Theorem dmi 5261

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 3178	. 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2		ax6ev 1877	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑥
3		vex 3176	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	ideq 5196	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5		equcom 1932	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	4, 5	bitri 263	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥)
7	6	exbii 1764	. . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
8	2, 7	mpbir 220	. . 3 ⊢ ∃𝑦 𝑥 I 𝑦
9		vex 3176	. . . 4 ⊢ 𝑥 ∈ V
10	9	eldm 5243	. . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
11	8, 10	mpbir 220	. 2 ⊢ 𝑥 ∈ dom I
12	1, 11	mpgbir 1717	1 ⊢ dom I = V

Syntax hints:   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583   I cid 4948  dom cdm 5038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-dm 5048

This theorem is referenced by:  dmv  5262  dmresi  5376  iprc  6993