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Theorem dmi 4651
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 3302 . 2 (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2 a9ev 1632 . . . 4 𝑦 𝑦 = 𝑥
3 vex 2622 . . . . . . 7 𝑦 ∈ V
43ideq 4588 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
5 equcom 1639 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
64, 5bitri 182 . . . . 5 (𝑥 I 𝑦𝑦 = 𝑥)
76exbii 1541 . . . 4 (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
82, 7mpbir 144 . . 3 𝑦 𝑥 I 𝑦
9 vex 2622 . . . 4 𝑥 ∈ V
109eldm 4633 . . 3 (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
118, 10mpbir 144 . 2 𝑥 ∈ dom I
121, 11mpgbir 1387 1 dom I = V
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619   class class class wbr 3845   I cid 4115  dom cdm 4438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-dm 4448
This theorem is referenced by:  dmv  4652  iprc  4701  dmresi  4767  climshft2  10691
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