Theorem opeldm 4828

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 3779	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
3	2	eleq1d 2246	. . 3 |- ( y = B -> ( <. A , y >. e. C <-> <. A , B >. e. C ) )
4	1, 3	spcev 2832	. 2 |- ( <. A , B >. e. C -> E. y <. A , y >. e. C )
5		opeldm.1	. . 3 |- A e. _V
6	5	eldm2 4823	. 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 1353   E.wex 1492   e. wcel 2148   _Vcvv 2737   <.cop 3595   dom cdm 4625

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-dm 4635

This theorem is referenced by:  breldm  4829  elreldm  4851  relssres  4943  iss  4951  imadmrn  4978  dfco2a  5127  funssres  5256  funun  5258  iinerm  6603