Theorem eldm2g 5320

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 5319	. 2 |- ( A e. V -> ( A e. dom B <-> E. y A B y ) )
2		df-br 4654	. . 3 |- ( A B y <-> <. A , y >. e. B )
3	2	exbii 1774	. 2 |- ( E. y A B y <-> E. y <. A , y >. e. B )
4	1, 3	syl6bb 276	1 |- ( A e. V -> ( A e. dom B <-> E. y <. A , y >. e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   E.wex 1704   e. wcel 1990   <.cop 4183   class class class wbr 4653   dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-dm 5124

This theorem is referenced by:  eldm2  5322  opeldmd  5327  dmfco  6272  releldm2  7218  tfrlem9  7481  climcau  14401  caucvgb  14410  lmff  21105  axhcompl-zf  27855