Theorem yoniso 16925

Description: If the codomain is recoverable from a hom-set, then the Yoneda embedding is injective on objects, and hence is an isomorphism from C into a full subcategory of a presheaf category. (Contributed by Mario Carneiro, 30-Jan-2017.)

Hypotheses

Ref	Expression
yoniso.y	|- Y = (Yon ` C )
yoniso.o	|- O = (oppCat ` C )
yoniso.s	|- S = ( SetCat ` U )
yoniso.d	|- D = (CatCat ` V )
yoniso.b	|- B = ( Base ` D )
yoniso.i	|- I = ( Iso ` D )
yoniso.q	|- Q = ( O FuncCat S )
yoniso.e	|- E = ( Q ↾s ran ( 1st ` Y ) )
yoniso.v	|- ( ph -> V e. X )
yoniso.c	|- ( ph -> C e. B )
yoniso.u	|- ( ph -> U e. W )
yoniso.h	|- ( ph -> ran ( Hom f ` C ) C_ U )
yoniso.eb	|- ( ph -> E e. B )
yoniso.1	|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( F ` ( x ( Hom ` C ) y ) ) = y )

Assertion

Ref	Expression
yoniso	|- ( ph -> Y e. ( C I E ) )

Distinct variable groups:   x, y, C   y, F   ph, x, y   x, Y, y

Allowed substitution hints:   B( x, y)   D( x, y)   Q( x, y)   S( x, y)   U( x, y)   E( x, y)   F( x)   I( x, y)   O( x, y)   V( x, y)   W( x, y)   X( x, y)

