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Theorem dmi 5372
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 3236 . 2 (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2 ax6ev 1947 . . . 4 𝑦 𝑦 = 𝑥
3 vex 3234 . . . . . . 7 𝑦 ∈ V
43ideq 5307 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
5 equcom 1991 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
64, 5bitri 264 . . . . 5 (𝑥 I 𝑦𝑦 = 𝑥)
76exbii 1814 . . . 4 (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
82, 7mpbir 221 . . 3 𝑦 𝑥 I 𝑦
9 vex 3234 . . . 4 𝑥 ∈ V
109eldm 5353 . . 3 (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
118, 10mpbir 221 . 2 𝑥 ∈ dom I
121, 11mpgbir 1766 1 dom I = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231   class class class wbr 4685   I cid 5052  dom cdm 5143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-dm 5153
This theorem is referenced by:  dmv  5373  dmresi  5492  iprc  7143
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