Theorem ressffth 16598

Description: The inclusion functor from a full subcategory is a full and faithful functor, see also remark 4.4(2) in [Adamek] p. 49. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
ressffth.d	|- D = ( C ↾s S )
ressffth.i	|- I = (idfunc ` D )

Assertion

Ref	Expression
ressffth	|- ( ( C e. Cat /\ S e. V ) -> I e. ( ( D Full C ) i^i ( D Faith C ) ) )

Proof of Theorem ressffth

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

Step	Hyp	Ref	Expression
1		relfunc 16522	. . 3 |- Rel ( D Func D )
2		ressffth.d	. . . . 5 |- D = ( C ↾s S )
3		resscat 16512	. . . . 5 |- ( ( C e. Cat /\ S e. V ) -> ( C ↾s S ) e. Cat )
4	2, 3	syl5eqel 2705	. . . 4 |- ( ( C e. Cat /\ S e. V ) -> D e. Cat )
5		ressffth.i	. . . . 5 |- I = (idfunc ` D )
6	5	idfucl 16541	. . . 4 |- ( D e. Cat -> I e. ( D Func D ) )
7	4, 6	syl 17	. . 3 |- ( ( C e. Cat /\ S e. V ) -> I e. ( D Func D ) )
8		1st2nd 7214	. . 3 |- ( ( Rel ( D Func D ) /\ I e. ( D Func D ) ) -> I = <. ( 1st ` I ) , ( 2nd ` I ) >. )
9	1, 7, 8	sylancr 695	. 2 |- ( ( C e. Cat /\ S e. V ) -> I = <. ( 1st ` I ) , ( 2nd ` I ) >. )
10		eqidd 2623	. . . . . . . . 9 |- ( ( C e. Cat /\ S e. V ) -> ( Hom f ` D ) = ( Hom f ` D ) )
11		eqidd 2623	. . . . . . . . 9 |- ( ( C e. Cat /\ S e. V ) -> (compf ` D ) = (compf ` D ) )
12		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` C ) = ( Base ` C )
13	12	ressinbas 15936	. . . . . . . . . . . . 13 |- ( S e. V -> ( C ↾s S ) = ( C ↾s ( S i^i ( Base ` C ) ) ) )
14	13	adantl 482	. . . . . . . . . . . 12 |- ( ( C e. Cat /\ S e. V ) -> ( C ↾s S ) = ( C ↾s ( S i^i ( Base ` C ) ) ) )
15	2, 14	syl5eq 2668	. . . . . . . . . . 11 |- ( ( C e. Cat /\ S e. V ) -> D = ( C ↾s ( S i^i ( Base ` C ) ) ) )
16	15	fveq2d 6195	. . . . . . . . . 10 |- ( ( C e. Cat /\ S e. V ) -> ( Hom f ` D ) = ( Hom f ` ( C ↾s ( S i^i ( Base ` C ) ) ) ) )
17		eqid 2622	. . . . . . . . . . . 12 |- ( Hom f ` C ) = ( Hom f ` C )
18		simpl 473	. . . . . . . . . . . 12 |- ( ( C e. Cat /\ S e. V ) -> C e. Cat )
19		inss2 3834	. . . . . . . . . . . . 13 |- ( S i^i ( Base ` C ) ) C_ ( Base ` C )
20	19	a1i 11	. . . . . . . . . . . 12 |- ( ( C e. Cat /\ S e. V ) -> ( S i^i ( Base ` C ) ) C_ ( Base ` C ) )
21		eqid 2622	. . . . . . . . . . . 12 |- ( C ↾s ( S i^i ( Base ` C ) ) ) = ( C ↾s ( S i^i ( Base ` C ) ) )
22		eqid 2622	. . . . . . . . . . . 12 |- ( C |`cat ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) = ( C |`cat ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) )
23	12, 17, 18, 20, 21, 22	fullresc 16511	. . . . . . . . . . 11 |- ( ( C e. Cat /\ S e. V ) -> ( ( Hom f ` ( C ↾s ( S i^i ( Base ` C ) ) ) ) = ( Hom f ` ( C |`cat ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) ) /\ (compf ` ( C ↾s ( S i^i ( Base ` C ) ) ) ) = (compf ` ( C |`cat ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) ) ) )
24	23	simpld 475	. . . . . . . . . 10 |- ( ( C e. Cat /\ S e. V ) -> ( Hom f ` ( C ↾s ( S i^i ( Base ` C ) ) ) ) = ( Hom f ` ( C |`cat ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) ) )
25	16, 24	eqtrd 2656	. . . . . . . . 9 |- ( ( C e. Cat /\ S e. V ) -> ( Hom f ` D ) = ( Hom f ` ( C |`cat ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) ) )
