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Theorem fncvm 24723
Description: Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.)
Assertion
Ref Expression
fncvm  |- CovMap  Fn  ( Top  X.  Top )

Proof of Theorem fncvm
Dummy variables  j 
c  f  x  k  s  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cvm 24722 . 2  |- CovMap  =  ( c  e.  Top , 
j  e.  Top  |->  { f  e.  ( c  Cn  j )  | 
A. x  e.  U. j E. k  e.  j  ( x  e.  k  /\  E. s  e.  ( ~P c  \  { (/) } ) ( U. s  =  ( `' f " k
)  /\  A. u  e.  s  ( A. v  e.  ( s  \  { u } ) ( u  i^i  v
)  =  (/)  /\  (
f  |`  u )  e.  ( ( ct  u ) 
Homeo  ( jt  k ) ) ) ) ) } )
2 ovex 6045 . . 3  |-  ( c  Cn  j )  e. 
_V
32rabex 4295 . 2  |-  { f  e.  ( c  Cn  j )  |  A. x  e.  U. j E. k  e.  j 
( x  e.  k  /\  E. s  e.  ( ~P c  \  { (/) } ) ( U. s  =  ( `' f " k
)  /\  A. u  e.  s  ( A. v  e.  ( s  \  { u } ) ( u  i^i  v
)  =  (/)  /\  (
f  |`  u )  e.  ( ( ct  u ) 
Homeo  ( jt  k ) ) ) ) ) }  e.  _V
41, 3fnmpt2i 6359 1  |- CovMap  Fn  ( Top  X.  Top )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650   {crab 2653    \ cdif 3260    i^i cin 3262   (/)c0 3571   ~Pcpw 3742   {csn 3757   U.cuni 3957    X. cxp 4816   `'ccnv 4817    |` cres 4820   "cima 4821    Fn wfn 5389  (class class class)co 6020   ↾t crest 13575   Topctop 16881    Cn ccn 17210    Homeo chmeo 17706   CovMap ccvm 24721
This theorem is referenced by:  cvmtop1  24726  cvmtop2  24727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-cvm 24722
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