Theorem opeldmg 4928

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 3858	. . . . 5 |- ( y = B -> <. A , y >. = <. A , B >. )
2	1	eleq1d 2298	. . . 4 |- ( y = B -> ( <. A , y >. e. C <-> <. A , B >. e. C ) )
3	2	spcegv 2891	. . 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 4919	. . 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 1395   E.wex 1538   e. wcel 2200   <.cop 3669   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:  tfr0dm  6468  tfrlemi14d  6479  tfr1onlemres  6495  tfrcllemres  6508  fnfi  7103  frecuzrdgtcl  10634  frecuzrdgdomlem  10639  hashennn  11002  imasaddfnlemg  13347