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Theorem ofeq 6541
Description: Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
Assertion
Ref Expression
ofeq  |-  ( R  =  S  ->  oF R  =  oF S )

Proof of Theorem ofeq
Dummy variables  f 
g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 996 . . . . 5  |-  ( ( R  =  S  /\  f  e.  _V  /\  g  e.  _V )  ->  R  =  S )
21oveqd 6313 . . . 4  |-  ( ( R  =  S  /\  f  e.  _V  /\  g  e.  _V )  ->  (
( f `  x
) R ( g `
 x ) )  =  ( ( f `
 x ) S ( g `  x
) ) )
32mpteq2dv 4544 . . 3  |-  ( ( R  =  S  /\  f  e.  _V  /\  g  e.  _V )  ->  (
x  e.  ( dom  f  i^i  dom  g
)  |->  ( ( f `
 x ) R ( g `  x
) ) )  =  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x ) S ( g `  x ) ) ) )
43mpt2eq3dva 6360 . 2  |-  ( R  =  S  ->  (
f  e.  _V , 
g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g
)  |->  ( ( f `
 x ) R ( g `  x
) ) ) )  =  ( f  e. 
_V ,  g  e. 
_V  |->  ( x  e.  ( dom  f  i^i 
dom  g )  |->  ( ( f `  x
) S ( g `
 x ) ) ) ) )
5 df-of 6539 . 2  |-  oF R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
6 df-of 6539 . 2  |-  oF S  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) S ( g `
 x ) ) ) )
74, 5, 63eqtr4g 2523 1  |-  ( R  =  S  ->  oF R  =  oF S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470    |-> cmpt 4515   dom cdm 5008   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298    oFcof 6537
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-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-iota 5557  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539
This theorem is referenced by:  psrval  18137  resspsradd  18197  resspsrvsca  18199  sitmval  28465  mendval  31294  mendplusgfval  31296  mendvscafval  31301  ldualset  34951
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