Theorem iss 4654

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 2939	. . . . . . 7 |- ( A C_ _I -> ( <. x , y >. e. A -> <. x , y >. e. _I ) )
2		vex 2560	. . . . . . . . 9 |- x e. _V
3		vex 2560	. . . . . . . . 9 |- y e. _V
4	2, 3	opeldm 4538	. . . . . . . 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 291	. . . . . 6 |- ( A C_ _I -> ( <. x , y >. e. A -> ( <. x , y >. e. _I /\ x e. dom A ) ) )
7		df-br 3765	. . . . . . . . 9 |- ( x _I y <-> <. x , y >. e. _I )
8	3	ideq 4488	. . . . . . . . 9 |- ( x _I y <-> x = y )
9	7, 8	bitr3i 175	. . . . . . . 8 |- ( <. x , y >. e. _I <-> x = y )
10	2	eldm2 4533	. . . . . . . . . 10 |- ( x e. dom A <-> E. y <. x , y >. e. A )
11		opeq2 3550	. . . . . . . . . . . . . . 15 |- ( x = y -> <. x , x >. = <. x , y >. )
12	11	eleq1d 2106	. . . . . . . . . . . . . 14 |- ( x = y -> ( <. x , x >. e. A <-> <. x , y >. e. A ) )
13	12	biimprcd 149	. . . . . . . . . . . . 13 |- ( <. x , y >. e. A -> ( x = y -> <. x , x >. e. A ) )
14	9, 13	syl5bi 141	. . . . . . . . . . . 12 |- ( <. x , y >. e. A -> ( <. x , y >. e. _I -> <. x , x >. e. A ) )
15	1, 14	sylcom 25	. . . . . . . . . . 11 |- ( A C_ _I -> ( <. x , y >. e. A -> <. x , x >. e. A ) )
16	15	exlimdv 1700	. . . . . . . . . 10 |- ( A C_ _I -> ( E. y <. x , y >. e. A -> <. x , x >. e. A ) )
17	10, 16	syl5bi 141	. . . . . . . . 9 |- ( A C_ _I -> ( x e. dom A -> <. x , x >. e. A ) )
18	12	imbi2d 219	. . . . . . . . 9 |- ( x = y -> ( ( x e. dom A -> <. x , x >. e. A ) <-> ( x e. dom A -> <. x , y >. e. A ) ) )
19	17, 18	syl5ibcom 144	. . . . . . . 8 |- ( A C_ _I -> ( x = y -> ( x e. dom A -> <. x , y >. e. A ) ) )
20	9, 19	syl5bi 141	. . . . . . 7 |- ( A C_ _I -> ( <. x , y >. e. _I -> ( x e. dom A -> <. x , y >. e. A ) ) )
21	20	impd 242	. . . . . 6 |- ( A C_ _I -> ( ( <. x , y >. e. _I /\ x e. dom A ) -> <. x , y >. e. A ) )
22	6, 21	impbid 120	. . . . 5 |- ( A C_ _I -> ( <. x , y >. e. A <-> ( <. x , y >. e. _I /\ x e. dom A ) ) )
23	3	opelres 4617	. . . . 5 |- ( <. x , y >. e. ( _I |` dom A ) <-> ( <. x , y >. e. _I /\ x e. dom A ) )
24	22, 23	syl6bbr 187	. . . 4 |- ( A C_ _I -> ( <. x , y >. e. A <-> <. x , y >. e. ( _I |` dom A ) ) )
25	24	alrimivv 1755	. . 3 |- ( A C_ _I -> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. ( _I |` dom A ) ) )
26		reli 4465	. . . . 5 |- Rel _I
27		relss 4427	. . . . 5 |- ( A C_ _I -> ( Rel _I -> Rel A ) )
28	26, 27	mpi 15	. . . 4 |- ( A C_ _I -> Rel A )
29		relres 4639	. . . 4 |- Rel ( _I |` dom A )
30		eqrel 4429	. . . 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 392	. . 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 156	. 2 |- ( A C_ _I -> A = ( _I |` dom A ) )
33		resss 4635	. . 3 |- ( _I |` dom A ) C_ _I
34		sseq1 2966	. . 3 |- ( A = ( _I |` dom A ) -> ( A C_ _I <-> ( _I |` dom A ) C_ _I ) )
35	33, 34	mpbiri 157	. 2 |- ( A = ( _I |` dom A ) -> A C_ _I )
36	32, 35	impbii 117	1 |- ( A C_ _I <-> A = ( _I |` dom A ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   C_ wss 2917   <.cop 3378   class class class wbr 3764   _I cid 4025   dom cdm 4345   |` cres 4347   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-dm 4355  df-res 4357

This theorem is referenced by:  funcocnv2  5151