Theorem eldm 2527

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 group(s):   y,A   y,B

Proof of Theorem eldm

Step	Hyp	Ref	Expression
1		eldm.1	. 2 |- A e. V
2		breq1 2065	. . 3 |- (x = A -> (xBy <-> ABy))
3	2	biexdv 936	. 2 |- (x = A -> (E.y xBy <-> E.y ABy))
4		df-dm 2428	. 2 |- dom B = {x | E.y xBy}
5	1, 3, 4	elab2 1419	1 |- (A e. dom B <-> E.y ABy)

Syntax hints:   <-> wb 127  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348   class class class wbr 2054  dom cdm 2410

This theorem is referenced by:  eldm2 2528  dmcosseq 2572  dminss 2648  dffun6 2687  fneu 2728  ndmfv 2848  cbvfo 2923  erref 3212  erdmrn 3213  ecdmn0 3217  aceq3lem 3555

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-dm 2428

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