Theorem eldm2g 4957

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 4956	. 2 |- ( A e. V -> ( A e. dom B <-> E. y A B y ) )
2		df-br 4115	. . 3 |- ( A B y <-> <. A , y >. e. B )
3	2	exbii 1654	. 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 1541   e. wcel 2205   <.cop 3697   class class class wbr 4114   dom cdm 4754

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-dm 4764

This theorem is referenced by:  eldm2  4959  opeldmg  4966  dmfco  5750  releldm2  6392  elmpom  6447  tfrlem9  6563  climcau  12057  lmff  15240