Theorem ofeq 6163

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 1000	. . . . 5 |- ( ( R = S /\ f e. _V /\ g e. _V ) -> R = S )
2	1	oveqd 5963	. . . 4 |- ( ( R = S /\ f e. _V /\ g e. _V ) -> ( ( f ` x ) R ( g ` x ) ) = ( ( f ` x ) S ( g ` x ) ) )
3	2	mpteq2dv 4136	. . 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 6011	. 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 6160	. 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 6160	. 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 2263	1 |- ( R = S -> oF R = oF S )

Syntax hints:   -> wi 4   /\ w3a 981   = wceq 1373   e. wcel 2176   _Vcvv 2772   i^i cin 3165   |-> cmpt 4106   dom cdm 4676   ` cfv 5272  (class class class)co 5946   e. cmpo 5948   oFcof 6158

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-iota 5233  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-of 6160

This theorem is referenced by: (None)