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Theorem dmi 5244
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 3173 . 2 (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2 ax6ev 1875 . . . 4 𝑦 𝑦 = 𝑥
3 vex 3171 . . . . . . 7 𝑦 ∈ V
43ideq 5180 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
5 equcom 1930 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
64, 5bitri 262 . . . . 5 (𝑥 I 𝑦𝑦 = 𝑥)
76exbii 1762 . . . 4 (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
82, 7mpbir 219 . . 3 𝑦 𝑥 I 𝑦
9 vex 3171 . . . 4 𝑥 ∈ V
109eldm 5226 . . 3 (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
118, 10mpbir 219 . 2 𝑥 ∈ dom I
121, 11mpgbir 1715 1 dom I = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wex 1694  wcel 1975  Vcvv 3168   class class class wbr 4573   I cid 4934  dom cdm 5024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-dm 5034
This theorem is referenced by:  dmv  5245  dmresi  5359  iprc  6966
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