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Theorem refbas 20537
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 20535 . . 3  |-  Rel  Ref
21brrelexi 4878 . 2  |-  ( A Ref B  ->  A  e.  _V )
3 refbas.1 . . . 4  |-  X  = 
U. A
4 refbas.2 . . . 4  |-  Y  = 
U. B
53, 4isref 20536 . . 3  |-  ( A  e.  _V  ->  ( A Ref B  <->  ( Y  =  X  /\  A. x  e.  A  E. y  e.  B  x  C_  y
) ) )
65simprbda 629 . 2  |-  ( ( A  e.  _V  /\  A Ref B )  ->  Y  =  X )
72, 6mpancom 676 1  |-  ( A Ref B  ->  Y  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = 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  locfinreflem  28679  cmpcref  28689  cmppcmp  28697  refssfne  31026
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