Theorem opeldm 4639

Description: Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)

Hypotheses

Ref	Expression
opeldm.1	|- A e. _V
opeldm.2	|- B e. _V

Assertion

Ref	Expression
opeldm	|- ( <. A , B >. e. C -> A e. dom C )

Proof of Theorem opeldm

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeldm.2	. . 3 |- B e. _V
2		opeq2 3623	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
3	2	eleq1d 2156	. . 3 |- ( y = B -> ( <. A , y >. e. C <-> <. A , B >. e. C ) )
4	1, 3	spcev 2713	. 2 |- ( <. A , B >. e. C -> E. y <. A , y >. e. C )
5		opeldm.1	. . 3 |- A e. _V
6	5	eldm2 4634	. 2 |- ( A e. dom C <-> E. y <. A , y >. e. C )
7	4, 6	sylibr 132	1 |- ( <. A , B >. e. C -> A e. dom C )

Syntax hints:   -> wi 4   = wceq 1289   E.wex 1426   e. wcel 1438   _Vcvv 2619   <.cop 3449   dom cdm 4438

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-dm 4448

This theorem is referenced by:  breldm  4640  elreldm  4661  relssres  4750  iss  4758  imadmrn  4784  dfco2a  4931  funssres  5056  funun  5058  iinerm  6362