Theorem class2seteq 2703

Description: Equality theorem based on class2set 2702. (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 1792	. 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 1689	. . . 4 |- (A e. V -> A.x e. A A e. V)
4		rabid2 1746	. . . 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 1456	. 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 1099   e. wcel 1105  A.wral 1621  {crab 1624  Vcvv 1786

This theorem is referenced by:  fsum1s 6898  fsump1s 6902

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-ral 1625  df-rab 1628  df-v 1787

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