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Theorem fncvm 27291
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 27290 . 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 6226 . . 3  |-  ( c  Cn  j )  e. 
_V
32rabex 4552 . 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 6754 1  |- CovMap  Fn  ( Top  X.  Top )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   {crab 2803    \ cdif 3434    i^i cin 3436   (/)c0 3746   ~Pcpw 3969   {csn 3986   U.cuni 4200    X. cxp 4947   `'ccnv 4948    |` cres 4951   "cima 4952    Fn wfn 5522  (class class class)co 6201   ↾t crest 14479   Topctop 18631    Cn ccn 18961   Homeochmeo 19459   CovMap ccvm 27289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-cvm 27290
This theorem is referenced by:  cvmtop1  27294  cvmtop2  27295
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