Theorem dmi 4762

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 3387	. 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2		a9ev 1676	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑥
3		vex 2692	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	ideq 4699	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5		equcom 1683	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	4, 5	bitri 183	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥)
7	6	exbii 1585	. . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
8	2, 7	mpbir 145	. . 3 ⊢ ∃𝑦 𝑥 I 𝑦
9		vex 2692	. . . 4 ⊢ 𝑥 ∈ V
10	9	eldm 4744	. . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
11	8, 10	mpbir 145	. 2 ⊢ 𝑥 ∈ dom I
12	1, 11	mpgbir 1430	1 ⊢ dom I = V

Syntax hints:   = wceq 1332  ∃wex 1469   ∈ wcel 1481  Vcvv 2689   class class class wbr 3937   I cid 4218  dom cdm 4547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-dm 4557

This theorem is referenced by:  dmv  4763  iprc  4815  dmresi  4882  climshft2  11107