Theorem dmi 4878

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 3467	. 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2		a9ev 1708	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑥
3		vex 2763	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	ideq 4815	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5		equcom 1717	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	4, 5	bitri 184	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥)
7	6	exbii 1616	. . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
8	2, 7	mpbir 146	. . 3 ⊢ ∃𝑦 𝑥 I 𝑦
9		vex 2763	. . . 4 ⊢ 𝑥 ∈ V
10	9	eldm 4860	. . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
11	8, 10	mpbir 146	. 2 ⊢ 𝑥 ∈ dom I
12	1, 11	mpgbir 1464	1 ⊢ dom I = V

Syntax hints:   = wceq 1364  ∃wex 1503   ∈ wcel 2164  Vcvv 2760   class class class wbr 4030   I cid 4320  dom cdm 4660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-dm 4670

This theorem is referenced by:  dmv  4879  iprc  4931  dmresi  4998  climshft2  11452