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Theorem ofceq 26685
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 6207 . . . . 5  |-  ( R  =  S  ->  (
( f `  x
) R c )  =  ( ( f `
 x ) S c ) )
21mpteq2dv 4488 . . . 4  |-  ( R  =  S  ->  (
x  e.  dom  f  |->  ( ( f `  x ) R c ) )  =  ( x  e.  dom  f  |->  ( ( f `  x ) S c ) ) )
323ad2ant1 1009 . . 3  |-  ( ( R  =  S  /\  f  e.  _V  /\  c  e.  _V )  ->  (
x  e.  dom  f  |->  ( ( f `  x ) R c ) )  =  ( x  e.  dom  f  |->  ( ( f `  x ) S c ) ) )
43mpt2eq3dva 6260 . 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 ) ) ) )
5 df-ofc 26684 . 2  |-𝑓/𝑐 R  =  ( f  e.  _V ,  c  e. 
_V  |->  ( x  e. 
dom  f  |->  ( ( f `  x ) R c ) ) )
6 df-ofc 26684 . 2  |-𝑓/𝑐 S  =  ( f  e.  _V ,  c  e. 
_V  |->  ( x  e. 
dom  f  |->  ( ( f `  x ) S c ) ) )
74, 5, 63eqtr4g 2520 1  |-  ( R  =  S  ->𝑓/𝑐 R  =𝑓/𝑐 S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078    |-> cmpt 4459   dom cdm 4949   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203  ∘𝑓/𝑐cofc 26683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-iota 5490  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-ofc 26684
This theorem is referenced by: (None)
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