Theorem opeldm 4979

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 3900	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
3	2	eleq1d 2307	. . 3 |- ( y = B -> ( <. A , y >. e. C <-> <. A , B >. e. C ) )
4	1, 3	spcev 2920	. 2 |- ( <. A , B >. e. C -> E. y <. A , y >. e. C )
5		opeldm.1	. . 3 |- A e. _V
6	5	eldm2 4974	. 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 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   <.cop 3708   dom cdm 4769

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-dm 4779

This theorem is referenced by:  breldm  4980  elreldm  5003  relssres  5096  iss  5104  imadmrn  5131  dfco2a  5283  funssres  5415  funun  5417  iinerm  6871