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Theorem opeldm 5154
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 4171 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
32eleq1d 2523 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  C  <->  <. A ,  B >.  e.  C ) )
41, 3spcev 3170 . 2  |-  ( <. A ,  B >.  e.  C  ->  E. y <. A ,  y >.  e.  C )
5 opeldm.1 . . 3  |-  A  e. 
_V
65eldm2 5149 . 2  |-  ( A  e.  dom  C  <->  E. y <. A ,  y >.  e.  C )
74, 6sylibr 212 1  |-  ( <. A ,  B >.  e.  C  ->  A  e.  dom  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   E.wex 1587    e. wcel 1758   _Vcvv 3078   <.cop 3994   dom cdm 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-dm 4961
This theorem is referenced by:  breldm  5155  elreldm  5175  relssres  5258  iss  5265  imadmrn  5290  dfco2a  5449  funssres  5569  funun  5571  tz7.48-1  7011  iiner  7285  r0weon  8294  axdc3lem2  8735  uzrdgfni  11902  imasaddfnlem  14589  imasvscafn  14598  gsum2d  16595  gsum2dOLD  16596  dfcnv2  26172  bnj1379  32179
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