Theorem dmi 4996

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 3541	. 2 |- ( dom _I = _V <-> A. x x e. dom _I )
2		a9ev 1749	. . . 4 |- E. y y = x
3		vex 2824	. . . . . . 7 |- y e. _V
4	3	ideq 4932	. . . . . 6 |- ( x _I y <-> x = y )
5		equcom 1758	. . . . . 6 |- ( x = y <-> y = x )
6	4, 5	bitri 184	. . . . 5 |- ( x _I y <-> y = x )
7	6	exbii 1658	. . . 4 |- ( E. y x _I y <-> E. y y = x )
8	2, 7	mpbir 146	. . 3 |- E. y x _I y
9		vex 2824	. . . 4 |- x e. _V
10	9	eldm 4978	. . 3 |- ( x e. dom _I <-> E. y x _I y )
11	8, 10	mpbir 146	. 2 |- x e. dom _I
12	1, 11	mpgbir 1506	1 |- dom _I = _V

This proof depends on syntax axioms:   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   _I cid 4433   dom cdm 4774

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-dm 4784

This theorem is used by:  dmv  4997  iprc  5051  dmresi  5118  climshft2  12072