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Theorem dmi 5543
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 3391 . 2 (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2 ax6ev 2074 . . . 4 𝑦 𝑦 = 𝑥
3 vex 3388 . . . . . . 7 𝑦 ∈ V
43ideq 5478 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
5 equcom 2117 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
64, 5bitri 267 . . . . 5 (𝑥 I 𝑦𝑦 = 𝑥)
76exbii 1944 . . . 4 (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
82, 7mpbir 223 . . 3 𝑦 𝑥 I 𝑦
9 vex 3388 . . . 4 𝑥 ∈ V
109eldm 5524 . . 3 (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
118, 10mpbir 223 . 2 𝑥 ∈ dom I
121, 11mpgbir 1895 1 dom I = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wex 1875  wcel 2157  Vcvv 3385   class class class wbr 4843   I cid 5219  dom cdm 5312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-dm 5322
This theorem is referenced by:  dmv  5544  dmresi  5676  iprc  7336
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