Theorem ofeq 6271

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.

Step	Hyp	Ref	Expression
1		simp1 1024	. . . . 5 |- ( ( R = S /\ f e. _V /\ g e. _V ) -> R = S )
2	1	oveqd 6069	. . . 4 |- ( ( R = S /\ f e. _V /\ g e. _V ) -> ( ( f ` x ) R ( g ` x ) ) = ( ( f ` x ) S ( g ` x ) ) )
3	2	mpteq2dv 4203	. . 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 ) ) ) )
4	3	mpoeq3dva 6119	. 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 6268	. 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 6268	. 2 |- oF S = ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) S ( g ` x ) ) ) )
7	4, 5, 6	3eqtr4g 2292	1 |- ( R = S -> oF R = oF S )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2205   _Vcvv 2815   i^i cin 3212   |-> cmpt 4173   dom cdm 4751   ` cfv 5354  (class class class)co 6052   e. cmpo 6054   oFcof 6266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-iota 5314  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-of 6268

This theorem is referenced by: (None)