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Theorem elreldm 5232
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

Proof of Theorem elreldm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rel 5011 . . . . 5
2 ssel 3497 . . . . 5
31, 2sylbi 195 . . . 4
4 elvv 5063 . . . 4
53, 4syl6ib 226 . . 3
6 eleq1 2529 . . . . . 6
7 vex 3112 . . . . . . 7
8 vex 3112 . . . . . . 7
97, 8opeldm 5211 . . . . . 6
106, 9syl6bi 228 . . . . 5
11 inteq 4289 . . . . . . . 8
1211inteqd 4291 . . . . . . 7
137, 8op1stb 4722 . . . . . . 7
1412, 13syl6eq 2514 . . . . . 6
1514eleq1d 2526 . . . . 5
1610, 15sylibrd 234 . . . 4
1716exlimivv 1723 . . 3
185, 17syli 37 . 2
1918imp 429 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818   cvv 3109  C_wss 3475  <.cop 4035  |^|cint 4286  X.cxp 5002  domcdm 5004  Relwrel 5009
This theorem is referenced by:  1stdm  6847  fundmen  7609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-int 4287  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-dm 5014
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