Proof of Theorem yoniso

Step	Hyp	Ref	Expression
1		relfunc 16522	. . . 4 |- Rel ( C Func Q )
2		yoniso.y	. . . . 5 |- Y = (Yon ` C )
3		yoniso.d	. . . . . . . 8 |- D = (CatCat ` V )
4		yoniso.b	. . . . . . . 8 |- B = ( Base ` D )
5		yoniso.v	. . . . . . . 8 |- ( ph -> V e. X )
6	3, 4, 5	catcbas 16747	. . . . . . 7 |- ( ph -> B = ( V i^i Cat ) )
7		inss2 3834	. . . . . . 7 |- ( V i^i Cat ) C_ Cat
8	6, 7	syl6eqss 3655	. . . . . 6 |- ( ph -> B C_ Cat )
9		yoniso.c	. . . . . 6 |- ( ph -> C e. B )
10	8, 9	sseldd 3604	. . . . 5 |- ( ph -> C e. Cat )
11		yoniso.o	. . . . 5 |- O = (oppCat ` C )
12		yoniso.s	. . . . 5 |- S = ( SetCat ` U )
13		yoniso.q	. . . . 5 |- Q = ( O FuncCat S )
14		yoniso.u	. . . . 5 |- ( ph -> U e. W )
15		yoniso.h	. . . . 5 |- ( ph -> ran ( Hom f ` C ) C_ U )
16	2, 10, 11, 12, 13, 14, 15	yoncl 16902	. . . 4 |- ( ph -> Y e. ( C Func Q ) )
17		1st2nd 7214	. . . 4 |- ( ( Rel ( C Func Q ) /\ Y e. ( C Func Q ) ) -> Y = <. ( 1st ` Y ) , ( 2nd ` Y ) >. )
18	1, 16, 17	sylancr 695	. . 3 |- ( ph -> Y = <. ( 1st ` Y ) , ( 2nd ` Y ) >. )
19	2, 11, 12, 13, 10, 14, 15	yonffth 16924	. . . . 5 |- ( ph -> Y e. ( ( C Full Q ) i^i ( C Faith Q ) ) )
20	18, 19	eqeltrrd 2702	. . . 4 |- ( ph -> <. ( 1st ` Y ) , ( 2nd ` Y ) >. e. ( ( C Full Q ) i^i ( C Faith Q ) ) )
21		eqid 2622	. . . . . 6 |- ( Base ` C ) = ( Base ` C )
22		yoniso.e	. . . . . 6 |- E = ( Q ↾s ran ( 1st ` Y ) )
23	11	oppccat 16382	. . . . . . . 8 |- ( C e. Cat -> O e. Cat )
24	10, 23	syl 17	. . . . . . 7 |- ( ph -> O e. Cat )
25	12	setccat 16735	. . . . . . . 8 |- ( U e. W -> S e. Cat )
26	14, 25	syl 17	. . . . . . 7 |- ( ph -> S e. Cat )
27	13, 24, 26	fuccat 16630	. . . . . 6 |- ( ph -> Q e. Cat )
28		fvex 6201	. . . . . . . 8 |- ( 1st ` Y ) e. _V
29	28	rnex 7100	. . . . . . 7 |- ran ( 1st ` Y ) e. _V
30	29	a1i 11	. . . . . 6 |- ( ph -> ran ( 1st ` Y ) e. _V )
31	13	fucbas 16620	. . . . . . . . 9 |- ( O Func S ) = ( Base ` Q )
32		1st2ndbr 7217	. . . . . . . . . 10 |- ( ( Rel ( C Func Q ) /\ Y e. ( C Func Q ) ) -> ( 1st ` Y ) ( C Func Q ) ( 2nd ` Y ) )
33	1, 16, 32	sylancr 695	. . . . . . . . 9 |- ( ph -> ( 1st ` Y ) ( C Func Q ) ( 2nd ` Y ) )
34	21, 31, 33	funcf1 16526	. . . . . . . 8 |- ( ph -> ( 1st ` Y ) : ( Base ` C ) --> ( O Func S ) )
35		ffn 6045	. . . . . . . 8 |- ( ( 1st ` Y ) : ( Base ` C ) --> ( O Func S ) -> ( 1st ` Y ) Fn ( Base ` C ) )
36	34, 35	syl 17	. . . . . . 7 |- ( ph -> ( 1st ` Y ) Fn ( Base ` C ) )
37		dffn3 6054	. . . . . . 7 |- ( ( 1st ` Y ) Fn ( Base ` C ) <-> ( 1st ` Y ) : ( Base ` C ) --> ran ( 1st ` Y ) )
38	36, 37	sylib 208	. . . . . 6 |- ( ph -> ( 1st ` Y ) : ( Base ` C ) --> ran ( 1st ` Y ) )
39	21, 22, 27, 30, 38	ffthres2c 16600	. . . . 5 |- ( ph -> ( ( 1st ` Y ) ( ( C Full Q ) i^i ( C Faith Q ) ) ( 2nd ` Y ) <-> ( 1st ` Y ) ( ( C Full E ) i^i ( C Faith E ) ) ( 2nd ` Y ) ) )
40		df-br 4654	. . . . 5 |- ( ( 1st ` Y ) ( ( C Full Q ) i^i ( C Faith Q ) ) ( 2nd ` Y ) <-> <. ( 1st ` Y ) , ( 2nd ` Y ) >. e. ( ( C Full Q ) i^i ( C Faith Q ) ) )
41		df-br 4654	. . . . 5 |- ( ( 1st ` Y ) ( ( C Full E ) i^i ( C Faith E ) ) ( 2nd ` Y ) <-> <. ( 1st ` Y ) , ( 2nd ` Y ) >. e. ( ( C Full E ) i^i ( C Faith E ) ) )
42	39, 40, 41	3bitr3g 302	. . . 4 |- ( ph -> ( <. ( 1st ` Y ) , ( 2nd ` Y ) >. e. ( ( C Full Q ) i^i ( C Faith Q ) ) <-> <. ( 1st ` Y ) , ( 2nd ` Y ) >. e. ( ( C Full E ) i^i ( C Faith E ) ) ) )
