Theorem dmi 5890

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 3458	. 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2		ax6ev 1983	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑥
3		vex 3452	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	ideq 5817	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5		equcom 2032	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	4, 5	bitri 277	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥)
7	6	exbii 1862	. . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
8	2, 7	mpbir 233	. . 3 ⊢ ∃𝑦 𝑥 I 𝑦
9		vex 3452	. . . 4 ⊢ 𝑥 ∈ V
10	9	eldm 5869	. . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
11	8, 10	mpbir 233	. 2 ⊢ 𝑥 ∈ dom I
12	1, 11	mpgbir 1813	1 ⊢ dom I = V

Syntax hints:   = wceq 1554  ∃wex 1793   ∈ wcel 2136  Vcvv 3448   class class class wbr 5094   I cid 5534  dom cdm 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-br 5095  df-opab 5157  df-id 5535  df-xp 5646  df-rel 5647  df-dm 5650

This theorem is referenced by:  dmv  5891  dmresi  6031  idfn  6638  iprc  7881  dfsucmap3  38910  dfsucmap2  38911