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Theorem cvmsrcl 23810
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 5185 . 2  |-  dom  S  C_  J
3 elfvdm 5570 . 2  |-  ( T  e.  ( S `  U )  ->  U  e.  dom  S )
42, 3sseldi 3191 1  |-  ( T  e.  ( S `  U )  ->  U  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162    i^i cin 3164   (/)c0 3468   ~Pcpw 3638   {csn 3653   U.cuni 3843    e. cmpt 4093   `'ccnv 4704   dom cdm 4705    |` cres 4707   "cima 4708   ` cfv 5271  (class class class)co 5874   ↾t crest 13341    Homeo chmeo 17460
This theorem is referenced by:  cvmsi  23811  cvmsf1o  23818  cvmsss2  23820  cvmopnlem  23824  cvmliftlem8  23838  cvmlift2lem9  23857  cvmlift2lem10  23858  cvmlift3lem6  23870  cvmlift3lem8  23872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279
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