43	20, 42	mpbid 222	. . 3 |- ( ph -> <. ( 1st ` Y ) , ( 2nd ` Y ) >. e. ( ( C Full E ) i^i ( C Faith E ) ) )
44	18, 43	eqeltrd 2701	. 2 |- ( ph -> Y e. ( ( C Full E ) i^i ( C Faith E ) ) )
45		fveq2 6191	. . . . . . . . 9 |- ( ( ( 1st ` Y ) ` x ) = ( ( 1st ` Y ) ` y ) -> ( 1st ` ( ( 1st ` Y ) ` x ) ) = ( 1st ` ( ( 1st ` Y ) ` y ) ) )
46	45	fveq1d 6193	. . . . . . . 8 |- ( ( ( 1st ` Y ) ` x ) = ( ( 1st ` Y ) ` y ) -> ( ( 1st ` ( ( 1st ` Y ) ` x ) ) ` x ) = ( ( 1st ` ( ( 1st ` Y ) ` y ) ) ` x ) )
47	46	fveq2d 6195	. . . . . . 7 |- ( ( ( 1st ` Y ) ` x ) = ( ( 1st ` Y ) ` y ) -> ( F ` ( ( 1st ` ( ( 1st ` Y ) ` x ) ) ` x ) ) = ( F ` ( ( 1st ` ( ( 1st ` Y ) ` y ) ) ` x ) ) )
48		simpl 473	. . . . . . . . . 10 |- ( ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> x e. ( Base ` C ) )
49	48, 48	jca 554	. . . . . . . . 9 |- ( ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( x e. ( Base ` C ) /\ x e. ( Base ` C ) ) )
50		eleq1 2689	. . . . . . . . . . . . 13 |- ( y = x -> ( y e. ( Base ` C ) <-> x e. ( Base ` C ) ) )
51	50	anbi2d 740	. . . . . . . . . . . 12 |- ( y = x -> ( ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) <-> ( x e. ( Base ` C ) /\ x e. ( Base ` C ) ) ) )
52	51	anbi2d 740	. . . . . . . . . . 11 |- ( y = x -> ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) <-> ( ph /\ ( x e. ( Base ` C ) /\ x e. ( Base ` C ) ) ) ) )
53		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( y = x -> ( ( 1st ` Y ) ` y ) = ( ( 1st ` Y ) ` x ) )
54	53	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( y = x -> ( 1st ` ( ( 1st ` Y ) ` y ) ) = ( 1st ` ( ( 1st ` Y ) ` x ) ) )
55	54	fveq1d 6193	. . . . . . . . . . . . 13 |- ( y = x -> ( ( 1st ` ( ( 1st ` Y ) ` y ) ) ` x ) = ( ( 1st ` ( ( 1st ` Y ) ` x ) ) ` x ) )
56	55	fveq2d 6195	. . . . . . . . . . . 12 |- ( y = x -> ( F ` ( ( 1st ` ( ( 1st ` Y ) ` y ) ) ` x ) ) = ( F ` ( ( 1st ` ( ( 1st ` Y ) ` x ) ) ` x ) ) )
57		id 22	. . . . . . . . . . . 12 |- ( y = x -> y = x )
58	56, 57	eqeq12d 2637	. . . . . . . . . . 11 |- ( y = x -> ( ( F ` ( ( 1st ` ( ( 1st ` Y ) ` y ) ) ` x ) ) = y <-> ( F ` ( ( 1st ` ( ( 1st ` Y ) ` x ) ) ` x ) ) = x ) )
59	52, 58	imbi12d 334	. . . . . . . . . 10 |- ( y = x -> ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( F ` ( ( 1st ` ( ( 1st ` Y ) ` y ) ) ` x ) ) = y ) <-> ( ( ph /\ ( x e. ( Base ` C ) /\ x e. ( Base ` C ) ) ) -> ( F ` ( ( 1st ` ( ( 1st ` Y ) ` x ) ) ` x ) ) = x ) ) )
60	10	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> C e. Cat )
61		simprr 796	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> y e. ( Base ` C ) )
62		eqid 2622	. . . . . . . . . . . . 13 |- ( Hom ` C ) = ( Hom ` C )
63		simprl 794	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> x e. ( Base ` C ) )
64	2, 21, 60, 61, 62, 63	yon11 16904	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` ( ( 1st ` Y ) ` y ) ) ` x ) = ( x ( Hom ` C ) y ) )
65	64	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( F ` ( ( 1st ` ( ( 1st ` Y ) ` y ) ) ` x ) ) = ( F ` ( x ( Hom ` C ) y ) ) )
66		yoniso.1	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( F ` ( x ( Hom ` C ) y ) ) = y )
67	65, 66	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( F ` ( ( 1st ` ( ( 1st ` Y ) ` y ) ) ` x ) ) = y )
68	59, 67	chvarv 2263	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ x e. ( Base ` C ) ) ) -> ( F ` ( ( 1st ` ( ( 1st ` Y ) ` x ) ) ` x ) ) = x )
