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Theorem dmi 4493
Description: The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmi dom I = V

Proof of Theorem dmi
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqv 3234 . 2 (dom I = V ↔ x x dom I )
2 a9ev 1584 . . . 4 y y = x
3 vex 2554 . . . . . . 7 y V
43ideq 4431 . . . . . 6 (x I yx = y)
5 equcom 1590 . . . . . 6 (x = yy = x)
64, 5bitri 173 . . . . 5 (x I yy = x)
76exbii 1493 . . . 4 (y x I yy y = x)
82, 7mpbir 134 . . 3 y x I y
9 vex 2554 . . . 4 x V
109eldm 4475 . . 3 (x dom I ↔ y x I y)
118, 10mpbir 134 . 2 x dom I
121, 11mpgbir 1339 1 dom I = V
Colors of variables: wff set class
Syntax hints:   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551   class class class wbr 3755   I cid 4016  dom cdm 4288
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-bnd 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-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  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-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-dm 4298
This theorem is referenced by:  dmv  4494  iprc  4543  dmresi  4604
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