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Theorem cvmsrcl 27152
Description: Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypothesis
Ref Expression
cvmcov.1  |-  S  =  ( k  e.  J  |->  { 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 ) ) ) ) } )
Assertion
Ref Expression
cvmsrcl  |-  ( T  e.  ( S `  U )  ->  U  e.  J )
Distinct variable groups:    k, s, u, v, C    k, F, s, u, v    k, J, s, u, v    U, k, s, u, v    T, s, u, v
Allowed substitution hints:    S( v, u, k, s)    T( k)

Proof of Theorem cvmsrcl
StepHypRef Expression
1 cvmcov.1 . . 3  |-  S  =  ( k  e.  J  |->  { 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 ) ) ) ) } )
21dmmptss 5333 . 2  |-  dom  S  C_  J
3 elfvdm 5715 . 2  |-  ( T  e.  ( S `  U )  ->  U  e.  dom  S )
42, 3sseldi 3353 1  |-  ( T  e.  ( S `  U )  ->  U  e.  J )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2714   {crab 2718    \ cdif 3324    i^i cin 3326   (/)c0 3636   ~Pcpw 3859   {csn 3876   U.cuni 4090    e. cmpt 4349   `'ccnv 4838   dom cdm 4839    |` cres 4841   "cima 4842   ` cfv 5417  (class class class)co 6090   ↾t crest 14358   Homeochmeo 19325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fv 5425
This theorem is referenced by:  cvmsi  27153  cvmsf1o  27160  cvmsss2  27162  cvmopnlem  27166  cvmliftlem8  27180  cvmlift2lem9  27199  cvmlift2lem10  27200  cvmlift3lem6  27212  cvmlift3lem8  27214
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