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Theorem dmi 5918
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 3478 . 2 (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2 ax6ev 1966 . . . 4 𝑦 𝑦 = 𝑥
3 vex 3473 . . . . . . 7 𝑦 ∈ V
43ideq 5849 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
5 equcom 2014 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
64, 5bitri 275 . . . . 5 (𝑥 I 𝑦𝑦 = 𝑥)
76exbii 1843 . . . 4 (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
82, 7mpbir 230 . . 3 𝑦 𝑥 I 𝑦
9 vex 3473 . . . 4 𝑥 ∈ V
109eldm 5897 . . 3 (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
118, 10mpbir 230 . 2 𝑥 ∈ dom I
121, 11mpgbir 1794 1 dom I = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wex 1774  wcel 2099  Vcvv 3469   class class class wbr 5142   I cid 5569  dom cdm 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-dm 5682
This theorem is referenced by:  dmv  5919  dmresi  6049  idfn  6677  iprc  7913
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