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Theorem dmi 5935
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 3488 . 2 (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2 ax6ev 1967 . . . 4 𝑦 𝑦 = 𝑥
3 vex 3482 . . . . . . 7 𝑦 ∈ V
43ideq 5866 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
5 equcom 2015 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
64, 5bitri 275 . . . . 5 (𝑥 I 𝑦𝑦 = 𝑥)
76exbii 1845 . . . 4 (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
82, 7mpbir 231 . . 3 𝑦 𝑥 I 𝑦
9 vex 3482 . . . 4 𝑥 ∈ V
109eldm 5914 . . 3 (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
118, 10mpbir 231 . 2 𝑥 ∈ dom I
121, 11mpgbir 1796 1 dom I = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478   class class class wbr 5148   I cid 5582  dom cdm 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-dm 5699
This theorem is referenced by:  dmv  5936  dmresi  6072  idfn  6697  iprc  7934
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