26	15	fveq2d 6195	. . . . . . . . . 10 |- ( ( C e. Cat /\ S e. V ) -> (compf ` D ) = (compf ` ( C ↾s ( S i^i ( Base ` C ) ) ) ) )
27	23	simprd 479	. . . . . . . . . 10 |- ( ( C e. Cat /\ S e. V ) -> (compf ` ( C ↾s ( S i^i ( Base ` C ) ) ) ) = (compf ` ( C |`cat ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) ) )
28	26, 27	eqtrd 2656	. . . . . . . . 9 |- ( ( C e. Cat /\ S e. V ) -> (compf ` D ) = (compf ` ( C |`cat ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) ) )
29		ovex 6678	. . . . . . . . . . 11 |- ( C ↾s S ) e. _V
30	2, 29	eqeltri 2697	. . . . . . . . . 10 |- D e. _V
31	30	a1i 11	. . . . . . . . 9 |- ( ( C e. Cat /\ S e. V ) -> D e. _V )
32		ovex 6678	. . . . . . . . . 10 |- ( C |`cat ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) e. _V
33	32	a1i 11	. . . . . . . . 9 |- ( ( C e. Cat /\ S e. V ) -> ( C |`cat ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) e. _V )
34	10, 11, 25, 28, 31, 31, 31, 33	funcpropd 16560	. . . . . . . 8 |- ( ( C e. Cat /\ S e. V ) -> ( D Func D ) = ( D Func ( C |`cat ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) ) )
35	12, 17, 18, 20	fullsubc 16510	. . . . . . . . 9 |- ( ( C e. Cat /\ S e. V ) -> ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) e. (Subcat ` C ) )
36		funcres2 16558	. . . . . . . . 9 |- ( ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) e. (Subcat ` C ) -> ( D Func ( C |`cat ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) ) C_ ( D Func C ) )
37	35, 36	syl 17	. . . . . . . 8 |- ( ( C e. Cat /\ S e. V ) -> ( D Func ( C |`cat ( ( Hom f ` C ) |` ( ( S i^i ( Base ` C ) ) X. ( S i^i ( Base ` C ) ) ) ) ) ) C_ ( D Func C ) )
38	34, 37	eqsstrd 3639	. . . . . . 7 |- ( ( C e. Cat /\ S e. V ) -> ( D Func D ) C_ ( D Func C ) )
39	38, 7	sseldd 3604	. . . . . 6 |- ( ( C e. Cat /\ S e. V ) -> I e. ( D Func C ) )
40	9, 39	eqeltrrd 2702	. . . . 5 |- ( ( C e. Cat /\ S e. V ) -> <. ( 1st ` I ) , ( 2nd ` I ) >. e. ( D Func C ) )
41		df-br 4654	. . . . 5 |- ( ( 1st ` I ) ( D Func C ) ( 2nd ` I ) <-> <. ( 1st ` I ) , ( 2nd ` I ) >. e. ( D Func C ) )
42	40, 41	sylibr 224	. . . 4 |- ( ( C e. Cat /\ S e. V ) -> ( 1st ` I ) ( D Func C ) ( 2nd ` I ) )
43		f1oi 6174	. . . . . 6 |- ( _I |` ( x ( Hom ` D ) y ) ) : ( x ( Hom ` D ) y ) -1-1-onto-> ( x ( Hom ` D ) y )
44		eqid 2622	. . . . . . . 8 |- ( Base ` D ) = ( Base ` D )
45	4	adantr 481	. . . . . . . 8 |- ( ( ( C e. Cat /\ S e. V ) /\ ( x e. ( Base ` D ) /\ y e. ( Base ` D ) ) ) -> D e. Cat )
46		eqid 2622	. . . . . . . 8 |- ( Hom ` D ) = ( Hom ` D )
47		simprl 794	. . . . . . . 8 |- ( ( ( C e. Cat /\ S e. V ) /\ ( x e. ( Base ` D ) /\ y e. ( Base ` D ) ) ) -> x e. ( Base ` D ) )
48		simprr 796	. . . . . . . 8 |- ( ( ( C e. Cat /\ S e. V ) /\ ( x e. ( Base ` D ) /\ y e. ( Base ` D ) ) ) -> y e. ( Base ` D ) )
49	5, 44, 45, 46, 47, 48	idfu2nd 16537	. . . . . . 7 |- ( ( ( C e. Cat /\ S e. V ) /\ ( x e. ( Base ` D ) /\ y e. ( Base ` D ) ) ) -> ( x ( 2nd ` I ) y ) = ( _I |` ( x ( Hom ` D ) y ) ) )
