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Theorem dmi 5932
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 3490 . 2 (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2 ax6ev 1969 . . . 4 𝑦 𝑦 = 𝑥
3 vex 3484 . . . . . . 7 𝑦 ∈ V
43ideq 5863 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
5 equcom 2017 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
64, 5bitri 275 . . . . 5 (𝑥 I 𝑦𝑦 = 𝑥)
76exbii 1848 . . . 4 (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
82, 7mpbir 231 . . 3 𝑦 𝑥 I 𝑦
9 vex 3484 . . . 4 𝑥 ∈ V
109eldm 5911 . . 3 (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
118, 10mpbir 231 . 2 𝑥 ∈ dom I
121, 11mpgbir 1799 1 dom I = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480   class class class wbr 5143   I cid 5577  dom cdm 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-dm 5695
This theorem is referenced by:  dmv  5933  dmresi  6070  idfn  6696  iprc  7933
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