Theorem dmi 5868

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 3440	. 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2		ax6ev 1971	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑥
3		vex 3434	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	ideq 5799	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5		equcom 2020	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	4, 5	bitri 275	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥)
7	6	exbii 1850	. . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
8	2, 7	mpbir 231	. . 3 ⊢ ∃𝑦 𝑥 I 𝑦
9		vex 3434	. . . 4 ⊢ 𝑥 ∈ V
10	9	eldm 5847	. . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
11	8, 10	mpbir 231	. 2 ⊢ 𝑥 ∈ dom I
12	1, 11	mpgbir 1801	1 ⊢ dom I = V

Syntax hints:   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430   class class class wbr 5086   I cid 5516  dom cdm 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5517  df-xp 5628  df-rel 5629  df-dm 5632

This theorem is referenced by:  dmv  5869  dmresi  6009  idfn  6618  iprc  7853  dfsucmap3  38795  dfsucmap2  38796