MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ofreq Structured version   Unicode version

Theorem ofreq 6542
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 4458 . . . 4  |-  ( R  =  S  ->  (
( f `  x
) R ( g `
 x )  <->  ( f `  x ) S ( g `  x ) ) )
21ralbidv 2896 . . 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 4520 . 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 6540 . 2  |-  oR R  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) R ( g `
 x ) }
5 df-ofr 6540 . 2  |-  oR S  =  { <. f ,  g >.  |  A. x  e.  ( dom  f  i^i  dom  g )
( f `  x
) S ( g `
 x ) }
63, 4, 53eqtr4g 2523 1  |-  ( R  =  S  ->  oR R  =  oR S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395   A.wral 2807    i^i cin 3470   class class class wbr 4456   {copab 4514   dom cdm 5008   ` cfv 5594    oRcofr 6538
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-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-ral 2812  df-br 4457  df-opab 4516  df-ofr 6540
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator