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Theorem refssex 25693
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 25690 . . . . 5  |-  Rel  Ref
21brrelex2i 4730 . . . 4  |-  ( A Ref B  ->  B  e.  _V )
3 eqid 2283 . . . . . 6  |-  U. A  =  U. A
4 eqid 2283 . . . . . 6  |-  U. B  =  U. B
53, 4isref 25691 . . . . 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 3199 . . . . 5  |-  ( y  =  S  ->  (
y  C_  x  <->  S  C_  x
) )
98rexbidv 2564 . . . 4  |-  ( y  =  S  ->  ( E. x  e.  A  y  C_  x  <->  E. x  e.  A  S  C_  x
) )
109rspccv 2881 . . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   U.cuni 3827   class class class wbr 4023   Refcref 25672
This theorem is referenced by:  reftr  25701  refssfne  25706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-ref 25676
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