Theorem opeldmg 4540

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 3550	. . . . 5 |- ( y = B -> <. A , y >. = <. A , B >. )
2	1	eleq1d 2106	. . . 4 |- ( y = B -> ( <. A , y >. e. C <-> <. A , B >. e. C ) )
3	2	spcegv 2641	. . 3 |- ( B e. W -> ( <. A , B >. e. C -> E. y <. A , y >. e. C ) )
4	3	adantl 262	. 2 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. C -> E. y <. A , y >. e. C ) )
5		eldm2g 4531	. . 3 |- ( A e. V -> ( A e. dom C <-> E. y <. A , y >. e. C ) )
6	5	adantr 261	. 2 |- ( ( A e. V /\ B e. W ) -> ( A e. dom C <-> E. y <. A , y >. e. C ) )
7	4, 6	sylibrd 158	1 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. C -> A e. dom C ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   <.cop 3378   dom cdm 4345

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355

This theorem is referenced by:  tfr0  5937  tfrlemi14d  5947  frecuzrdgfn  9198