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Theorem ofeq 6080
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 955 . . . . 5  |-  ( ( R  =  S  /\  f  e.  _V  /\  g  e.  _V )  ->  R  =  S )
21oveqd 5875 . . . 4  |-  ( ( R  =  S  /\  f  e.  _V  /\  g  e.  _V )  ->  (
( f `  x
) R ( g `
 x ) )  =  ( ( f `
 x ) S ( g `  x
) ) )
32mpteq2dv 4107 . . 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 5912 . 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 6078 . 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 6078 . 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 2340 1  |-  ( R  =  S  ->  o F R  =  o F S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    e. cmpt 4077   dom cdm 4689   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    o Fcof 6076
This theorem is referenced by:  psrval  16110  resspsradd  16160  resspsrvsca  16162  mendval  27491  mendplusgfval  27493  mendvscafval  27498  ldualset  29315
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-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-iota 5219  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078
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