Theorem brab1 4064

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 2754	. . 3 |- x e. _V
2		breq1 4020	. . . 4 |- ( z = y -> ( z R A <-> y R A ) )
3		breq1 4020	. . . 4 |- ( y = x -> ( y R A <-> x R A ) )
4	2, 3	sbcie2g 3010	. . 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 2977	. 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 2159   {cab 2174   _Vcvv 2751   [.wsbc 2976   class class class wbr 4017

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-v 2753  df-sbc 2977  df-un 3147  df-sn 3612  df-pr 3613  df-op 3615  df-br 4018

This theorem is referenced by: (None)