Theorem brab1 3970

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 2684	. . 3 |- x e. _V
2		breq1 3927	. . . 4 |- ( z = y -> ( z R A <-> y R A ) )
3		breq1 3927	. . . 4 |- ( y = x -> ( y R A <-> x R A ) )
4	2, 3	sbcie2g 2937	. . 3 |- ( x e. _V -> ( [. x / z ]. z R A <-> x R A ) )
5	1, 4	ax-mp 5	. 2 |- ( [. x / z ]. z R A <-> x R A )
6		df-sbc 2905	. 2 |- ( [. x / z ]. z R A <-> x e. { z | z R A } )
7	5, 6	bitr3i 185	1 |- ( x R A <-> x e. { z | z R A } )

Syntax hints:   <-> wb 104   e. wcel 1480   {cab 2123   _Vcvv 2681   [.wsbc 2904   class class class wbr 3924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925

This theorem is referenced by: (None)