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Theorem dmi 5924
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 3470 . 2 (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2 ax6ev 1965 . . . 4 𝑦 𝑦 = 𝑥
3 vex 3465 . . . . . . 7 𝑦 ∈ V
43ideq 5855 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
5 equcom 2013 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
64, 5bitri 274 . . . . 5 (𝑥 I 𝑦𝑦 = 𝑥)
76exbii 1842 . . . 4 (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
82, 7mpbir 230 . . 3 𝑦 𝑥 I 𝑦
9 vex 3465 . . . 4 𝑥 ∈ V
109eldm 5903 . . 3 (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
118, 10mpbir 230 . 2 𝑥 ∈ dom I
121, 11mpgbir 1793 1 dom I = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wex 1773  wcel 2098  Vcvv 3461   class class class wbr 5149   I cid 5575  dom cdm 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-dm 5688
This theorem is referenced by:  dmv  5925  dmresi  6056  idfn  6684  iprc  7919
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