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Theorem refssex 20538
Description: Every set in a refinement has a superset in the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Assertion
Ref Expression
refssex  |-  ( ( A Ref B  /\  S  e.  A )  ->  E. x  e.  B  S  C_  x )
Distinct variable groups:    x, B    x, S
Allowed substitution hint:    A( x)

Proof of Theorem refssex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 refrel 20535 . . . . 5  |-  Rel  Ref
21brrelexi 4878 . . . 4  |-  ( A Ref B  ->  A  e.  _V )
3 eqid 2453 . . . . . 6  |-  U. A  =  U. A
4 eqid 2453 . . . . . 6  |-  U. B  =  U. B
53, 4isref 20536 . . . . 5  |-  ( A  e.  _V  ->  ( A Ref B  <->  ( U. B  =  U. A  /\  A. y  e.  A  E. x  e.  B  y  C_  x ) ) )
65simplbda 630 . . . 4  |-  ( ( A  e.  _V  /\  A Ref B )  ->  A. y  e.  A  E. x  e.  B  y  C_  x )
72, 6mpancom 676 . . 3  |-  ( A Ref B  ->  A. y  e.  A  E. x  e.  B  y  C_  x )
8 sseq1 3455 . . . . 5  |-  ( y  =  S  ->  (
y  C_  x  <->  S  C_  x
) )
98rexbidv 2903 . . . 4  |-  ( y  =  S  ->  ( E. x  e.  B  y  C_  x  <->  E. x  e.  B  S  C_  x
) )
109rspccv 3149 . . 3  |-  ( A. y  e.  A  E. x  e.  B  y  C_  x  ->  ( S  e.  A  ->  E. x  e.  B  S  C_  x
) )
117, 10syl 17 . 2  |-  ( A Ref B  ->  ( S  e.  A  ->  E. x  e.  B  S  C_  x ) )
1211imp 431 1  |-  ( ( A Ref B  /\  S  e.  A )  ->  E. x  e.  B  S  C_  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1446    e. wcel 1889   A.wral 2739   E.wrex 2740   _Vcvv 3047    C_ wss 3406   U.cuni 4201   class class class wbr 4405   Refcref 20529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-xp 4843  df-rel 4844  df-ref 20532
This theorem is referenced by:  reftr  20541  refun0  20542  refssfne  31026
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