Theorem brab1 4045

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 2738	. . 3 |- x e. _V
2		breq1 4001	. . . 4 |- ( z = y -> ( z R A <-> y R A ) )
3		breq1 4001	. . . 4 |- ( y = x -> ( y R A <-> x R A ) )
4	2, 3	sbcie2g 2994	. . 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 2961	. 2 |- ( [. x / z ]. z R A <-> x e. { z | z R A } )
7	5, 6	bitr3i 186	1 |- ( x R A <-> x e. { z | z R A } )

Syntax hints:   <-> wb 105   e. wcel 2146   {cab 2161   _Vcvv 2735   [.wsbc 2960   class class class wbr 3998

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-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-sbc 2961  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999

This theorem is referenced by: (None)