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Theorem elreldm 5094
 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 4885 . . . . 5
2 ssel 3342 . . . . 5
31, 2sylbi 188 . . . 4
4 elvv 4936 . . . 4
53, 4syl6ib 218 . . 3
6 eleq1 2496 . . . . . 6
7 vex 2959 . . . . . . 7
8 vex 2959 . . . . . . 7
97, 8opeldm 5073 . . . . . 6
106, 9syl6bi 220 . . . . 5
11 inteq 4053 . . . . . . . 8
1211inteqd 4055 . . . . . . 7
137, 8op1stb 4758 . . . . . . 7
1412, 13syl6eq 2484 . . . . . 6
1514eleq1d 2502 . . . . 5
1610, 15sylibrd 226 . . . 4
1716exlimivv 1645 . . 3
185, 17syli 35 . 2
1918imp 419 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wex 1550   wceq 1652   wcel 1725  cvv 2956   wss 3320  cop 3817  cint 4050   cxp 4876   cdm 4878   wrel 4883 This theorem is referenced by:  1stdm  6394  fundmen  7180 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-int 4051  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-dm 4888
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