Theorem eldm 3307

Description: Membership in a domain. Theorem 4 of [Suppes] p. 59.

Hypothesis

Ref	Expression
eldm.1	|- A e. V

Assertion

Ref	Expression
eldm	|- (A e. dom B <-> E.y ABy)

Distinct variable groups:   y,A   y,B

Proof of Theorem eldm

Step	Hyp	Ref	Expression
1		eldm.1	. 2 |- A e. V
2		breq1 2622	. . 3 |- (x = A -> (xBy <-> ABy))
3	2	exbidv 1279	. 2 |- (x = A -> (E.y xBy <-> E.y ABy))
4		df-dm 3188	. 2 |- dom B = {x | E.y xBy}
5	1, 3, 4	elab2 1901	1 |- (A e. dom B <-> E.y ABy)

Syntax hints:   <-> wb 146   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811   class class class wbr 2619  dom cdm 3170

This theorem is referenced by:  eldm2 3308  dmcosseq 3365  dminss 3462  dffun6 3539  fneu 3592  ndmfv 3745  dff2 3817  cbvfo 3885  erref 4275  erdmrn 4276  ecdmn0 4280  aceq3lem 4732

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-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-dm 3188

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