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Theorem opeldm 4964
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 3878 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
32eleq1d 2424 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  C  <->  <. A ,  B >.  e.  C ) )
41, 3spcev 2951 . 2  |-  ( <. A ,  B >.  e.  C  ->  E. y <. A ,  y >.  e.  C )
5 opeldm.1 . . 3  |-  A  e. 
_V
65eldm2 4959 . 2  |-  ( A  e.  dom  C  <->  E. y <. A ,  y >.  e.  C )
74, 6sylibr 203 1  |-  ( <. A ,  B >.  e.  C  ->  A  e.  dom  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1541    = wceq 1642    e. wcel 1710   _Vcvv 2864   <.cop 3719   dom cdm 4771
This theorem is referenced by:  breldm  4965  elreldm  4985  relssres  5074  iss  5080  imadmrn  5106  dfco2a  5255  funssres  5376  funun  5378  tz7.48-1  6542  iiner  6818  r0weon  7730  axdc3lem2  8167  uzrdgfni  11113  imasaddfnlem  13529  imasvscafn  13538  gsum2d  15322  bnj1379  28625
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-dm 4781
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