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Theorem ofceq 28544
Description: Equality theorem for function/constant operation. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Assertion
Ref Expression
ofceq  |-  ( R  =  S  ->𝑓/𝑐 R  =𝑓/𝑐 S )

Proof of Theorem ofceq
Dummy variables  f 
c  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq 6284 . . . 4  |-  ( R  =  S  ->  (
( f `  x
) R c )  =  ( ( f `
 x ) S c ) )
21mpteq2dv 4482 . . 3  |-  ( R  =  S  ->  (
x  e.  dom  f  |->  ( ( f `  x ) R c ) )  =  ( x  e.  dom  f  |->  ( ( f `  x ) S c ) ) )
32mpt2eq3dv 6344 . 2  |-  ( R  =  S  ->  (
f  e.  _V , 
c  e.  _V  |->  ( x  e.  dom  f  |->  ( ( f `  x ) R c ) ) )  =  ( f  e.  _V ,  c  e.  _V  |->  ( x  e.  dom  f  |->  ( ( f `
 x ) S c ) ) ) )
4 df-ofc 28543 . 2  |-𝑓/𝑐 R  =  ( f  e.  _V ,  c  e. 
_V  |->  ( x  e. 
dom  f  |->  ( ( f `  x ) R c ) ) )
5 df-ofc 28543 . 2  |-𝑓/𝑐 S  =  ( f  e.  _V ,  c  e. 
_V  |->  ( x  e. 
dom  f  |->  ( ( f `  x ) S c ) ) )
63, 4, 53eqtr4g 2468 1  |-  ( R  =  S  ->𝑓/𝑐 R  =𝑓/𝑐 S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405   _Vcvv 3059    |-> cmpt 4453   dom cdm 4823   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280  ∘𝑓/𝑐cofc 28542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-iota 5533  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-ofc 28543
This theorem is referenced by: (None)
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