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Theorem refssex 26353
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 26350 . . . . 5  |-  Rel  Ref
21brrelex2i 4912 . . . 4  |-  ( A Ref B  ->  B  e.  _V )
3 eqid 2436 . . . . . 6  |-  U. A  =  U. A
4 eqid 2436 . . . . . 6  |-  U. B  =  U. B
53, 4isref 26351 . . . . 5  |-  ( B  e.  _V  ->  ( A Ref B  <->  ( U. A  =  U. B  /\  A. y  e.  B  E. x  e.  A  y  C_  x ) ) )
65simplbda 608 . . . 4  |-  ( ( B  e.  _V  /\  A Ref B )  ->  A. y  e.  B  E. x  e.  A  y  C_  x )
72, 6mpancom 651 . . 3  |-  ( A Ref B  ->  A. y  e.  B  E. x  e.  A  y  C_  x )
8 sseq1 3362 . . . . 5  |-  ( y  =  S  ->  (
y  C_  x  <->  S  C_  x
) )
98rexbidv 2719 . . . 4  |-  ( y  =  S  ->  ( E. x  e.  A  y  C_  x  <->  E. x  e.  A  S  C_  x
) )
109rspccv 3042 . . 3  |-  ( A. y  e.  B  E. x  e.  A  y  C_  x  ->  ( S  e.  B  ->  E. x  e.  A  S  C_  x
) )
117, 10syl 16 . 2  |-  ( A Ref B  ->  ( S  e.  B  ->  E. x  e.  A  S  C_  x ) )
1211imp 419 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 359    = wceq 1652    e. wcel 1725   A.wral 2698   E.wrex 2699   _Vcvv 2949    C_ wss 3313   U.cuni 4008   class class class wbr 4205   Refcref 26332
This theorem is referenced by:  reftr  26361  refssfne  26366
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 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-xp 4877  df-rel 4878  df-ref 26336
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