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Theorem ofreq 5746
Description: Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Assertion
Ref Expression
ofreq  |-  ( R  =  S  ->  oR R  =  oR S )

Proof of Theorem ofreq
Dummy variables  f  g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 3795 . . . 4  |-  ( R  =  S  ->  (
( f `  x
) R ( g `
 x )  <->  ( f `  x ) S ( g `  x ) ) )
21ralbidv 2369 . . 3  |-  ( R  =  S  ->  ( A. x  e.  ( dom  f  i^i  dom  g
) ( f `  x ) R ( g `  x )  <->  A. x  e.  ( dom  f  i^i  dom  g
) ( f `  x ) S ( g `  x ) ) )
32opabbidv 3852 . 2  |-  ( R  =  S  ->  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) R ( g `
 x ) }  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) S ( g `
 x ) } )
4 df-ofr 5744 . 2  |-  oR R  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) R ( g `
 x ) }
5 df-ofr 5744 . 2  |-  oR S  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) S ( g `
 x ) }
63, 4, 53eqtr4g 2139 1  |-  ( R  =  S  ->  oR R  =  oR S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   A.wral 2349    i^i cin 2973   class class class wbr 3793   {copab 3846   dom cdm 4371   ` cfv 4932    oRcofr 5742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-br 3794  df-opab 3848  df-ofr 5744
This theorem is referenced by: (None)
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