Theorem ofeq 6133

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 999	. . . . 5 |- ( ( R = S /\ f e. _V /\ g e. _V ) -> R = S )
2	1	oveqd 5935	. . . 4 |- ( ( R = S /\ f e. _V /\ g e. _V ) -> ( ( f ` x ) R ( g ` x ) ) = ( ( f ` x ) S ( g ` x ) ) )
3	2	mpteq2dv 4120	. . 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 5982	. 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 6130	. 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 6130	. 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 2251	1 |- ( R = S -> oF R = oF S )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2164   _Vcvv 2760   i^i cin 3152   |-> cmpt 4090   dom cdm 4659   ` cfv 5254  (class class class)co 5918   e. cmpo 5920   oFcof 6128

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-iota 5215  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-of 6130

This theorem is referenced by: (None)