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Theorem ofeq 6484
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 1005 . . . . 5  |-  ( ( R  =  S  /\  f  e.  _V  /\  g  e.  _V )  ->  R  =  S )
21oveqd 6259 . . . 4  |-  ( ( R  =  S  /\  f  e.  _V  /\  g  e.  _V )  ->  (
( f `  x
) R ( g `
 x ) )  =  ( ( f `
 x ) S ( g `  x
) ) )
32mpteq2dv 4447 . . 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 6306 . 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 6482 . 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 6482 . 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 2481 1  |-  ( R  =  S  ->  oF R  =  oF S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1872   _Vcvv 3016    i^i cin 3371    |-> cmpt 4418   dom cdm 4789   ` cfv 5537  (class class class)co 6242    |-> cmpt2 6244    oFcof 6480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402
This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ral 2713  df-rex 2714  df-uni 4156  df-br 4360  df-opab 4419  df-mpt 4420  df-iota 5501  df-fv 5545  df-ov 6245  df-oprab 6246  df-mpt2 6247  df-of 6482
This theorem is referenced by:  psrval  18522  resspsradd  18576  resspsrvsca  18578  sitmval  29127  ldualset  32597  mendval  35956  mendplusgfval  35958  mendvscafval  35963
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