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Theorem dmi 4722
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 3350 . 2 (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2 a9ev 1658 . . . 4 𝑦 𝑦 = 𝑥
3 vex 2661 . . . . . . 7 𝑦 ∈ V
43ideq 4659 . . . . . 6 (𝑥 I 𝑦𝑥 = 𝑦)
5 equcom 1665 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
64, 5bitri 183 . . . . 5 (𝑥 I 𝑦𝑦 = 𝑥)
76exbii 1567 . . . 4 (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
82, 7mpbir 145 . . 3 𝑦 𝑥 I 𝑦
9 vex 2661 . . . 4 𝑥 ∈ V
109eldm 4704 . . 3 (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
118, 10mpbir 145 . 2 𝑥 ∈ dom I
121, 11mpgbir 1412 1 dom I = V
Colors of variables: wff set class
Syntax hints:   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658   class class class wbr 3897   I cid 4178  dom cdm 4507
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-dm 4517
This theorem is referenced by:  dmv  4723  iprc  4775  dmresi  4842  climshft2  11015
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