Theorem dmi 4976

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 3532	. 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2		a9ev 1745	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑥
3		vex 2818	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	ideq 4912	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5		equcom 1754	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	4, 5	bitri 184	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥)
7	6	exbii 1654	. . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
8	2, 7	mpbir 146	. . 3 ⊢ ∃𝑦 𝑥 I 𝑦
9		vex 2818	. . . 4 ⊢ 𝑥 ∈ V
10	9	eldm 4958	. . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
11	8, 10	mpbir 146	. 2 ⊢ 𝑥 ∈ dom I
12	1, 11	mpgbir 1502	1 ⊢ dom I = V

Syntax hints:   = wceq 1398  ∃wex 1541   ∈ wcel 2205  Vcvv 2815   class class class wbr 4114   I cid 4414  dom cdm 4754

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-dm 4764

This theorem is referenced by:  dmv  4977  iprc  5031  dmresi  5098  climshft2  12019