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Theorem ofeq 6307
Description: Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
Assertion
Ref Expression
ofeq  |-  ( R  =  S  ->  o F R  =  o F S )

Proof of Theorem ofeq
Dummy variables  f 
g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 957 . . . . 5  |-  ( ( R  =  S  /\  f  e.  _V  /\  g  e.  _V )  ->  R  =  S )
21oveqd 6098 . . . 4  |-  ( ( R  =  S  /\  f  e.  _V  /\  g  e.  _V )  ->  (
( f `  x
) R ( g `
 x ) )  =  ( ( f `
 x ) S ( g `  x
) ) )
32mpteq2dv 4296 . . 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 6138 . 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 6305 . 2  |-  o F R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
6 df-of 6305 . 2  |-  o F S  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) S ( g `
 x ) ) ) )
74, 5, 63eqtr4g 2493 1  |-  ( R  =  S  ->  o F R  =  o F S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319    e. cmpt 4266   dom cdm 4878   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083    o Fcof 6303
This theorem is referenced by:  psrval  16429  resspsradd  16479  resspsrvsca  16481  sitmval  24661  mendval  27468  mendplusgfval  27470  mendvscafval  27475  ldualset  29923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-iota 5418  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305
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