov2  30965  cvmopnlem  30968  cvmfolem  30969  cvmliftmolem2  30972  cvmliftlem15  30988  cvmlift2lem10  31002  cvmlift3lem8  31016
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