Theorem iss 5152

Description: A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
iss	|- ( A C_ _I <-> A = ( _I |` dom A ) )

Proof of Theorem iss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3348	. . . . . . 7 |- ( A C_ _I -> ( <. x , y >. e. A -> <. x , y >. e. _I ) )
2		vex 2973	. . . . . . . . 9 |- x e. _V
3		vex 2973	. . . . . . . . 9 |- y e. _V
4	2, 3	opeldm 5041	. . . . . . . 8 |- ( <. x , y >. e. A -> x e. dom A )
5	4	a1i 11	. . . . . . 7 |- ( A C_ _I -> ( <. x , y >. e. A -> x e. dom A ) )
6	1, 5	jcad 533	. . . . . 6 |- ( A C_ _I -> ( <. x , y >. e. A -> ( <. x , y >. e. _I /\ x e. dom A ) ) )
7		df-br 4291	. . . . . . . . 9 |- ( x _I y <-> <. x , y >. e. _I )
8	3	ideq 4990	. . . . . . . . 9 |- ( x _I y <-> x = y )
9	7, 8	bitr3i 251	. . . . . . . 8 |- ( <. x , y >. e. _I <-> x = y )
10	2	eldm2 5036	. . . . . . . . . 10 |- ( x e. dom A <-> E. y <. x , y >. e. A )
11		opeq2 4058	. . . . . . . . . . . . . . 15 |- ( x = y -> <. x , x >. = <. x , y >. )
12	11	eleq1d 2507	. . . . . . . . . . . . . 14 |- ( x = y -> ( <. x , x >. e. A <-> <. x , y >. e. A ) )
13	12	biimprcd 225	. . . . . . . . . . . . 13 |- ( <. x , y >. e. A -> ( x = y -> <. x , x >. e. A ) )
14	9, 13	syl5bi 217	. . . . . . . . . . . 12 |- ( <. x , y >. e. A -> ( <. x , y >. e. _I -> <. x , x >. e. A ) )
15	1, 14	sylcom 29	. . . . . . . . . . 11 |- ( A C_ _I -> ( <. x , y >. e. A -> <. x , x >. e. A ) )
16	15	exlimdv 1690	. . . . . . . . . 10 |- ( A C_ _I -> ( E. y <. x , y >. e. A -> <. x , x >. e. A ) )
17	10, 16	syl5bi 217	. . . . . . . . 9 |- ( A C_ _I -> ( x e. dom A -> <. x , x >. e. A ) )
18	12	imbi2d 316	. . . . . . . . 9 |- ( x = y -> ( ( x e. dom A -> <. x , x >. e. A ) <-> ( x e. dom A -> <. x , y >. e. A ) ) )
19	17, 18	syl5ibcom 220	. . . . . . . 8 |- ( A C_ _I -> ( x = y -> ( x e. dom A -> <. x , y >. e. A ) ) )
20	9, 19	syl5bi 217	. . . . . . 7 |- ( A C_ _I -> ( <. x , y >. e. _I -> ( x e. dom A -> <. x , y >. e. A ) ) )
21	20	impd 431	. . . . . 6 |- ( A C_ _I -> ( ( <. x , y >. e. _I /\ x e. dom A ) -> <. x , y >. e. A ) )
22	6, 21	impbid 191	. . . . 5 |- ( A C_ _I -> ( <. x , y >. e. A <-> ( <. x , y >. e. _I /\ x e. dom A ) ) )
23	3	opelres 5114	. . . . 5 |- ( <. x , y >. e. ( _I |` dom A ) <-> ( <. x , y >. e. _I /\ x e. dom A ) )
24	22, 23	syl6bbr 263	. . . 4 |- ( A C_ _I -> ( <. x , y >. e. A <-> <. x , y >. e. ( _I |` dom A ) ) )
25	24	alrimivv 1686	. . 3 |- ( A C_ _I -> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. ( _I |` dom A ) ) )
26		reli 4965	. . . . 5 |- Rel _I
27		relss 4925	. . . . 5 |- ( A C_ _I -> ( Rel _I -> Rel A ) )
28	26, 27	mpi 17	. . . 4 |- ( A C_ _I -> Rel A )
29		relres 5136	. . . 4 |- Rel ( _I |` dom A )
30		eqrel 4927	. . . 4 |- ( ( Rel A /\ Rel ( _I |` dom A ) ) -> ( A = ( _I |` dom A ) <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. ( _I |` dom A ) ) ) )
31	28, 29, 30	sylancl 662	. . 3 |- ( A C_ _I -> ( A = ( _I |` dom A ) <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. ( _I |` dom A ) ) ) )
32	25, 31	mpbird 232	. 2 |- ( A C_ _I -> A = ( _I |` dom A ) )
33		resss 5132	. . 3 |- ( _I |` dom A ) C_ _I
34		sseq1 3375	. . 3 |- ( A = ( _I |` dom A ) -> ( A C_ _I <-> ( _I |` dom A ) C_ _I ) )
35	33, 34	mpbiri 233	. 2 |- ( A = ( _I |` dom A ) -> A C_ _I )
36	32, 35	impbii 188	1 |- ( A C_ _I <-> A = ( _I |` dom A ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1367   = wceq 1369   E.wex 1586   e. wcel 1756   C_ wss 3326   <.cop 3881   class class class wbr 4290   _I cid 4629   dom cdm 4838   |` cres 4840   Rel wrel 4843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-dm 4848  df-res 4850

This theorem is referenced by:  funcocnv2  5663  trust  19802