Theorem eldmg 4861

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 4036	. . 3 |- ( x = A -> ( x B y <-> A B y ) )
2	1	exbidv 1839	. 2 |- ( x = A -> ( E. y x B y <-> E. y A B y ) )
3		df-dm 4673	. 2 |- dom B = { x | E. y x B y }
4	2, 3	elab2g 2911	1 |- ( A e. V -> ( A e. dom B <-> E. y A B y ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   E.wex 1506   e. wcel 2167   class class class wbr 4033   dom cdm 4663

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  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-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-dm 4673

This theorem is referenced by:  eldm2g  4862  eldm  4863  breldmg  4872  releldmb  4903  funeu  5283  fneu  5362  ndmfvg  5589  erref  6612  ecdmn0m  6636  shftdm  10987  dvcnp2cntop  14935