Theorem iss

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)

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

Proof

Step	Hyp	Ref	Expression
1		vex	|- x e. _V
2		vex	|- y e. _V
3	1 2	opeldm	|- ( <. x , y >. e. A -> x e. dom A )
4	3	a1i	|- ( A C_ _I -> ( <. x , y >. e. A -> x e. dom A ) )
5		ssel	|- ( A C_ _I -> ( <. x , y >. e. A -> <. x , y >. e. _I ) )
6	4 5	jcad	|- ( A C_ _I -> ( <. x , y >. e. A -> ( x e. dom A /\ <. x , y >. e. _I ) ) )
7		df-br	|- ( x _I y <-> <. x , y >. e. _I )
8	2	ideq	|- ( x _I y <-> x = y )
9	7 8	bitr3i	|- ( <. x , y >. e. _I <-> x = y )
10	1	eldm2	|- ( x e. dom A <-> E. y <. x , y >. e. A )
11		opeq2	|- ( x = y -> <. x , x >. = <. x , y >. )
12	11	eleq1d	|- ( x = y -> ( <. x , x >. e. A <-> <. x , y >. e. A ) )
13	12	biimprcd	|- ( <. x , y >. e. A -> ( x = y -> <. x , x >. e. A ) )
14	9 13	syl5bi	|- ( <. x , y >. e. A -> ( <. x , y >. e. _I -> <. x , x >. e. A ) )
15	5 14	sylcom	|- ( A C_ _I -> ( <. x , y >. e. A -> <. x , x >. e. A ) )
16	15	exlimdv	|- ( A C_ _I -> ( E. y <. x , y >. e. A -> <. x , x >. e. A ) )
17	10 16	syl5bi	|- ( A C_ _I -> ( x e. dom A -> <. x , x >. e. A ) )
18	12	imbi2d	|- ( x = y -> ( ( x e. dom A -> <. x , x >. e. A ) <-> ( x e. dom A -> <. x , y >. e. A ) ) )
19	17 18	syl5ibcom	|- ( A C_ _I -> ( x = y -> ( x e. dom A -> <. x , y >. e. A ) ) )
20	9 19	syl5bi	|- ( A C_ _I -> ( <. x , y >. e. _I -> ( x e. dom A -> <. x , y >. e. A ) ) )
21	20	impcomd	|- ( A C_ _I -> ( ( x e. dom A /\ <. x , y >. e. _I ) -> <. x , y >. e. A ) )
22	6 21	impbid	|- ( A C_ _I -> ( <. x , y >. e. A <-> ( x e. dom A /\ <. x , y >. e. _I ) ) )
23	2	opelresi	|- ( <. x , y >. e. ( _I |` dom A ) <-> ( x e. dom A /\ <. x , y >. e. _I ) )
24	22 23	syl6bbr	|- ( A C_ _I -> ( <. x , y >. e. A <-> <. x , y >. e. ( _I |` dom A ) ) )
25	24	alrimivv	|- ( A C_ _I -> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. ( _I |` dom A ) ) )
26		reli	|- Rel _I
27		relss	|- ( A C_ _I -> ( Rel _I -> Rel A ) )
28	26 27	mpi	|- ( A C_ _I -> Rel A )
29		relres	|- Rel ( _I |` dom A )
30		eqrel	|- ( ( 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	|- ( 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	|- ( A C_ _I -> A = ( _I |` dom A ) )
33		resss	|- ( _I |` dom A ) C_ _I
34		sseq1	|- ( A = ( _I |` dom A ) -> ( A C_ _I <-> ( _I |` dom A ) C_ _I ) )
35	33 34	mpbiri	|- ( A = ( _I |` dom A ) -> A C_ _I )
36	32 35	impbii	|- ( A C_ _I <-> A = ( _I |` dom A ) )