Theorem eldmg 4874

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 4048	. . 3 |- ( x = A -> ( x B y <-> A B y ) )
2	1	exbidv 1848	. 2 |- ( x = A -> ( E. y x B y <-> E. y A B y ) )
3		df-dm 4686	. 2 |- dom B = { x | E. y x B y }
4	2, 3	elab2g 2920	1 |- ( A e. V -> ( A e. dom B <-> E. y A B y ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   E.wex 1515   e. wcel 2176   class class class wbr 4045   dom cdm 4676

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-dm 4686

This theorem is referenced by:  eldm2g  4875  eldm  4876  breldmg  4885  releldmb  4916  funeu  5297  fneu  5381  ndmfvg  5609  erref  6642  ecdmn0m  6666  shftdm  11166  dvcnp2cntop  15204