Theorem brab1 2096

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

Assertion

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

Distinct variable group(s):   x,z   y,z   z,R

Proof of Theorem brab1

Step	Hyp	Ref	Expression
1		visset 1350	. . 3 |- x e. V
2		breq1 2065	. . 3 |- (z = x -> (zRy <-> xRy))
3	1, 2	elab 1415	. 2 |- (x e. {z | zRy} <-> xRy)
4	3	bicomi 150	1 |- (xRy <-> x e. {z | zRy})

Syntax hints:   <-> wb 127  {cab 1090   e. wcel 1092   class class class wbr 2054

This theorem is referenced by:  fr2nr 2177  fr3nr 2178

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063

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