Theorem ofreq 6134

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 4031	. . . 4 |- ( R = S -> ( ( f ` x ) R ( g ` x ) <-> ( f ` x ) S ( g ` x ) ) )
2	1	ralbidv 2494	. . 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 4095	. 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 6131	. 2 |- oR R = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) }
5		df-ofr 6131	. 2 |- oR S = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) S ( g ` x ) }
6	3, 4, 5	3eqtr4g 2251	1 |- ( R = S -> oR R = oR S )

Syntax hints:   -> wi 4   = wceq 1364   A.wral 2472   i^i cin 3152   class class class wbr 4029   {copab 4089   dom cdm 4659   ` cfv 5254   oRcofr 6129

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  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-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-ral 2477  df-br 4030  df-opab 4091  df-ofr 6131

This theorem is referenced by: (None)