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Theorem cvmcov 26298
 Description: Property of a covering map. In order to make the covering property more manageable, we define here the set S( ) of all even coverings of an open set 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
cvmcov.2
Assertion
Ref Expression
cvmcov
Distinct variable groups:   ,,,,,   ,,,,,   P,,   ,J,,,,   ,S   ,
Allowed substitution hints:   P(,,)   S(,,,)   (,,,)

Proof of Theorem cvmcov
StepHypRef Expression
1 cvmcov.1 . . . . 5
2 cvmcov.2 . . . . 5
31, 2iscvm 26294 . . . 4
43simprbi 452 . . 3
5 eleq1 2549 . . . . . 6
65anbi1d 687 . . . . 5
76rexbidv 2780 . . . 4
87rspcv 3109 . . 3
94, 8mpan9 457 . 2
10 nfv 1647 . . . 4
11 nfmpt1 4407 . . . . . . 7
121, 11nfcxfr 2622 . . . . . 6
13 nfcv 2625 . . . . . 6
1412, 13nffv 5715 . . . . 5
15 nfcv 2625 . . . . 5
1614, 15nfne 2748 . . . 4
1710, 16nfan 1850 . . 3
18 nfv 1647 . . 3
19 eleq2 2550 . . . 4
20 fveq2 5708 . . . . 5
2120neeq1d 2667 . . . 4
2219, 21anbi12d 693 . . 3
2317, 18, 22cbvrex 2987 . 2
249, 23sylibr 205 1
 Colors of variables: wff set class Syntax hints:  ->wi 4  /\wa 360  /\w3a 939  =wceq 1670  e.wcel 1732  =/=wne 2652  A.wral 2759  E.wrex 2760  {crab 2763  \cdif 3362  i^icin 3364   c0 3673  ~Pcpw 3893  {csn 3909  U.cuni 4117  e.cmpt 4376  'ccnv 4861  |cres 4864  "cima 4865  `cfv 5438  (class class class)co 6103   crest 14202   ctop 17972   ccn 18302   chmeo 18800   ccvm 26290 This theorem is referenced by:  cvmcov2  26310  cvmopnlem  26313  cvmfolem  26314  cvmliftmolem2  26317  cvmliftlem15  26333  cvmlift2lem10  26347  cvmlift3lem8  26361 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1570  ax-4 1581  ax-5 1644  ax-6 1685  ax-7 1705  ax-8 1734  ax-9 1736  ax-10 1751  ax-11 1756  ax-12 1768  ax-13 1955  ax-ext 2470  ax-sep 4439  ax-nul 4447  ax-pow 4493  ax-pr 4554 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1338  df-ex 1566  df-nf 1569  df-sb 1677  df-eu 2317  df-mo 2318  df-clab 2476  df-cleq 2482  df-clel 2485  df-nfc 2614  df-ne 2654  df-ral 2764  df-rex 2765  df-rab 2768  df-v 3017  df-sbc 3225  df-csb 3326  df-dif 3368  df-un 3370  df-in 3372  df-ss 3379  df-nul 3674  df-if 3826  df-pw 3895  df-sn 3915  df-pr 3916  df-op 3918  df-uni 4118  df-br 4319  df-opab 4377  df-mpt 4378  df-id 4657  df-xp 4868  df-rel 4869  df-cnv 4870  df-co 4871  df-dm 4872  df-rn 4873  df-res 4874  df-ima 4875  df-iota 5401  df-fun 5440  df-fv 5446  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-cvm 26291
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