Theorem dmi 4898

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.

Step	Hyp	Ref	Expression
1		eqv 3481	. 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2		a9ev 1721	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑥
3		vex 2776	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	ideq 4834	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5		equcom 1730	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	4, 5	bitri 184	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥)
7	6	exbii 1629	. . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
8	2, 7	mpbir 146	. . 3 ⊢ ∃𝑦 𝑥 I 𝑦
9		vex 2776	. . . 4 ⊢ 𝑥 ∈ V
10	9	eldm 4880	. . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
11	8, 10	mpbir 146	. 2 ⊢ 𝑥 ∈ dom I
12	1, 11	mpgbir 1477	1 ⊢ dom I = V

Syntax hints:   = wceq 1373  ∃wex 1516   ∈ wcel 2177  Vcvv 2773   class class class wbr 4047   I cid 4339  dom cdm 4679

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-br 4048  df-opab 4110  df-id 4344  df-xp 4685  df-rel 4686  df-dm 4689

This theorem is referenced by:  dmv  4899  iprc  4952  dmresi  5019  climshft2  11661