Theorem class2seteq 3472

Description: Equality theorem based on class2set 3471. (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 2299	. 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 2175	. . . 4 |- (A e. _V -> A.x e. A A e. _V)
4		rabid2 2254	. . . 4 |- (A = {x e. A | A e. _V} <-> A.x e. A A e. _V)
5	3, 4	sylibr 217	. . 3 |- (A e. _V -> A = {x e. A | A e. _V})
6	5	eqcomd 1889	. 2 |- (A e. _V -> {x e. A | A e. _V} = A)
7	1, 6	syl 12	1 |- (A e. B -> {x e. A | A e. _V} = A)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108  _Vcvv 2292

This theorem is referenced by:  fsum1s 8269  fsump1s 8273  fprod1s 14677  fprodp1s 14682

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rab 2112  df-v 2294

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