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Theorem yoniso 15532
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  =  ( Qs  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
StepHypRef Expression
1 relfunc 15209 . . . 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 )
63, 4, 5catcbas 15402 . . . . . . 7  |-  ( ph  ->  B  =  ( V  i^i  Cat ) )
7 inss2 3704 . . . . . . 7  |-  ( V  i^i  Cat )  C_  Cat
86, 7syl6eqss 3539 . . . . . 6  |-  ( ph  ->  B  C_  Cat )
9 yoniso.c . . . . . 6  |-  ( ph  ->  C  e.  B )
108, 9sseldd 3490 . . . . 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 )
162, 10, 11, 12, 13, 14, 15yoncl 15509 . . . 4  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
17 1st2nd 6831 . . . 4  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  Y  =  <. ( 1st `  Y
) ,  ( 2nd `  Y ) >. )
181, 16, 17sylancr 663 . . 3  |-  ( ph  ->  Y  =  <. ( 1st `  Y ) ,  ( 2nd `  Y
) >. )
192, 11, 12, 13, 10, 14, 15yonffth 15531 . . . . 5  |-  ( ph  ->  Y  e.  ( ( C Full  Q )  i^i  ( C Faith  Q ) ) )
2018, 19eqeltrrd 2532 . . . 4  |-  ( ph  -> 
<. ( 1st `  Y
) ,  ( 2nd `  Y ) >.  e.  ( ( C Full  Q )  i^i  ( C Faith  Q
) ) )
21 eqid 2443 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
22 yoniso.e . . . . . 6  |-  E  =  ( Qs  ran  ( 1st `  Y
) )
2311oppccat 15098 . . . . . . . 8  |-  ( C  e.  Cat  ->  O  e.  Cat )
2410, 23syl 16 . . . . . . 7  |-  ( ph  ->  O  e.  Cat )
2512setccat 15390 . . . . . . . 8  |-  ( U  e.  W  ->  S  e.  Cat )
2614, 25syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  Cat )
2713, 24, 26fuccat 15317 . . . . . 6  |-  ( ph  ->  Q  e.  Cat )
28 fvex 5866 . . . . . . . 8  |-  ( 1st `  Y )  e.  _V
2928rnex 6719 . . . . . . 7  |-  ran  ( 1st `  Y )  e. 
_V
3029a1i 11 . . . . . 6  |-  ( ph  ->  ran  ( 1st `  Y
)  e.  _V )
3113fucbas 15307 . . . . . . . . 9  |-  ( O 
Func  S )  =  (
Base `  Q )
32 1st2ndbr 6834 . . . . . . . . . 10  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
331, 16, 32sylancr 663 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
3421, 31, 33funcf1 15213 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) : ( Base `  C ) --> ( O 
Func  S ) )
35 ffn 5721 . . . . . . . 8  |-  ( ( 1st `  Y ) : ( Base `  C
) --> ( O  Func  S )  ->  ( 1st `  Y )  Fn  ( Base `  C ) )
3634, 35syl 16 . . . . . . 7  |-  ( ph  ->  ( 1st `  Y
)  Fn  ( Base `  C ) )
37 dffn3 5728 . . . . . . 7  |-  ( ( 1st `  Y )  Fn  ( Base `  C
)  <->  ( 1st `  Y
) : ( Base `  C ) --> ran  ( 1st `  Y ) )
3836, 37sylib 196 . . . . . 6  |-  ( ph  ->  ( 1st `  Y
) : ( Base `  C ) --> ran  ( 1st `  Y ) )
3921, 22, 27, 30, 38ffthres2c 15287 . . . . 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 4438 . . . . 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 4438 . . . . 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 ) ) )
4239, 40, 413bitr3g 287 . . . 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 ) ) ) )
4320, 42mpbid 210 . . 3  |-  ( ph  -> 
<. ( 1st `  Y
) ,  ( 2nd `  Y ) >.  e.  ( ( C Full  E )  i^i  ( C Faith  E
) ) )
4418, 43eqeltrd 2531 . 2  |-  ( ph  ->  Y  e.  ( ( C Full  E )  i^i  ( C Faith  E ) ) )