50		eqidd 2623	. . . . . . 7 |- ( ( ( C e. Cat /\ S e. V ) /\ ( x e. ( Base ` D ) /\ y e. ( Base ` D ) ) ) -> ( x ( Hom ` D ) y ) = ( x ( Hom ` D ) y ) )
51		eqid 2622	. . . . . . . . . 10 |- ( Hom ` C ) = ( Hom ` C )
52	2, 51	resshom 16078	. . . . . . . . 9 |- ( S e. V -> ( Hom ` C ) = ( Hom ` D ) )
53	52	ad2antlr 763	. . . . . . . 8 |- ( ( ( C e. Cat /\ S e. V ) /\ ( x e. ( Base ` D ) /\ y e. ( Base ` D ) ) ) -> ( Hom ` C ) = ( Hom ` D ) )
54	5, 44, 45, 47	idfu1 16540	. . . . . . . 8 |- ( ( ( C e. Cat /\ S e. V ) /\ ( x e. ( Base ` D ) /\ y e. ( Base ` D ) ) ) -> ( ( 1st ` I ) ` x ) = x )
55	5, 44, 45, 48	idfu1 16540	. . . . . . . 8 |- ( ( ( C e. Cat /\ S e. V ) /\ ( x e. ( Base ` D ) /\ y e. ( Base ` D ) ) ) -> ( ( 1st ` I ) ` y ) = y )
56	53, 54, 55	oveq123d 6671	. . . . . . 7 |- ( ( ( C e. Cat /\ S e. V ) /\ ( x e. ( Base ` D ) /\ y e. ( Base ` D ) ) ) -> ( ( ( 1st ` I ) ` x ) ( Hom ` C ) ( ( 1st ` I ) ` y ) ) = ( x ( Hom ` D ) y ) )
57	49, 50, 56	f1oeq123d 6133	. . . . . 6 |- ( ( ( C e. Cat /\ S e. V ) /\ ( x e. ( Base ` D ) /\ y e. ( Base ` D ) ) ) -> ( ( x ( 2nd ` I ) y ) : ( x ( Hom ` D ) y ) -1-1-onto-> ( ( ( 1st ` I ) ` x ) ( Hom ` C ) ( ( 1st ` I ) ` y ) ) <-> ( _I |` ( x ( Hom ` D ) y ) ) : ( x ( Hom ` D ) y ) -1-1-onto-> ( x ( Hom ` D ) y ) ) )
58	43, 57	mpbiri 248	. . . . 5 |- ( ( ( C e. Cat /\ S e. V ) /\ ( x e. ( Base ` D ) /\ y e. ( Base ` D ) ) ) -> ( x ( 2nd ` I ) y ) : ( x ( Hom ` D ) y ) -1-1-onto-> ( ( ( 1st ` I ) ` x ) ( Hom ` C ) ( ( 1st ` I ) ` y ) ) )
59	58	ralrimivva 2971	. . . 4 |- ( ( C e. Cat /\ S e. V ) -> A. x e. ( Base ` D ) A. y e. ( Base ` D ) ( x ( 2nd ` I ) y ) : ( x ( Hom ` D ) y ) -1-1-onto-> ( ( ( 1st ` I ) ` x ) ( Hom ` C ) ( ( 1st ` I ) ` y ) ) )
60	44, 46, 51	isffth2 16576	. . . 4 |- ( ( 1st ` I ) ( ( D Full C ) i^i ( D Faith C ) ) ( 2nd ` I ) <-> ( ( 1st ` I ) ( D Func C ) ( 2nd ` I ) /\ A. x e. ( Base ` D ) A. y e. ( Base ` D ) ( x ( 2nd ` I ) y ) : ( x ( Hom ` D ) y ) -1-1-onto-> ( ( ( 1st ` I ) ` x ) ( Hom ` C ) ( ( 1st ` I ) ` y ) ) ) )
61	42, 59, 60	sylanbrc 698	. . 3 |- ( ( C e. Cat /\ S e. V ) -> ( 1st ` I ) ( ( D Full C ) i^i ( D Faith C ) ) ( 2nd ` I ) )
62		df-br 4654	. . 3 |- ( ( 1st ` I ) ( ( D Full C ) i^i ( D Faith C ) ) ( 2nd ` I ) <-> <. ( 1st ` I ) , ( 2nd ` I ) >. e. ( ( D Full C ) i^i ( D Faith C ) ) )
63	61, 62	sylib 208	. 2 |- ( ( C e. Cat /\ S e. V ) -> <. ( 1st ` I ) , ( 2nd ` I ) >. e. ( ( D Full C ) i^i ( D Faith C ) ) )
64	9, 63	eqeltrd 2701	1 |- ( ( C e. Cat /\ S e. V ) -> I e. ( ( D Full C ) i^i ( D Faith C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   <.cop 4183   class class class wbr 4653   _I cid 5023   X. cxp 5112   |` cres 5116   Rel wrel 5119   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   ↾_s cress 15858   Hom chom 15952   Catccat 16325   Hom _f chomf 16327  comp_fccomf 16328   |`_cat cresc 16468  Subcatcsubc 16469   Func cfunc 16514  id_funccidfu 16515   Full cful 16562   Faith cfth 16563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-homf 16331  df-comf 16332  df-ssc 16470  df-resc 16471  df-subc 16472  df-func 16518  df-idfu 16519  df-full 16564  df-fth 16565

This theorem is referenced by: (None)