Theorem eldmg 4780

Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
eldmg	|- ( A e. V -> ( A e. dom B <-> E. y A B y ) )

Distinct variable groups:   y, A   y, B

Allowed substitution hint:   V( y)

Proof of Theorem eldmg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 3968	. . 3 |- ( x = A -> ( x B y <-> A B y ) )
2	1	exbidv 1805	. 2 |- ( x = A -> ( E. y x B y <-> E. y A B y ) )
3		df-dm 4595	. 2 |- dom B = { x | E. y x B y }
4	2, 3	elab2g 2859	1 |- ( A e. V -> ( A e. dom B <-> E. y A B y ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1335   E.wex 1472   e. wcel 2128   class class class wbr 3965   dom cdm 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-dm 4595

This theorem is referenced by:  eldm2g  4781  eldm  4782  breldmg  4791  releldmb  4822  funeu  5194  fneu  5273  ndmfvg  5498  erref  6497  ecdmn0m  6519  shftdm  10715  dvcnp2cntop  13034