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Theorem dmi 4724
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 3352 . 2 (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2 a9ev 1660 . . . 4 𝑦 𝑦 = 𝑥
3 vex 2663 . . . . . . 7 𝑦 ∈ V
43ideq 4661 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
5 equcom 1667 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
64, 5bitri 183 . . . . 5 (𝑥 I 𝑦𝑦 = 𝑥)
76exbii 1569 . . . 4 (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
82, 7mpbir 145 . . 3 𝑦 𝑥 I 𝑦
9 vex 2663 . . . 4 𝑥 ∈ V
109eldm 4706 . . 3 (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
118, 10mpbir 145 . 2 𝑥 ∈ dom I
121, 11mpgbir 1414 1 dom I = V
Colors of variables: wff set class
Syntax hints:   = wceq 1316  wex 1453  wcel 1465  Vcvv 2660   class class class wbr 3899   I cid 4180  dom cdm 4509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-dm 4519
This theorem is referenced by:  dmv  4725  iprc  4777  dmresi  4844  climshft2  11043
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