Theorem dmi 3315

Description: The domain of the identity relation is the universe.

Assertion

Ref	Expression
dmi	|- dom I = V

Proof of Theorem dmi

Step	Hyp	Ref	Expression
1		a9e 1121	. . . . 5 |- E.y y = x
2		visset 1804	. . . . . . . 8 |- y e. V
3	2	ideq 3267	. . . . . . 7 |- (xIy <-> x = y)
4		eqcom 1469	. . . . . . 7 |- (x = y <-> y = x)
5	3, 4	bitr 173	. . . . . 6 |- (xIy <-> y = x)
6	5	exbii 1047	. . . . 5 |- (E.y xIy <-> E.y y = x)
7	1, 6	mpbir 190	. . . 4 |- E.y xIy
8		eqid 1468	. . . 4 |- x = x
9	7, 8	2th 716	. . 3 |- (E.y xIy <-> x = x)
10	9	abbii 1567	. 2 |- {x | E.y xIy} = {x | x = x}
11		df-dm 3178	. 2 |- dom I = {x | E.y xIy}
12		df-v 1803	. 2 |- V = {x | x = x}
13	10, 11, 12	3eqtr4 1497	1 |- dom I = V

Syntax hints:   = wceq 953  E.wex 977  {cab 1456  Vcvv 1802   class class class wbr 2609  Icid 2820  dom cdm 3160

This theorem is referenced by:  dmv 3316  inelv 3346  dmresi 3383  fvi 3827  dmen 4388

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-dm 3178

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