Theorem opeldmg 4871

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 3809	. . . . 5 |- ( y = B -> <. A , y >. = <. A , B >. )
2	1	eleq1d 2265	. . . 4 |- ( y = B -> ( <. A , y >. e. C <-> <. A , B >. e. C ) )
3	2	spcegv 2852	. . 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 4862	. . 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 1364   E.wex 1506   e. wcel 2167   <.cop 3625   dom cdm 4663

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-dm 4673

This theorem is referenced by:  tfr0dm  6380  tfrlemi14d  6391  tfr1onlemres  6407  tfrcllemres  6420  fnfi  7002  frecuzrdgtcl  10504  frecuzrdgdomlem  10509  hashennn  10872  imasaddfnlemg  12957