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Theorem cvmcov 29774
 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 29770 . . . 4 CovMap
43simprbi 465 . . 3 CovMap
5 eleq1 2501 . . . . . 6
65anbi1d 709 . . . . 5
76rexbidv 2946 . . . 4
87rspcv 3184 . . 3
94, 8mpan9 471 . 2 CovMap
10 nfv 1754 . . . 4
11 nfmpt1 4515 . . . . . . 7 t t
121, 11nfcxfr 2589 . . . . . 6
13 nfcv 2591 . . . . . 6
1412, 13nffv 5888 . . . . 5
15 nfcv 2591 . . . . 5
1614, 15nfne 2763 . . . 4
1710, 16nfan 1986 . . 3
18 nfv 1754 . . 3
19 eleq2 2502 . . . 4
20 fveq2 5881 . . . . 5
2120neeq1d 2708 . . . 4
2219, 21anbi12d 715 . . 3
2317, 18, 22cbvrex 3059 . 2
249, 23sylibr 215 1 CovMap
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   w3a 982   wceq 1437   wcel 1870   wne 2625  wral 2782  wrex 2783  crab 2786   cdif 3439   cin 3441  c0 3767  cpw 3985  csn 4002  cuni 4222   cmpt 4484  ccnv 4853   cres 4856  cima 4857  cfv 5601  (class class class)co 6305   ↾t crest 15278  ctop 19848   ccn 20171  chmeo 20699   CovMap ccvm 29766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-cvm 29767 This theorem is referenced by:  cvmcov2  29786  cvmopnlem  29789  cvmfolem  29790  cvmliftmolem2  29793  cvmliftlem15  29809  cvmlift2lem10  29823  cvmlift3lem8  29837
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