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Theorem elreldm 4503
Description: The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)
Assertion
Ref Expression
elreldm  Rel  |^| |^|  dom

Proof of Theorem elreldm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rel 4295 . . . . 5  Rel  C_  _V  X.  _V
2 ssel 2933 . . . . 5 
C_  _V  X.  _V  _V  X.  _V
31, 2sylbi 114 . . . 4  Rel  _V  X.  _V
4 elvv 4345 . . . 4  _V  X.  _V  <. ,  >.
53, 4syl6ib 150 . . 3  Rel  <. ,  >.
6 eleq1 2097 . . . . . 6  <. , 
>. 
<. ,  >.
7 vex 2554 . . . . . . 7 
_V
8 vex 2554 . . . . . . 7 
_V
97, 8opeldm 4481 . . . . . 6  <. ,  >.  dom
106, 9syl6bi 152 . . . . 5  <. , 
>.  dom
11 inteq 3609 . . . . . . . 8  <. , 
>.  |^|  |^|
<. ,  >.
1211inteqd 3611 . . . . . . 7  <. , 
>.  |^| |^|  |^| |^| <. , 
>.
137, 8op1stb 4175 . . . . . . 7  |^| |^| <. ,  >.
1412, 13syl6eq 2085 . . . . . 6  <. , 
>.  |^| |^|
1514eleq1d 2103 . . . . 5  <. , 
>.  |^| |^|  dom 
dom
1610, 15sylibrd 158 . . . 4  <. , 
>.  |^| |^|  dom
1716exlimivv 1773 . . 3  <. ,  >.  |^| |^|  dom
185, 17syli 33 . 2  Rel  |^| |^|  dom
1918imp 115 1  Rel  |^| |^|  dom
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390   _Vcvv 2551    C_ wss 2911   <.cop 3370   |^|cint 3606    X. cxp 4286   dom cdm 4288   Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-int 3607  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298
This theorem is referenced by:  1stdm  5750  fundmen  6222
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