Theorem ofceq 30159

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.

Step	Hyp	Ref	Expression
1		oveq 6656	. . . 4 |- ( R = S -> ( ( f ` x ) R c ) = ( ( f ` x ) S c ) )
2	1	mpteq2dv 4745	. . 3 |- ( R = S -> ( x e. dom f |-> ( ( f ` x ) R c ) ) = ( x e. dom f |-> ( ( f ` x ) S c ) ) )
3	2	mpt2eq3dv 6721	. 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 30158	. 2 |- ∘𝑓/𝑐 R = ( f e. _V , c e. _V |-> ( x e. dom f |-> ( ( f ` x ) R c ) ) )
5		df-ofc 30158	. 2 |- ∘𝑓/𝑐 S = ( f e. _V , c e. _V |-> ( x e. dom f |-> ( ( f ` x ) S c ) ) )
6	3, 4, 5	3eqtr4g 2681	1 |- ( R = S -> ∘𝑓/𝑐 R = ∘𝑓/𝑐 S )

Syntax hints:   -> wi 4   = wceq 1483   _Vcvv 3200   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652  ∘_𝑓/𝑐cofc 30157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-ofc 30158

This theorem is referenced by: (None)