Theorem class2seteq 2735

Description: Equality theorem based on class2set 2734. (The proof was shortened by Raph Levien, 30-Jun-2006.)

Assertion

Ref	Expression
class2seteq	|- (A e. B -> {x e. A | A e. V} = A)

Distinct variable group:   x,A

Proof of Theorem class2seteq

Step	Hyp	Ref	Expression
1		elisset 1817	. 2 |- (A e. B -> A e. V)
2		ax-1 4	. . . . 5 |- (A e. V -> (x e. A -> A e. V))
3	2	r19.21aiv 1713	. . . 4 |- (A e. V -> A.x e. A A e. V)
4		rabid2 1770	. . . 4 |- (A = {x e. A | A e. V} <-> A.x e. A A e. V)
5	3, 4	sylibr 200	. . 3 |- (A e. V -> A = {x e. A | A e. V})
6	5	eqcomd 1480	. 2 |- (A e. V -> {x e. A | A e. V} = A)
7	1, 6	syl 10	1 |- (A e. B -> {x e. A | A e. V} = A)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  A.wral 1645  {crab 1648  Vcvv 1811

This theorem is referenced by:  fsum1s 7009  fsump1s 7013

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rab 1652  df-v 1812

Copyright terms: Public domain