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Theorem opeldmg 4562
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 3573 . . . . 5  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
21eleq1d 2148 . . . 4  |-  ( y  =  B  ->  ( <. A ,  y >.  e.  C  <->  <. A ,  B >.  e.  C ) )
32spcegv 2687 . . 3  |-  ( B  e.  W  ->  ( <. A ,  B >.  e.  C  ->  E. y <. A ,  y >.  e.  C ) )
43adantl 271 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  C  ->  E. y <. A ,  y >.  e.  C ) )
5 eldm2g 4553 . . 3  |-  ( A  e.  V  ->  ( A  e.  dom  C  <->  E. y <. A ,  y >.  e.  C ) )
65adantr 270 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  dom  C  <->  E. y <. A ,  y
>.  e.  C ) )
74, 6sylibrd 167 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 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   <.cop 3403   dom cdm 4365
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-dm 4375
This theorem is referenced by:  tfr0dm  5965  tfrlemi14d  5976  tfr1onlemres  5992  tfrcllemres  6005  fnfi  6436  frecuzrdgtcl  9483  frecuzrdgdomlem  9488  sizeennn  9793
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