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Theorem opeldmg 4704
Description: Membership of first of an ordered pair in a domain. (Contributed by Jim Kingdon, 9-Jul-2019.)
Assertion
Ref Expression
opeldmg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  C  ->  A  e.  dom  C ) )

Proof of Theorem opeldmg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 opeq2 3672 . . . . 5  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
21eleq1d 2183 . . . 4  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  C  <->  <. A ,  B >.  e.  C ) )
32spcegv 2745 . . 3  |-  ( B  e.  W  ->  ( <. A ,  B >.  e.  C  ->  E. y <. A ,  y >.  e.  C ) )
43adantl 273 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  C  ->  E. y <. A ,  y >.  e.  C ) )
5 eldm2g 4695 . . 3  |-  ( A  e.  V  ->  ( A  e.  dom  C  <->  E. y <. A ,  y >.  e.  C ) )
65adantr 272 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  dom  C  <->  E. y <. A ,  y
>.  e.  C ) )
74, 6sylibrd 168 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  C  ->  A  e.  dom  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   <.cop 3496   dom cdm 4499
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-dm 4509
This theorem is referenced by:  tfr0dm  6173  tfrlemi14d  6184  tfr1onlemres  6200  tfrcllemres  6213  fnfi  6777  frecuzrdgtcl  10078  frecuzrdgdomlem  10083  hashennn  10419
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