Theorem dmi 5899

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 3460	. 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2		ax6ev 2002	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑥
3		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	ideq 5826	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5		equcom 2051	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	4, 5	bitri 278	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥)
7	6	exbii 1881	. . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
8	2, 7	mpbir 234	. . 3 ⊢ ∃𝑦 𝑥 I 𝑦
9		vex 3454	. . . 4 ⊢ 𝑥 ∈ V
10	9	eldm 5878	. . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
11	8, 10	mpbir 234	. 2 ⊢ 𝑥 ∈ dom I
12	1, 11	mpgbir 1832	1 ⊢ dom I = V

This proof depends on syntax axioms:   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3450   class class class wbr 5102   I cid 5541  dom cdm 5647

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5542  df-xp 5653  df-rel 5654  df-dm 5657

This theorem is used by:  dmv  5900  dmresi  6042  idfn  6655  iprc  7906  dfsucmap3  39315  dfsucmap2  39316