Theorem eldmg 4530

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 3767	. . 3 |- ( x = A -> ( x B y <-> A B y ) )
2	1	exbidv 1706	. 2 |- ( x = A -> ( E. y x B y <-> E. y A B y ) )
3		df-dm 4355	. 2 |- dom B = { x | E. y x B y }
4	2, 3	elab2g 2689	1 |- ( A e. V -> ( A e. dom B <-> E. y A B y ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   class class class wbr 3764   dom cdm 4345

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355

This theorem is referenced by:  eldm2g  4531  eldm  4532  breldmg  4541  releldmb  4571  funeu  4926  fneu  5003  ndmfvg  5204  erref  6126  ecdmn0m  6148  shftdm  9423