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Theorem dmi 4477
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 3217 . 2 (dom I = V ↔ x x dom I )
2 a9ev 1569 . . . 4 y y = x
3 vex 2538 . . . . . . 7 y V
43ideq 4415 . . . . . 6 (x I yx = y)
5 equcom 1575 . . . . . 6 (x = yy = x)
64, 5bitri 173 . . . . 5 (x I yy = x)
76exbii 1478 . . . 4 (y x I yy y = x)
82, 7mpbir 134 . . 3 y x I y
9 vex 2538 . . . 4 x V
109eldm 4459 . . 3 (x dom I ↔ y x I y)
118, 10mpbir 134 . 2 x dom I
121, 11mpgbir 1322 1 dom I = V
Colors of variables: wff set class
Syntax hints:   = wceq 1228  wex 1362   wcel 1374  Vcvv 2535   class class class wbr 3738   I cid 3999  dom cdm 4272
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-dm 4282
This theorem is referenced by:  dmv  4478  iprc  4527  dmresi  4588
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