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Theorem dmi 4860
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 3457 . 2 (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2 a9ev 1708 . . . 4 𝑦 𝑦 = 𝑥
3 vex 2755 . . . . . . 7 𝑦 ∈ V
43ideq 4797 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
5 equcom 1717 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
64, 5bitri 184 . . . . 5 (𝑥 I 𝑦𝑦 = 𝑥)
76exbii 1616 . . . 4 (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
82, 7mpbir 146 . . 3 𝑦 𝑥 I 𝑦
9 vex 2755 . . . 4 𝑥 ∈ V
109eldm 4842 . . 3 (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
118, 10mpbir 146 . 2 𝑥 ∈ dom I
121, 11mpgbir 1464 1 dom I = V
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wex 1503  wcel 2160  Vcvv 2752   class class class wbr 4018   I cid 4306  dom cdm 4644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-dm 4654
This theorem is referenced by:  dmv  4861  iprc  4913  dmresi  4980  climshft2  11346
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