Theorem ofreq 5715

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 3766	. . . 4 |- ( R = S -> ( ( f ` x ) R ( g ` x ) <-> ( f ` x ) S ( g ` x ) ) )
2	1	ralbidv 2326	. . 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 3823	. 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 5713	. 2 |- oR R = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) }
5		df-ofr 5713	. 2 |- oR S = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) S ( g ` x ) }
6	3, 4, 5	3eqtr4g 2097	1 |- ( R = S -> oR R = oR S )

Syntax hints:   -> wi 4   = wceq 1243   A.wral 2306   i^i cin 2916   class class class wbr 3764   {copab 3817   dom cdm 4345   ` cfv 4902   oRcofr 5711

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-br 3765  df-opab 3819  df-ofr 5713

This theorem is referenced by: (None)