Theorem eldmg 4558

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 3796	. . 3 |- ( x = A -> ( x B y <-> A B y ) )
2	1	exbidv 1747	. 2 |- ( x = A -> ( E. y x B y <-> E. y A B y ) )
3		df-dm 4381	. 2 |- dom B = { x | E. y x B y }
4	2, 3	elab2g 2741	1 |- ( A e. V -> ( A e. dom B <-> E. y A B y ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1285   E.wex 1422   e. wcel 1434   class class class wbr 3793   dom cdm 4371

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-dm 4381

This theorem is referenced by:  eldm2g  4559  eldm  4560  breldmg  4569  releldmb  4599  funeu  4956  fneu  5034  ndmfvg  5236  erref  6192  ecdmn0m  6214  shftdm  9848