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Theorem refbas 28695
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 28693 . . 3  |-  Rel  Ref
21brrelex2i 4983 . 2  |-  ( A Ref B  ->  B  e.  _V )
3 refbas.1 . . . 4  |-  X  = 
U. A
4 refbas.2 . . . 4  |-  Y  = 
U. B
53, 4isref 28694 . . 3  |-  ( B  e.  _V  ->  ( A Ref B  <->  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y
) ) )
65simprbda 623 . 2  |-  ( ( B  e.  _V  /\  A Ref B )  ->  X  =  Y )
72, 6mpancom 669 1  |-  ( A Ref B  ->  X  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   A.wral 2796   E.wrex 2797   _Vcvv 3072    C_ wss 3431   U.cuni 4194   class class class wbr 4395   Refcref 28675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-xp 4949  df-rel 4950  df-ref 28679
This theorem is referenced by:  reftr  28704  refssfne  28709
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