Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmi GIF version

Theorem dmi 4550
 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 3240 . 2 (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2 a9ev 1587 . . . 4 𝑦 𝑦 = 𝑥
3 vex 2560 . . . . . . 7 𝑦 ∈ V
43ideq 4488 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
5 equcom 1593 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
64, 5bitri 173 . . . . 5 (𝑥 I 𝑦𝑦 = 𝑥)
76exbii 1496 . . . 4 (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
82, 7mpbir 134 . . 3 𝑦 𝑥 I 𝑦
9 vex 2560 . . . 4 𝑥 ∈ V
109eldm 4532 . . 3 (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
118, 10mpbir 134 . 2 𝑥 ∈ dom I
121, 11mpgbir 1342 1 dom I = V
 Colors of variables: wff set class Syntax hints:   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557   class class class wbr 3764   I cid 4025  dom cdm 4345 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-dm 4355 This theorem is referenced by:  dmv  4551  iprc  4600  dmresi  4661  climshft2  9827
 Copyright terms: Public domain W3C validator