69	49, 68	sylan2 491	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( F ` ( ( 1st ` ( ( 1st ` Y ) ` x ) ) ` x ) ) = x )
70	69, 67	eqeq12d 2637	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( F ` ( ( 1st ` ( ( 1st ` Y ) ` x ) ) ` x ) ) = ( F ` ( ( 1st ` ( ( 1st ` Y ) ` y ) ) ` x ) ) <-> x = y ) )
71	47, 70	syl5ib 234	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( 1st ` Y ) ` x ) = ( ( 1st ` Y ) ` y ) -> x = y ) )
72	71	ralrimivva 2971	. . . . 5 |- ( ph -> A. x e. ( Base ` C ) A. y e. ( Base ` C ) ( ( ( 1st ` Y ) ` x ) = ( ( 1st ` Y ) ` y ) -> x = y ) )
73		dff13 6512	. . . . 5 |- ( ( 1st ` Y ) : ( Base ` C ) -1-1-> ( O Func S ) <-> ( ( 1st ` Y ) : ( Base ` C ) --> ( O Func S ) /\ A. x e. ( Base ` C ) A. y e. ( Base ` C ) ( ( ( 1st ` Y ) ` x ) = ( ( 1st ` Y ) ` y ) -> x = y ) ) )
74	34, 72, 73	sylanbrc 698	. . . 4 |- ( ph -> ( 1st ` Y ) : ( Base ` C ) -1-1-> ( O Func S ) )
75		f1f1orn 6148	. . . 4 |- ( ( 1st ` Y ) : ( Base ` C ) -1-1-> ( O Func S ) -> ( 1st ` Y ) : ( Base ` C ) -1-1-onto-> ran ( 1st ` Y ) )
76	74, 75	syl 17	. . 3 |- ( ph -> ( 1st ` Y ) : ( Base ` C ) -1-1-onto-> ran ( 1st ` Y ) )
77		frn 6053	. . . . . 6 |- ( ( 1st ` Y ) : ( Base ` C ) --> ( O Func S ) -> ran ( 1st ` Y ) C_ ( O Func S ) )
78	34, 77	syl 17	. . . . 5 |- ( ph -> ran ( 1st ` Y ) C_ ( O Func S ) )
79	22, 31	ressbas2 15931	. . . . 5 |- ( ran ( 1st ` Y ) C_ ( O Func S ) -> ran ( 1st ` Y ) = ( Base ` E ) )
80	78, 79	syl 17	. . . 4 |- ( ph -> ran ( 1st ` Y ) = ( Base ` E ) )
81		f1oeq3 6129	. . . 4 |- ( ran ( 1st ` Y ) = ( Base ` E ) -> ( ( 1st ` Y ) : ( Base ` C ) -1-1-onto-> ran ( 1st ` Y ) <-> ( 1st ` Y ) : ( Base ` C ) -1-1-onto-> ( Base ` E ) ) )
82	80, 81	syl 17	. . 3 |- ( ph -> ( ( 1st ` Y ) : ( Base ` C ) -1-1-onto-> ran ( 1st ` Y ) <-> ( 1st ` Y ) : ( Base ` C ) -1-1-onto-> ( Base ` E ) ) )
83	76, 82	mpbid 222	. 2 |- ( ph -> ( 1st ` Y ) : ( Base ` C ) -1-1-onto-> ( Base ` E ) )
84		eqid 2622	. . 3 |- ( Base ` E ) = ( Base ` E )
85		yoniso.eb	. . 3 |- ( ph -> E e. B )
86		yoniso.i	. . 3 |- I = ( Iso ` D )
87	3, 4, 21, 84, 5, 9, 85, 86	catciso 16757	. 2 |- ( ph -> ( Y e. ( C I E ) <-> ( Y e. ( ( C Full E ) i^i ( C Faith E ) ) /\ ( 1st ` Y ) : ( Base ` C ) -1-1-onto-> ( Base ` E ) ) ) )
88	44, 83, 87	mpbir2and 957	1 |- ( ph -> Y e. ( C I E ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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   ran crn 5115   Rel wrel 5119   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -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  oppCatcoppc 16371   Iso ciso 16406   Func cfunc 16514   Full cful 16562   Faith cfth 16563   FuncCat cfuc 16602   SetCatcsetc 16725  CatCatccatc 16744  Yoncyon 16889

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-int 4476  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-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-uz 11688  df-fz 12327  df-struct 15859  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-oppc 16372  df-sect 16407  df-inv 16408  df-iso 16409  df-ssc 16470  df-resc 16471  df-subc 16472  df-func 16518  df-idfu 16519  df-cofu 16520  df-full 16564  df-fth 16565  df-nat 16603  df-fuc 16604  df-setc 16726  df-catc 16745  df-xpc 16812  df-1stf 16813  df-2ndf 16814  df-prf 16815  df-evlf 16853  df-curf 16854  df-hof 16890  df-yon 16891

This theorem is referenced by: (None)