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Theorem opeldm 4639
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 3623 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
32eleq1d 2156 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  C  <->  <. A ,  B >.  e.  C ) )
41, 3spcev 2713 . 2  |-  ( <. A ,  B >.  e.  C  ->  E. y <. A ,  y >.  e.  C )
5 opeldm.1 . . 3  |-  A  e. 
_V
65eldm2 4634 . 2  |-  ( A  e.  dom  C  <->  E. y <. A ,  y >.  e.  C )
74, 6sylibr 132 1  |-  ( <. A ,  B >.  e.  C  ->  A  e.  dom  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619   <.cop 3449   dom cdm 4438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-dm 4448
This theorem is referenced by:  breldm  4640  elreldm  4661  relssres  4750  iss  4758  imadmrn  4784  dfco2a  4931  funssres  5056  funun  5058  iinerm  6362
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