Theorem ofreq 6296

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 4127	. . . 4 |- ( R = S -> ( ( f ` x ) R ( g ` x ) <-> ( f ` x ) S ( g ` x ) ) )
2	1	ralbidv 2550	. . 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 4192	. 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 6293	. 2 |- oR R = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) }
5		df-ofr 6293	. 2 |- oR S = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) S ( g ` x ) }
6	3, 4, 5	3eqtr4g 2296	1 |- ( R = S -> oR R = oR S )

Syntax hints:   -> wi 4   = wceq 1402   A.wral 2528   i^i cin 3219   class class class wbr 4125   {copab 4186   dom cdm 4769   ` cfv 5372   oRcofr 6291

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-br 4126  df-opab 4188  df-ofr 6293

This theorem is referenced by: (None)