Theorem dmi 4664

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 3306	. 2 |- ( dom _I = _V <-> A. x x e. dom _I )
2		a9ev 1633	. . . 4 |- E. y y = x
3		vex 2623	. . . . . . 7 |- y e. _V
4	3	ideq 4601	. . . . . 6 |- ( x _I y <-> x = y )
5		equcom 1640	. . . . . 6 |- ( x = y <-> y = x )
6	4, 5	bitri 183	. . . . 5 |- ( x _I y <-> y = x )
7	6	exbii 1542	. . . 4 |- ( E. y x _I y <-> E. y y = x )
8	2, 7	mpbir 145	. . 3 |- E. y x _I y
9		vex 2623	. . . 4 |- x e. _V
10	9	eldm 4646	. . 3 |- ( x e. dom _I <-> E. y x _I y )
11	8, 10	mpbir 145	. 2 |- x e. dom _I
12	1, 11	mpgbir 1388	1 |- dom _I = _V

Syntax hints:   = wceq 1290   E.wex 1427   e. wcel 1439   _Vcvv 2620   class class class wbr 3851   _I cid 4124   dom cdm 4452

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-dm 4462

This theorem is referenced by:  dmv  4665  iprc  4714  dmresi  4780  climshft2  10756