Theorem ofreq 6089

Description: Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Assertion

Ref	Expression
ofreq	|- ( R = S -> oR R = oR S )

Proof of Theorem ofreq

Dummy variables f g x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq 4007	. . . 4 |- ( R = S -> ( ( f ` x ) R ( g ` x ) <-> ( f ` x ) S ( g ` x ) ) )
2	1	ralbidv 2477	. . 3 |- ( R = S -> ( A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) <-> A. x e. ( dom f i^i dom g ) ( f ` x ) S ( g ` x ) ) )
3	2	opabbidv 4071	. 2 |- ( R = S -> { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) } = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) S ( g ` x ) } )
4		df-ofr 6087	. 2 |- oR R = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) }
5		df-ofr 6087	. 2 |- oR S = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) S ( g ` x ) }
6	3, 4, 5	3eqtr4g 2235	1 |- ( R = S -> oR R = oR S )

Syntax hints:   -> wi 4   = wceq 1353   A.wral 2455   i^i cin 3130   class class class wbr 4005   {copab 4065   dom cdm 4628   ` cfv 5218   oRcofr 6085

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-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-br 4006  df-opab 4067  df-ofr 6087

This theorem is referenced by: (None)