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Theorem dmi 5778
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 1973 . . . 4 𝑦 𝑦 = 𝑥
3 vex 3483 . . . . . . 7 𝑦 ∈ V
43ideq 5710 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
5 equcom 2026 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
64, 5bitri 278 . . . . 5 (𝑥 I 𝑦𝑦 = 𝑥)
76exbii 1849 . . . 4 (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
82, 7mpbir 234 . . 3 𝑦 𝑥 I 𝑦
9 vex 3483 . . . 4 𝑥 ∈ V
109eldm 5756 . . 3 (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
118, 10mpbir 234 . 2 𝑥 ∈ dom I
121, 11mpgbir 1801 1 dom I = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wex 1781  wcel 2115  Vcvv 3480   class class class wbr 5052   I cid 5446  dom cdm 5542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-dm 5552
This theorem is referenced by:  dmv  5779  dmresi  5908  idfn  6464  iprc  7613
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