Theorem brab1 3830

Description: Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)

Assertion

Ref	Expression
brab1	|- ( x R A <-> x e. { z | z R A } )

Distinct variable groups:   z, A   z, R

Allowed substitution hints:   A( x)   R( x)

Proof of Theorem brab1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . 3 |- x e. _V
2		breq1 3788	. . . 4 |- ( z = y -> ( z R A <-> y R A ) )
3		breq1 3788	. . . 4 |- ( y = x -> ( y R A <-> x R A ) )
4	2, 3	sbcie2g 2847	. . 3 |- ( x e. _V -> ( [. x / z ]. z R A <-> x R A ) )
5	1, 4	ax-mp 7	. 2 |- ( [. x / z ]. z R A <-> x R A )
6		df-sbc 2816	. 2 |- ( [. x / z ]. z R A <-> x e. { z | z R A } )
7	5, 6	bitr3i 184	1 |- ( x R A <-> x e. { z | z R A } )

Syntax hints:   <-> wb 103   e. wcel 1433   {cab 2067   _Vcvv 2601   [.wsbc 2815   class class class wbr 3785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786

This theorem is referenced by: (None)