Theorem class2seteq 4196

Description: Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   V( x)

Proof of Theorem class2seteq

Step	Hyp	Ref	Expression
1		elex 2774	. 2 |- ( A e. V -> A e. _V )
2		ax-1 6	. . . . 5 |- ( A e. _V -> ( x e. A -> A e. _V ) )
3	2	ralrimiv 2569	. . . 4 |- ( A e. _V -> A. x e. A A e. _V )
4		rabid2 2674	. . . 4 |- ( A = { x e. A | A e. _V } <-> A. x e. A A e. _V )
5	3, 4	sylibr 134	. . 3 |- ( A e. _V -> A = { x e. A | A e. _V } )
6	5	eqcomd 2202	. 2 |- ( A e. _V -> { x e. A | A e. _V } = A )
7	1, 6	syl 14	1 |- ( A e. V -> { x e. A | A e. _V } = A )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   A.wral 2475   {crab 2479   _Vcvv 2763

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-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-rab 2484  df-v 2765

This theorem is referenced by: (None)