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Theorem refbas 26351
Description: A refinement covers the same set. (Contributed by Jeff Hankins, 18-Jan-2010.)
Hypotheses
Ref Expression
refbas.1  |-  X  = 
U. A
refbas.2  |-  Y  = 
U. B
Assertion
Ref Expression
refbas  |-  ( A Ref B  ->  X  =  Y )

Proof of Theorem refbas
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 refrel 26349 . . 3  |-  Rel  Ref
21brrelex2i 4911 . 2  |-  ( A Ref B  ->  B  e.  _V )
3 refbas.1 . . . 4  |-  X  = 
U. A
4 refbas.2 . . . 4  |-  Y  = 
U. B
53, 4isref 26350 . . 3  |-  ( B  e.  _V  ->  ( A Ref B  <->  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y
) ) )
65simprbda 607 . 2  |-  ( ( B  e.  _V  /\  A Ref B )  ->  X  =  Y )
72, 6mpancom 651 1  |-  ( A Ref B  ->  X  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    C_ wss 3312   U.cuni 4007   class class class wbr 4204   Refcref 26331
This theorem is referenced by:  reftr  26360  refssfne  26365
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-ref 26335
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