Theorem iss 4860

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 3086	. . . . . . 7 |- ( A C_ _I -> ( <. x , y >. e. A -> <. x , y >. e. _I ) )
2		vex 2684	. . . . . . . . 9 |- x e. _V
3		vex 2684	. . . . . . . . 9 |- y e. _V
4	2, 3	opeldm 4737	. . . . . . . 8 |- ( <. x , y >. e. A -> x e. dom A )
5	4	a1i 9	. . . . . . 7 |- ( A C_ _I -> ( <. x , y >. e. A -> x e. dom A ) )
6	1, 5	jcad 305	. . . . . 6 |- ( A C_ _I -> ( <. x , y >. e. A -> ( <. x , y >. e. _I /\ x e. dom A ) ) )
7		df-br 3925	. . . . . . . . 9 |- ( x _I y <-> <. x , y >. e. _I )
8	3	ideq 4686	. . . . . . . . 9 |- ( x _I y <-> x = y )
9	7, 8	bitr3i 185	. . . . . . . 8 |- ( <. x , y >. e. _I <-> x = y )
10	2	eldm2 4732	. . . . . . . . . 10 |- ( x e. dom A <-> E. y <. x , y >. e. A )
11		opeq2 3701	. . . . . . . . . . . . . . 15 |- ( x = y -> <. x , x >. = <. x , y >. )
12	11	eleq1d 2206	. . . . . . . . . . . . . 14 |- ( x = y -> ( <. x , x >. e. A <-> <. x , y >. e. A ) )
13	12	biimprcd 159	. . . . . . . . . . . . 13 |- ( <. x , y >. e. A -> ( x = y -> <. x , x >. e. A ) )
14	9, 13	syl5bi 151	. . . . . . . . . . . 12 |- ( <. x , y >. e. A -> ( <. x , y >. e. _I -> <. x , x >. e. A ) )
15	1, 14	sylcom 28	. . . . . . . . . . 11 |- ( A C_ _I -> ( <. x , y >. e. A -> <. x , x >. e. A ) )
16	15	exlimdv 1791	. . . . . . . . . 10 |- ( A C_ _I -> ( E. y <. x , y >. e. A -> <. x , x >. e. A ) )
17	10, 16	syl5bi 151	. . . . . . . . 9 |- ( A C_ _I -> ( x e. dom A -> <. x , x >. e. A ) )
18	12	imbi2d 229	. . . . . . . . 9 |- ( x = y -> ( ( x e. dom A -> <. x , x >. e. A ) <-> ( x e. dom A -> <. x , y >. e. A ) ) )
19	17, 18	syl5ibcom 154	. . . . . . . 8 |- ( A C_ _I -> ( x = y -> ( x e. dom A -> <. x , y >. e. A ) ) )
20	9, 19	syl5bi 151	. . . . . . 7 |- ( A C_ _I -> ( <. x , y >. e. _I -> ( x e. dom A -> <. x , y >. e. A ) ) )
21	20	impd 252	. . . . . 6 |- ( A C_ _I -> ( ( <. x , y >. e. _I /\ x e. dom A ) -> <. x , y >. e. A ) )
22	6, 21	impbid 128	. . . . 5 |- ( A C_ _I -> ( <. x , y >. e. A <-> ( <. x , y >. e. _I /\ x e. dom A ) ) )
23	3	opelres 4819	. . . . 5 |- ( <. x , y >. e. ( _I |` dom A ) <-> ( <. x , y >. e. _I /\ x e. dom A ) )
24	22, 23	syl6bbr 197	. . . 4 |- ( A C_ _I -> ( <. x , y >. e. A <-> <. x , y >. e. ( _I |` dom A ) ) )
25	24	alrimivv 1847	. . 3 |- ( A C_ _I -> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. ( _I |` dom A ) ) )
26		reli 4663	. . . . 5 |- Rel _I
27		relss 4621	. . . . 5 |- ( A C_ _I -> ( Rel _I -> Rel A ) )
28	26, 27	mpi 15	. . . 4 |- ( A C_ _I -> Rel A )
29		relres 4842	. . . 4 |- Rel ( _I |` dom A )
30		eqrel 4623	. . . 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 409	. . 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 166	. 2 |- ( A C_ _I -> A = ( _I |` dom A ) )
33		resss 4838	. . 3 |- ( _I |` dom A ) C_ _I
34		sseq1 3115	. . 3 |- ( A = ( _I |` dom A ) -> ( A C_ _I <-> ( _I |` dom A ) C_ _I ) )
35	33, 34	mpbiri 167	. 2 |- ( A = ( _I |` dom A ) -> A C_ _I )
36	32, 35	impbii 125	1 |- ( A C_ _I <-> A = ( _I |` dom A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1329   = wceq 1331   E.wex 1468   e. wcel 1480   C_ wss 3066   <.cop 3525   class class class wbr 3924   _I cid 4205   dom cdm 4534   |` cres 4536   Rel wrel 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-dm 4544  df-res 4546

This theorem is referenced by:  funcocnv2  5385