Theorem ofreq 6139

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 4035	. . . 4 |- ( R = S -> ( ( f ` x ) R ( g ` x ) <-> ( f ` x ) S ( g ` x ) ) )
2	1	ralbidv 2497	. . 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 4099	. 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 6136	. 2 |- oR R = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) }
5		df-ofr 6136	. 2 |- oR S = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) S ( g ` x ) }
6	3, 4, 5	3eqtr4g 2254	1 |- ( R = S -> oR R = oR S )

Syntax hints:   -> wi 4   = wceq 1364   A.wral 2475   i^i cin 3156   class class class wbr 4033   {copab 4093   dom cdm 4663   ` cfv 5258   oRcofr 6134

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-br 4034  df-opab 4095  df-ofr 6136

This theorem is referenced by: (None)