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Theorem cvmcov 24981
 Description: Property of a covering map. In order to make the covering property more manageable, we define here the set 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 t t
cvmcov.2
Assertion
Ref Expression
cvmcov CovMap
Distinct variable groups:   ,,,,,   ,,,,,   ,,   ,,,,,   ,   ,
Allowed substitution hints:   (,,)   (,,,)   (,,,)

Proof of Theorem cvmcov
StepHypRef Expression
1 cvmcov.1 . . . . 5 t t
2 cvmcov.2 . . . . 5
31, 2iscvm 24977 . . . 4 CovMap
43simprbi 452 . . 3 CovMap
5 eleq1 2502 . . . . . 6
65anbi1d 687 . . . . 5
76rexbidv 2732 . . . 4
87rspcv 3054 . . 3
94, 8mpan9 457 . 2 CovMap
10 nfv 1630 . . . 4
11 nfmpt1 4323 . . . . . . 7 t t
121, 11nfcxfr 2575 . . . . . 6
13 nfcv 2578 . . . . . 6
1412, 13nffv 5764 . . . . 5
15 nfcv 2578 . . . . 5
1614, 15nfne 2701 . . . 4
1710, 16nfan 1848 . . 3
18 nfv 1630 . . 3
19 eleq2 2503 . . . 4
20 fveq2 5757 . . . . 5
2120neeq1d 2620 . . . 4
2219, 21anbi12d 693 . . 3
2317, 18, 22cbvrex 2935 . 2
249, 23sylibr 205 1 CovMap
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1727   wne 2605  wral 2711  wrex 2712  crab 2715   cdif 3303   cin 3305  c0 3613  cpw 3823  csn 3838  cuni 4039   cmpt 4291  ccnv 4906   cres 4909  cima 4910  cfv 5483  (class class class)co 6110   ↾t crest 13679  ctop 16989   ccn 17319   chmeo 17816   CovMap ccvm 24973 This theorem is referenced by:  cvmcov2  24993  cvmopnlem  24996  cvmfolem  24997  cvmliftmolem2  25000  cvmliftlem15  25016  cvmlift2lem10  25030  cvmlift3lem8  25044 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-cvm 24974
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