Theorem opeldmg 4936

Description: Membership of first of an ordered pair in a domain. (Contributed by Jim Kingdon, 9-Jul-2019.)

Assertion

Ref	Expression
opeldmg	|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. C -> A e. dom C ) )

Proof of Theorem opeldmg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 3863	. . . . 5 |- ( y = B -> <. A , y >. = <. A , B >. )
2	1	eleq1d 2300	. . . 4 |- ( y = B -> ( <. A , y >. e. C <-> <. A , B >. e. C ) )
3	2	spcegv 2894	. . 3 |- ( B e. W -> ( <. A , B >. e. C -> E. y <. A , y >. e. C ) )
4	3	adantl 277	. 2 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. C -> E. y <. A , y >. e. C ) )
5		eldm2g 4927	. . 3 |- ( A e. V -> ( A e. dom C <-> E. y <. A , y >. e. C ) )
6	5	adantr 276	. 2 |- ( ( A e. V /\ B e. W ) -> ( A e. dom C <-> E. y <. A , y >. e. C ) )
7	4, 6	sylibrd 169	1 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. C -> A e. dom C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   e. wcel 2202   <.cop 3672   dom cdm 4725

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-dm 4735

This theorem is referenced by:  tfr0dm  6487  tfrlemi14d  6498  tfr1onlemres  6514  tfrcllemres  6527  fnfi  7134  frecuzrdgtcl  10673  frecuzrdgdomlem  10678  hashennn  11041  imasaddfnlemg  13396