Theorem brab1 2665

Description: Relationship between a binary relation and a class abstraction.

Assertion

Ref	Expression
brab1	|- (xRy <-> x e. {z | zRy})

Distinct variable groups:   x,z   y,z   z,R

Proof of Theorem brab1

Step	Hyp	Ref	Expression
1		visset 1816	. . 3 |- x e. V
2		breq1 2627	. . 3 |- (z = x -> (zRy <-> xRy))
3	1, 2	elab 1900	. 2 |- (x e. {z | zRy} <-> xRy)
4	3	bicomi 172	1 |- (xRy <-> x e. {z | zRy})

Syntax hints:   <-> wb 146   e. wcel 960  {cab 1466   class class class wbr 2624

This theorem is referenced by:  fr2nr 2931  fr3nr 2932

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625

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