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Theorem refbas 20137
Description: A refinement covers the same set. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Hypotheses
Ref Expression
refbas.1  |-  X  = 
U. A
refbas.2  |-  Y  = 
U. B
Assertion
Ref Expression
refbas  |-  ( A Ref B  ->  Y  =  X )

Proof of Theorem refbas
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 refrel 20135 . . 3  |-  Rel  Ref
21brrelexi 5049 . 2  |-  ( A Ref B  ->  A  e.  _V )
3 refbas.1 . . . 4  |-  X  = 
U. A
4 refbas.2 . . . 4  |-  Y  = 
U. B
53, 4isref 20136 . . 3  |-  ( A  e.  _V  ->  ( A Ref B  <->  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y
) ) )
65simprbda 623 . 2  |-  ( ( A  e.  _V  /\  A Ref B )  ->  Y  =  X )
72, 6mpancom 669 1  |-  ( A Ref B  ->  Y  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109    C_ wss 3471   U.cuni 4251   class class class wbr 4456   Refcref 20129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-ref 20132
This theorem is referenced by:  reftr  20141  refun0  20142  locfinreflem  28004  cmpcref  28014  cmppcmp  28022  refssfne  30381
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