45 fveq2 5856 . . . . . . . . 9  |-  ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  ( 1st `  ( ( 1st `  Y
) `  x )
)  =  ( 1st `  ( ( 1st `  Y
) `  y )
) )
4645fveq1d 5858 . . . . . . . 8  |-  ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  ( ( 1st `  ( ( 1st `  Y ) `  x
) ) `  x
)  =  ( ( 1st `  ( ( 1st `  Y ) `
 y ) ) `
 x ) )
4746fveq2d 5860 . . . . . . 7  |-  ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  ( F `  ( ( 1st `  (
( 1st `  Y
) `  x )
) `  x )
)  =  ( F `
 ( ( 1st `  ( ( 1st `  Y
) `  y )
) `  x )
) )
48 simpl 457 . . . . . . . . . 10  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) )  ->  x  e.  ( Base `  C
) )
4948, 48jca 532 . . . . . . . . 9  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) )  ->  (
x  e.  ( Base `  C )  /\  x  e.  ( Base `  C
) ) )
50 eleq1 2515 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  (
y  e.  ( Base `  C )  <->  x  e.  ( Base `  C )
) )
5150anbi2d 703 . . . . . . . . . . . 12  |-  ( y  =  x  ->  (
( x  e.  (
Base `  C )  /\  y  e.  ( Base `  C ) )  <-> 
( x  e.  (
Base `  C )  /\  x  e.  ( Base `  C ) ) ) )
5251anbi2d 703 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  <->  ( ph  /\  ( x  e.  (
Base `  C )  /\  x  e.  ( Base `  C ) ) ) ) )
53 fveq2 5856 . . . . . . . . . . . . . . 15  |-  ( y  =  x  ->  (
( 1st `  Y
) `  y )  =  ( ( 1st `  Y ) `  x
) )
5453fveq2d 5860 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  ( 1st `  ( ( 1st `  Y ) `  y
) )  =  ( 1st `  ( ( 1st `  Y ) `
 x ) ) )
5554fveq1d 5858 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  (
( 1st `  (
( 1st `  Y
) `  y )
) `  x )  =  ( ( 1st `  ( ( 1st `  Y
) `  x )
) `  x )
)
5655fveq2d 5860 . . . . . . . . . . . 12  |-  ( y  =  x  ->  ( F `  ( ( 1st `  ( ( 1st `  Y ) `  y
) ) `  x
) )  =  ( F `  ( ( 1st `  ( ( 1st `  Y ) `
 x ) ) `
 x ) ) )
57 id 22 . . . . . . . . . . . 12  |-  ( y  =  x  ->  y  =  x )
5856, 57eqeq12d 2465 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
( F `  (
( 1st `  (
( 1st `  Y
) `  y )
) `  x )
)  =  y  <->  ( F `  ( ( 1st `  (
( 1st `  Y
) `  x )
) `  x )
)  =  x ) )
5952, 58imbi12d 320 . . . . . . . . . 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 ) ) )
6010adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  C  e.  Cat )
61 simprr 757 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
62 eqid 2443 . . . . . . . . . . . . 13  |-  ( Hom  `  C )  =  ( Hom  `  C )
63 simprl 756 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
642, 21, 60, 61, 62, 63yon11 15511 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  (
( 1st `  Y
) `  y )
) `  x )  =  ( x ( Hom  `  C )
y ) )
6564fveq2d 5860 . . . . . . . . . . 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 )
6765, 66eqtrd 2484 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( F `  ( ( 1st `  ( ( 1st `  Y ) `  y
) ) `  x
) )  =  y )
6859, 67chvarv 2000 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  x  e.  ( Base `  C )
) )  ->  ( F `  ( ( 1st `  ( ( 1st `  Y ) `  x
) ) `  x
) )  =  x )
