Theorem brab1 4036

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 2733	. . 3 |- x e. _V
2		breq1 3992	. . . 4 |- ( z = y -> ( z R A <-> y R A ) )
3		breq1 3992	. . . 4 |- ( y = x -> ( y R A <-> x R A ) )
4	2, 3	sbcie2g 2988	. . 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 2956	. 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 2141   {cab 2156   _Vcvv 2730   [.wsbc 2955   class class class wbr 3989

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990

This theorem is referenced by: (None)