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Theorem fncvm 28453
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 28452 . 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 6310 . . 3  |-  ( c  Cn  j )  e. 
_V
32rabex 4598 . 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 6854 1  |- CovMap  Fn  ( Top  X.  Top )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818    \ cdif 3473    i^i cin 3475   (/)c0 3785   ~Pcpw 4010   {csn 4027   U.cuni 4245    X. cxp 4997   `'ccnv 4998    |` cres 5001   "cima 5002    Fn wfn 5583  (class class class)co 6285   ↾t crest 14679   Topctop 19201    Cn ccn 19531   Homeochmeo 20081   CovMap ccvm 28451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-cvm 28452
This theorem is referenced by:  cvmtop1  28456  cvmtop2  28457
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