Theorem dmi 4952

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 3516	. 2 |- ( dom _I = _V <-> A. x x e. dom _I )
2		a9ev 1745	. . . 4 |- E. y y = x
3		vex 2806	. . . . . . 7 |- y e. _V
4	3	ideq 4888	. . . . . 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 |- ( E. y x _I y <-> E. y y = x )
8	2, 7	mpbir 146	. . 3 |- E. y x _I y
9		vex 2806	. . . 4 |- x e. _V
10	9	eldm 4934	. . 3 |- ( x e. dom _I <-> E. y x _I y )
11	8, 10	mpbir 146	. 2 |- x e. dom _I
12	1, 11	mpgbir 1502	1 |- dom _I = _V

Syntax hints:   = wceq 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803   class class class wbr 4093   _I cid 4391   dom cdm 4731

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-dm 4741

This theorem is referenced by:  dmv  4953  iprc  5007  dmresi  5074  climshft2  11929