6949, 68sylan2 474 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( F `  ( ( 1st `  ( ( 1st `  Y ) `  x
) ) `  x
) )  =  x )
7069, 67eqeq12d 2465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( F `  (
( 1st `  (
( 1st `  Y
) `  x )
) `  x )
)  =  ( F `
 ( ( 1st `  ( ( 1st `  Y
) `  y )
) `  x )
)  <->  x  =  y
) )
7147, 70syl5ib 219 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  x  =  y ) )
7271ralrimivva 2864 . . . . 5  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  x  =  y ) )
73 dff13 6151 . . . . 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 ) ) )
7434, 72, 73sylanbrc 664 . . . 4  |-  ( ph  ->  ( 1st `  Y
) : ( Base `  C ) -1-1-> ( O 
Func  S ) )
75 f1f1orn 5817 . . . 4  |-  ( ( 1st `  Y ) : ( Base `  C
) -1-1-> ( O  Func  S )  ->  ( 1st `  Y ) : (
Base `  C ) -1-1-onto-> ran  ( 1st `  Y ) )
7674, 75syl 16 . . 3  |-  ( ph  ->  ( 1st `  Y
) : ( Base `  C ) -1-1-onto-> ran  ( 1st `  Y
) )
77 frn 5727 . . . . . 6  |-  ( ( 1st `  Y ) : ( Base `  C
) --> ( O  Func  S )  ->  ran  ( 1st `  Y )  C_  ( O  Func  S ) )
7834, 77syl 16 . . . . 5  |-  ( ph  ->  ran  ( 1st `  Y
)  C_  ( O  Func  S ) )
7922, 31ressbas2 14669 . . . . 5  |-  ( ran  ( 1st `  Y
)  C_  ( O  Func  S )  ->  ran  ( 1st `  Y )  =  ( Base `  E
) )
8078, 79syl 16 . . . 4  |-  ( ph  ->  ran  ( 1st `  Y
)  =  ( Base `  E ) )
81 f1oeq3 5799 . . . 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
) ) )
8280, 81syl 16 . . 3  |-  ( ph  ->  ( ( 1st `  Y
) : ( Base `  C ) -1-1-onto-> ran  ( 1st `  Y
)  <->  ( 1st `  Y
) : ( Base `  C ) -1-1-onto-> ( Base `  E
) ) )
8376, 82mpbid 210 . 2  |-  ( ph  ->  ( 1st `  Y
) : ( Base `  C ) -1-1-onto-> ( Base `  E
) )
84 eqid 2443 . . 3  |-  ( Base `  E )  =  (
Base `  E )
85 yoniso.eb . . 3  |-  ( ph  ->  E  e.  B )
86 yoniso.i . . 3  |-  I  =  (  Iso  `  D
)
873, 4, 21, 84, 5, 9, 85, 86catciso 15412 . 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 )
) ) )
8844, 83, 87mpbir2and 922 1  |-  ( ph  ->  Y  e.  ( C I E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095    i^i cin 3460    C_ wss 3461   <.cop 4020   class class class wbr 4437   ran crn 4990   Rel wrel 4994    Fn wfn 5573   -->wf 5574   -1-1->wf1 5575   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   Basecbs 14613   ↾s cress 14614   Hom chom 14689   Catccat 15042   Hom f chomf 15044  oppCatcoppc 15087    Iso ciso 15122    Func cfunc 15201   Full cful 15249   Faith cfth 15250   FuncCat cfuc 15289   SetCatcsetc 15380  CatCatccatc 15399  Yoncyon 15496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-fz 11683  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-hom 14702  df-cco 14703  df-cat 15046  df-cid 15047  df-homf 15048  df-comf 15049  df-oppc 15088  df-sect 15123  df-inv 15124  df-iso 15125  df-ssc 15160  df-resc 15161  df-subc 15162  df-func 15205  df-idfu 15206  df-cofu 15207  df-full 15251  df-fth 15252  df-nat 15290  df-fuc 15291  df-setc 15381  df-catc 15400  df-xpc 15419  df-1stf 15420  df-2ndf 15421  df-prf 15422  df-evlf 15460  df-curf 15461  df-hof 15497  df-yon 15498
This theorem is referenced by: (None)
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