Theorem iss 5005

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 3187	. . . . . . 7 |- ( A C_ _I -> ( <. x , y >. e. A -> <. x , y >. e. _I ) )
2		vex 2775	. . . . . . . . 9 |- x e. _V
3		vex 2775	. . . . . . . . 9 |- y e. _V
4	2, 3	opeldm 4881	. . . . . . . 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 307	. . . . . 6 |- ( A C_ _I -> ( <. x , y >. e. A -> ( <. x , y >. e. _I /\ x e. dom A ) ) )
7		df-br 4045	. . . . . . . . 9 |- ( x _I y <-> <. x , y >. e. _I )
8	3	ideq 4830	. . . . . . . . 9 |- ( x _I y <-> x = y )
9	7, 8	bitr3i 186	. . . . . . . 8 |- ( <. x , y >. e. _I <-> x = y )
10	2	eldm2 4876	. . . . . . . . . 10 |- ( x e. dom A <-> E. y <. x , y >. e. A )
11		opeq2 3820	. . . . . . . . . . . . . . 15 |- ( x = y -> <. x , x >. = <. x , y >. )
12	11	eleq1d 2274	. . . . . . . . . . . . . 14 |- ( x = y -> ( <. x , x >. e. A <-> <. x , y >. e. A ) )
13	12	biimprcd 160	. . . . . . . . . . . . 13 |- ( <. x , y >. e. A -> ( x = y -> <. x , x >. e. A ) )
14	9, 13	biimtrid 152	. . . . . . . . . . . 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 1842	. . . . . . . . . 10 |- ( A C_ _I -> ( E. y <. x , y >. e. A -> <. x , x >. e. A ) )
17	10, 16	biimtrid 152	. . . . . . . . 9 |- ( A C_ _I -> ( x e. dom A -> <. x , x >. e. A ) )
18	12	imbi2d 230	. . . . . . . . 9 |- ( x = y -> ( ( x e. dom A -> <. x , x >. e. A ) <-> ( x e. dom A -> <. x , y >. e. A ) ) )
19	17, 18	syl5ibcom 155	. . . . . . . 8 |- ( A C_ _I -> ( x = y -> ( x e. dom A -> <. x , y >. e. A ) ) )
20	9, 19	biimtrid 152	. . . . . . 7 |- ( A C_ _I -> ( <. x , y >. e. _I -> ( x e. dom A -> <. x , y >. e. A ) ) )
21	20	impd 254	. . . . . 6 |- ( A C_ _I -> ( ( <. x , y >. e. _I /\ x e. dom A ) -> <. x , y >. e. A ) )
22	6, 21	impbid 129	. . . . 5 |- ( A C_ _I -> ( <. x , y >. e. A <-> ( <. x , y >. e. _I /\ x e. dom A ) ) )
23	3	opelres 4964	. . . . 5 |- ( <. x , y >. e. ( _I |` dom A ) <-> ( <. x , y >. e. _I /\ x e. dom A ) )
24	22, 23	bitr4di 198	. . . 4 |- ( A C_ _I -> ( <. x , y >. e. A <-> <. x , y >. e. ( _I |` dom A ) ) )
25	24	alrimivv 1898	. . 3 |- ( A C_ _I -> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. ( _I |` dom A ) ) )
26		reli 4807	. . . . 5 |- Rel _I
27		relss 4762	. . . . 5 |- ( A C_ _I -> ( Rel _I -> Rel A ) )
28	26, 27	mpi 15	. . . 4 |- ( A C_ _I -> Rel A )
29		relres 4987	. . . 4 |- Rel ( _I |` dom A )
30		eqrel 4764	. . . 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 413	. . 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 167	. 2 |- ( A C_ _I -> A = ( _I |` dom A ) )
33		resss 4983	. . 3 |- ( _I |` dom A ) C_ _I
34		sseq1 3216	. . 3 |- ( A = ( _I |` dom A ) -> ( A C_ _I <-> ( _I |` dom A ) C_ _I ) )
35	33, 34	mpbiri 168	. 2 |- ( A = ( _I |` dom A ) -> A C_ _I )
36	32, 35	impbii 126	1 |- ( A C_ _I <-> A = ( _I |` dom A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1515   e. wcel 2176   C_ wss 3166   <.cop 3636   class class class wbr 4044   _I cid 4335   dom cdm 4675   |` cres 4677   Rel wrel 4680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-dm 4685  df-res 4687

This theorem is referenced by:  funcocnv2  5547