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Theorem fncvm 23788
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 23787 . 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 5883 . . 3  |-  ( c  Cn  j )  e. 
_V
32rabex 4165 . 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 6193 1  |- CovMap  Fn  ( Top  X.  Top )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    \ cdif 3149    i^i cin 3151   (/)c0 3455   ~Pcpw 3625   {csn 3640   U.cuni 3827    X. cxp 4687   `'ccnv 4688    |` cres 4691   "cima 4692    Fn wfn 5250  (class class class)co 5858   ↾t crest 13325   Topctop 16631    Cn ccn 16954    Homeo chmeo 17444   CovMap ccvm 23786
This theorem is referenced by:  cvmtop1  23791  cvmtop2  23792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-cvm 23787
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