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Theorem fncvm 24936
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 24935 . 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 6098 . . 3  |-  ( c  Cn  j )  e. 
_V
32rabex 4346 . 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 6412 1  |- CovMap  Fn  ( Top  X.  Top )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   {crab 2701    \ cdif 3309    i^i cin 3311   (/)c0 3620   ~Pcpw 3791   {csn 3806   U.cuni 4007    X. cxp 4868   `'ccnv 4869    |` cres 4872   "cima 4873    Fn wfn 5441  (class class class)co 6073   ↾t crest 13640   Topctop 16950    Cn ccn 17280    Homeo chmeo 17777   CovMap ccvm 24934
This theorem is referenced by:  cvmtop1  24939  cvmtop2  24940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-cvm 24935
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