Theorem eldm2g 4919

Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
eldm2g	|- ( A e. V -> ( A e. dom B <-> E. y <. A , y >. e. B ) )

Distinct variable groups:   y, A   y, B

Allowed substitution hint:   V( y)

Proof of Theorem eldm2g

Step	Hyp	Ref	Expression
1		eldmg 4918	. 2 |- ( A e. V -> ( A e. dom B <-> E. y A B y ) )
2		df-br 4084	. . 3 |- ( A B y <-> <. A , y >. e. B )
3	2	exbii 1651	. 2 |- ( E. y A B y <-> E. y <. A , y >. e. B )
4	1, 3	bitrdi 196	1 |- ( A e. V -> ( A e. dom B <-> E. y <. A , y >. e. B ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1538   e. wcel 2200   <.cop 3669   class class class wbr 4083   dom cdm 4719

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-dm 4729

This theorem is referenced by:  eldm2  4921  opeldmg  4928  dmfco  5702  releldm2  6331  tfrlem9  6465  climcau  11858  lmff  14923