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Theorem ofreq 6081
Description: Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Assertion
Ref Expression
ofreq  |-  ( R  =  S  ->  o R R  =  o R S )

Proof of Theorem ofreq
Dummy variables  f 
g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 4025 . . . 4  |-  ( R  =  S  ->  (
( f `  x
) R ( g `
 x )  <->  ( f `  x ) S ( g `  x ) ) )
21ralbidv 2563 . . 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 4082 . 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 6079 . 2  |-  o R R  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) R ( g `
 x ) }
5 df-ofr 6079 . 2  |-  o R S  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) S ( g `
 x ) }
63, 4, 53eqtr4g 2340 1  |-  ( R  =  S  ->  o R R  =  o R S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   A.wral 2543    i^i cin 3151   class class class wbr 4023   {copab 4076   dom cdm 4689   ` cfv 5255    o Rcofr 6077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-br 4024  df-opab 4078  df-ofr 6079
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