Theorem opeldm 5328

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 4403	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
3	2	eleq1d 2686	. . 3 |- ( y = B -> ( <. A , y >. e. C <-> <. A , B >. e. C ) )
4	1, 3	spcev 3300	. 2 |- ( <. A , B >. e. C -> E. y <. A , y >. e. C )
5		opeldm.1	. . 3 |- A e. _V
6	5	eldm2 5322	. 2 |- ( A e. dom C <-> E. y <. A , y >. e. C )
7	4, 6	sylibr 224	1 |- ( <. A , B >. e. C -> A e. dom C )

Syntax hints:   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   <.cop 4183   dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-dm 5124

This theorem is referenced by:  breldm  5329  elreldm  5350  relssres  5437  iss  5447  imadmrn  5476  dfco2a  5635  funssres  5930  funun  5932  tz7.48-1  7538  iiner  7819  r0weon  8835  axdc3lem2  9273  uzrdgfni  12757  imasaddfnlem  16188  imasvscafn  16197  cicsym  16464  gsum2d  18371  cffldtocusgr  26343  dfcnv2  29476  bnj1379  30901  iss2  34112  rfovcnvf1od  38298