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Theorem ofeq 5742
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 915 . . . . 5  |-  ( ( R  =  S  /\  f  e.  _V  /\  g  e.  _V )  ->  R  =  S )
21oveqd 5557 . . . 4  |-  ( ( R  =  S  /\  f  e.  _V  /\  g  e.  _V )  ->  (
( f `  x
) R ( g `
 x ) )  =  ( ( f `
 x ) S ( g `  x
) ) )
32mpteq2dv 3876 . . 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 5597 . 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 5740 . 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 5740 . 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 2113 1  |-  ( R  =  S  ->  oF R  =  oF S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 896    = wceq 1259    e. wcel 1409   _Vcvv 2574    i^i cin 2944    |-> cmpt 3846   dom cdm 4373   ` cfv 4930  (class class class)co 5540    |-> cmpt2 5542    oFcof 5738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-iota 4895  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-of 5740
This theorem is referenced by: (None)
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