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Theorem opeldm 5102
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.
StepHypRef Expression
1 opeldm.2 . . 3  |-  B  e. 
_V
2 opeq2 4009 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
32eleq1d 2508 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  C  <->  <. A ,  B >.  e.  C ) )
41, 3spcev 3049 . 2  |-  ( <. A ,  B >.  e.  C  ->  E. y <. A ,  y >.  e.  C )
5 opeldm.1 . . 3  |-  A  e. 
_V
65eldm2 5097 . 2  |-  ( A  e.  dom  C  <->  E. y <. A ,  y >.  e.  C )
74, 6sylibr 205 1  |-  ( <. A ,  B >.  e.  C  ->  A  e.  dom  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1551    = wceq 1653    e. wcel 1727   _Vcvv 2962   <.cop 3841   dom cdm 4907
This theorem is referenced by:  breldm  5103  elreldm  5123  relssres  5212  iss  5218  imadmrn  5244  dfco2a  5399  funssres  5522  funun  5524  tz7.48-1  6729  iiner  7005  r0weon  7925  axdc3lem2  8362  uzrdgfni  11329  imasaddfnlem  13784  imasvscafn  13793  gsum2d  15577  bnj1379  29300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-dm 4917
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