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Theorem elreldm 5078
 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 4860 . . . . 5
2 ssel 3458 . . . . 5
31, 2sylbi 198 . . . 4
4 elvv 4912 . . . 4
53, 4syl6ib 229 . . 3
6 eleq1 2495 . . . . . 6
7 vex 3083 . . . . . . 7
8 vex 3083 . . . . . . 7
97, 8opeldm 5057 . . . . . 6
106, 9syl6bi 231 . . . . 5
11 inteq 4258 . . . . . . . 8
1211inteqd 4260 . . . . . . 7
137, 8op1stb 4691 . . . . . . 7
1412, 13syl6eq 2479 . . . . . 6
1514eleq1d 2491 . . . . 5
1610, 15sylibrd 237 . . . 4
1716exlimivv 1771 . . 3
185, 17syli 38 . 2
1918imp 430 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   wceq 1437  wex 1657   wcel 1872  cvv 3080   wss 3436  cop 4004  cint 4255   cxp 4851   cdm 4853   wrel 4858 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-int 4256  df-br 4424  df-opab 4483  df-xp 4859  df-rel 4860  df-dm 4863 This theorem is referenced by:  1stdm  6854  fundmen  7653
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