Theorem dmi 5888

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 3460	. 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I )
2		ax6ev 1969	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑥
3		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	ideq 5819	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5		equcom 2018	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	4, 5	bitri 275	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥)
7	6	exbii 1848	. . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
8	2, 7	mpbir 231	. . 3 ⊢ ∃𝑦 𝑥 I 𝑦
9		vex 3454	. . . 4 ⊢ 𝑥 ∈ V
10	9	eldm 5867	. . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦)
11	8, 10	mpbir 231	. 2 ⊢ 𝑥 ∈ dom I
12	1, 11	mpgbir 1799	1 ⊢ dom I = V

Syntax hints:   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3450   class class class wbr 5110   I cid 5535  dom cdm 5641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-dm 5651

This theorem is referenced by:  dmv  5889  dmresi  6026  idfn  6649  iprc  7890