Theorem eldmg 4926

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 4091	. . 3 |- ( x = A -> ( x B y <-> A B y ) )
2	1	exbidv 1873	. 2 |- ( x = A -> ( E. y x B y <-> E. y A B y ) )
3		df-dm 4735	. 2 |- dom B = { x | E. y x B y }
4	2, 3	elab2g 2953	1 |- ( A e. V -> ( A e. dom B <-> E. y A B y ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   E.wex 1540   e. wcel 2202   class class class wbr 4088   dom cdm 4725

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-dm 4735

This theorem is referenced by:  eldm2g  4927  eldm  4928  breldmg  4937  releldmb  4969  funeu  5351  fneu  5436  ndmfvg  5670  erref  6721  ecdmn0m  6745  shftdm  11382  dvcnp2cntop  15422