Theorem dmi 5870

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 3443	. 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2		ax6ev 1977	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑥
3		vex 3437	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	ideq 5797	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5		equcom 2026	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	4, 5	bitri 277	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥)
7	6	exbii 1856	. . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
8	2, 7	mpbir 233	. . 3 ⊢ ∃𝑦 𝑥 I 𝑦
9		vex 3437	. . . 4 ⊢ 𝑥 ∈ V
10	9	eldm 5849	. . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
11	8, 10	mpbir 233	. 2 ⊢ 𝑥 ∈ dom I
12	1, 11	mpgbir 1807	1 ⊢ dom I = V

Syntax hints:   = wceq 1548  ∃wex 1787   ∈ wcel 2121  Vcvv 3433   class class class wbr 5075   I cid 5515  dom cdm 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-dm 5631

This theorem is referenced by:  dmv  5871  dmresi  6011  idfn  6617  iprc  7855  dfsucmap3  38845  dfsucmap2  38846