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Theorem cvmcov 30499
 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 30495 . . . 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 2676 . . . . . 6 (𝑥 = 𝑃 → (𝑥𝑘𝑃𝑘))
65anbi1d 737 . . . . 5 (𝑥 = 𝑃 → ((𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) ↔ (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
76rexbidv 3034 . . . 4 (𝑥 = 𝑃 → (∃𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) ↔ ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
87rspcv 3278 . . 3 (𝑃𝑋 → (∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) → ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
94, 8mpan9 485 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃𝑋) → ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅))
10 nfv 1830 . . . 4 𝑘 𝑃𝑥
11 nfmpt1 4675 . . . . . . 7 𝑘(𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
121, 11nfcxfr 2749 . . . . . 6 𝑘𝑆
13 nfcv 2751 . . . . . 6 𝑘𝑥
1412, 13nffv 6110 . . . . 5 𝑘(𝑆𝑥)
15 nfcv 2751 . . . . 5 𝑘
1614, 15nfne 2882 . . . 4 𝑘(𝑆𝑥) ≠ ∅
1710, 16nfan 1816 . . 3 𝑘(𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅)
18 nfv 1830 . . 3 𝑥(𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)
19 eleq2 2677 . . . 4 (𝑥 = 𝑘 → (𝑃𝑥𝑃𝑘))
20 fveq2 6103 . . . . 5 (𝑥 = 𝑘 → (𝑆𝑥) = (𝑆𝑘))
2120neeq1d 2841 . . . 4 (𝑥 = 𝑘 → ((𝑆𝑥) ≠ ∅ ↔ (𝑆𝑘) ≠ ∅))
2219, 21anbi12d 743 . . 3 (𝑥 = 𝑘 → ((𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅) ↔ (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅)))
2317, 18, 22cbvrex 3144 . 2 (∃𝑥𝐽 (𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅) ↔ ∃𝑘𝐽 (𝑃𝑘 ∧ (𝑆𝑘) ≠ ∅))
249, 23sylibr 223 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃𝑋) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ∩ cin 3539  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   ↦ cmpt 4643  ◡ccnv 5037   ↾ cres 5040   “ cima 5041  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517   Cn ccn 20838  Homeochmeo 21366   CovMap ccvm 30491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-cvm 30492 This theorem is referenced by:  cvmcov2  30511  cvmopnlem  30514  cvmfolem  30515  cvmliftmolem2  30518  cvmliftlem15  30534  cvmlift2lem10  30548  cvmlift3lem8  30562
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