Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ofceq Structured version   Unicode version

Theorem ofceq 27733
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 6288 . . . . 5  |-  ( R  =  S  ->  (
( f `  x
) R c )  =  ( ( f `
 x ) S c ) )
21mpteq2dv 4534 . . . 4  |-  ( R  =  S  ->  (
x  e.  dom  f  |->  ( ( f `  x ) R c ) )  =  ( x  e.  dom  f  |->  ( ( f `  x ) S c ) ) )
323ad2ant1 1017 . . 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 6343 . 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 27732 . 2  |-𝑓/𝑐 R  =  ( f  e.  _V ,  c  e. 
_V  |->  ( x  e. 
dom  f  |->  ( ( f `  x ) R c ) ) )
6 df-ofc 27732 . 2  |-𝑓/𝑐 S  =  ( f  e.  _V ,  c  e. 
_V  |->  ( x  e. 
dom  f  |->  ( ( f `  x ) S c ) ) )
74, 5, 63eqtr4g 2533 1  |-  ( R  =  S  ->𝑓/𝑐 R  =𝑓/𝑐 S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113    |-> cmpt 4505   dom cdm 4999   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284  ∘𝑓/𝑐cofc 27731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-iota 5549  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-ofc 27732
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator