Theorem dmi 4598

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 3283	. 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2		a9ev 1628	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑥
3		vex 2613	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	ideq 4536	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5		equcom 1635	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	4, 5	bitri 182	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥)
7	6	exbii 1537	. . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
8	2, 7	mpbir 144	. . 3 ⊢ ∃𝑦 𝑥 I 𝑦
9		vex 2613	. . . 4 ⊢ 𝑥 ∈ V
10	9	eldm 4580	. . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
11	8, 10	mpbir 144	. 2 ⊢ 𝑥 ∈ dom I
12	1, 11	mpgbir 1383	1 ⊢ dom I = V

Syntax hints:   = wceq 1285  ∃wex 1422   ∈ wcel 1434  Vcvv 2610   class class class wbr 3805   I cid 4071  dom cdm 4391

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-dm 4401

This theorem is referenced by:  dmv  4599  iprc  4648  dmresi  4711  climshft2  10346