Theorem opeldm 4932

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 3861	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
3	2	eleq1d 2298	. . 3 |- ( y = B -> ( <. A , y >. e. C <-> <. A , B >. e. C ) )
4	1, 3	spcev 2899	. 2 |- ( <. A , B >. e. C -> E. y <. A , y >. e. C )
5		opeldm.1	. . 3 |- A e. _V
6	5	eldm2 4927	. 2 |- ( A e. dom C <-> E. y <. A , y >. e. C )
7	4, 6	sylibr 134	1 |- ( <. A , B >. e. C -> A e. dom C )

Syntax hints:   -> wi 4   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2800   <.cop 3670   dom cdm 4723

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 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-dm 4733

This theorem is referenced by:  breldm  4933  elreldm  4956  relssres  5049  iss  5057  imadmrn  5084  dfco2a  5235  funssres  5366  funun  5368  iinerm  6771