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Theorem refssex 26384
Description: Every set in a refinement has a superset in the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.)
Assertion
Ref Expression
refssex  |-  ( ( A Ref B  /\  S  e.  B )  ->  E. x  e.  A  S  C_  x )
Distinct variable groups:    x, A    x, S
Allowed substitution hint:    B( x)

Proof of Theorem refssex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 refrel 26381 . . . . 5  |-  Rel  Ref
21brrelex2i 4746 . . . 4  |-  ( A Ref B  ->  B  e.  _V )
3 eqid 2296 . . . . . 6  |-  U. A  =  U. A
4 eqid 2296 . . . . . 6  |-  U. B  =  U. B
53, 4isref 26382 . . . . 5  |-  ( B  e.  _V  ->  ( A Ref B  <->  ( U. A  =  U. B  /\  A. y  e.  B  E. x  e.  A  y  C_  x ) ) )
65simplbda 607 . . . 4  |-  ( ( B  e.  _V  /\  A Ref B )  ->  A. y  e.  B  E. x  e.  A  y  C_  x )
72, 6mpancom 650 . . 3  |-  ( A Ref B  ->  A. y  e.  B  E. x  e.  A  y  C_  x )
8 sseq1 3212 . . . . 5  |-  ( y  =  S  ->  (
y  C_  x  <->  S  C_  x
) )
98rexbidv 2577 . . . 4  |-  ( y  =  S  ->  ( E. x  e.  A  y  C_  x  <->  E. x  e.  A  S  C_  x
) )
109rspccv 2894 . . 3  |-  ( A. y  e.  B  E. x  e.  A  y  C_  x  ->  ( S  e.  B  ->  E. x  e.  A  S  C_  x
) )
117, 10syl 15 . 2  |-  ( A Ref B  ->  ( S  e.  B  ->  E. x  e.  A  S  C_  x ) )
1211imp 418 1  |-  ( ( A Ref B  /\  S  e.  B )  ->  E. x  e.  A  S  C_  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   U.cuni 3843   class class class wbr 4039   Refcref 26363
This theorem is referenced by:  reftr  26392  refssfne  26397
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  ax-un 4528
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-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-ref 26367
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