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Theorem ofreq 5989
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 3935 . . . 4  |-  ( R  =  S  ->  (
( f `  x
) R ( g `
 x )  <->  ( f `  x ) S ( g `  x ) ) )
21ralbidv 2438 . . 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 3998 . 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 5987 . 2  |-  oR R  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) R ( g `
 x ) }
5 df-ofr 5987 . 2  |-  oR S  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) S ( g `
 x ) }
63, 4, 53eqtr4g 2198 1  |-  ( R  =  S  ->  oR R  =  oR S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332   A.wral 2417    i^i cin 3071   class class class wbr 3933   {copab 3992   dom cdm 4543   ` cfv 5127    oRcofr 5985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-br 3934  df-opab 3994  df-ofr 5987
This theorem is referenced by: (None)
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