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Theorem yoniso 14374
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  (  Homf  `  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 14051 . . . 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 14244 . . . . . . 7  |-  ( ph  ->  B  =  ( V  i^i  Cat ) )
7 inss2 3554 . . . . . . 7  |-  ( V  i^i  Cat )  C_  Cat
86, 7syl6eqss 3390 . . . . . 6  |-  ( ph  ->  B  C_  Cat )
9 yoniso.c . . . . . 6  |-  ( ph  ->  C  e.  B )
108, 9sseldd 3341 . . . . 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  (  Homf  `  C ) 
C_  U )
162, 10, 11, 12, 13, 14, 15yoncl 14351 . . . 4  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
17 1st2nd 6385 . . . 4  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  Y  =  <. ( 1st `  Y
) ,  ( 2nd `  Y ) >. )
181, 16, 17sylancr 645 . . 3  |-  ( ph  ->  Y  =  <. ( 1st `  Y ) ,  ( 2nd `  Y
) >. )
192, 11, 12, 13, 10, 14, 15yonffth 14373 . . . . 5  |-  ( ph  ->  Y  e.  ( ( C Full  Q )  i^i  ( C Faith  Q ) ) )
2018, 19eqeltrrd 2510 . . . 4  |-  ( ph  -> 
<. ( 1st `  Y
) ,  ( 2nd `  Y ) >.  e.  ( ( C Full  Q )  i^i  ( C Faith  Q
) ) )
21 eqid 2435 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
22 yoniso.e . . . . . 6  |-  E  =  ( Qs  ran  ( 1st `  Y
) )
2311oppccat 13940 . . . . . . . 8  |-  ( C  e.  Cat  ->  O  e.  Cat )
2410, 23syl 16 . . . . . . 7  |-  ( ph  ->  O  e.  Cat )
2512setccat 14232 . . . . . . . 8  |-  ( U  e.  W  ->  S  e.  Cat )
2614, 25syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  Cat )
2713, 24, 26fuccat 14159 . . . . . 6  |-  ( ph  ->  Q  e.  Cat )
28 fvex 5734 . . . . . . . 8  |-  ( 1st `  Y )  e.  _V
2928rnex 5125 . . . . . . 7  |-  ran  ( 1st `  Y )  e. 
_V
3029a1i 11 . . . . . 6  |-  ( ph  ->  ran  ( 1st `  Y
)  e.  _V )
3113fucbas 14149 . . . . . . . . 9  |-  ( O 
Func  S )  =  (
Base `  Q )
32 1st2ndbr 6388 . . . . . . . . . 10  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
331, 16, 32sylancr 645 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
3421, 31, 33funcf1 14055 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) : ( Base `  C ) --> ( O 
Func  S ) )
35 ffn 5583 . . . . . . . 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 5590 . . . . . . 7  |-  ( ( 1st `  Y )  Fn  ( Base `  C
)  <->  ( 1st `  Y
) : ( Base `  C ) --> ran  ( 1st `  Y ) )
3836, 37sylib 189 . . . . . 6  |-  ( ph  ->  ( 1st `  Y
) : ( Base `  C ) --> ran  ( 1st `  Y ) )
3921, 22, 27, 30, 38ffthres2c 14129 . . . . 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 4205 . . . . 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 4205 . . . . 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 279 . . . 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 202 . . 3  |-  ( ph  -> 
<. ( 1st `  Y
) ,  ( 2nd `  Y ) >.  e.  ( ( C Full  E )  i^i  ( C Faith  E
) ) )
4418, 43eqeltrd 2509 . 2  |-  ( ph  ->  Y  e.  ( ( C Full  E )  i^i  ( C Faith  E ) ) )
45 fveq2 5720 . . . . . . . . 9  |-  ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  ( 1st `  ( ( 1st `  Y
) `  x )
)  =  ( 1st `  ( ( 1st `  Y
) `  y )
) )
4645fveq1d 5722 . . . . . . . 8  |-  ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  ( ( 1st `  ( ( 1st `  Y ) `  x
) ) `  x
)  =  ( ( 1st `  ( ( 1st `  Y ) `
 y ) ) `
 x ) )
4746fveq2d 5724 . . . . . . 7  |-  ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  ( F `  ( ( 1st `  (
( 1st `  Y
) `  x )
) `  x )
)  =  ( F `
 ( ( 1st `  ( ( 1st `  Y
) `  y )
) `  x )
) )
48 simpl 444 . . . . . . . . . 10  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) )  ->  x  e.  ( Base `  C
) )
4948, 48jca 519 . . . . . . . . 9  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) )  ->  (
x  e.  ( Base `  C )  /\  x  e.  ( Base `  C
) ) )
50 eleq1 2495 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  (
y  e.  ( Base `  C )  <->  x  e.  ( Base `  C )
) )
5150anbi2d 685 . . . . . . . . . . . 12  |-  ( y  =  x  ->  (
( x  e.  (
Base `  C )  /\  y  e.  ( Base `  C ) )  <-> 
( x  e.  (
Base `  C )  /\  x  e.  ( Base `  C ) ) ) )
5251anbi2d 685 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  <->  ( ph  /\  ( x  e.  (
Base `  C )  /\  x  e.  ( Base `  C ) ) ) ) )
53 fveq2 5720 . . . . . . . . . . . . . . 15  |-  ( y  =  x  ->  (
( 1st `  Y
) `  y )  =  ( ( 1st `  Y ) `  x
) )
5453fveq2d 5724 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  ( 1st `  ( ( 1st `  Y ) `  y
) )  =  ( 1st `  ( ( 1st `  Y ) `
 x ) ) )
5554fveq1d 5722 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  (
( 1st `  (
( 1st `  Y
) `  y )
) `  x )  =  ( ( 1st `  ( ( 1st `  Y
) `  x )
) `  x )
)
5655fveq2d 5724 . . . . . . . . . . . 12  |-  ( y  =  x  ->  ( F `  ( ( 1st `  ( ( 1st `  Y ) `  y
) ) `  x
) )  =  ( F `  ( ( 1st `  ( ( 1st `  Y ) `
 x ) ) `
 x ) ) )
57 id 20 . . . . . . . . . . . 12  |-  ( y  =  x  ->  y  =  x )
5856, 57eqeq12d 2449 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
( F `  (
( 1st `  (
( 1st `  Y
) `  y )
) `  x )
)  =  y  <->  ( F `  ( ( 1st `  (
( 1st `  Y
) `  x )
) `  x )
)  =  x ) )
5952, 58imbi12d 312 . . . . . . . . . 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 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  C  e.  Cat )
61 simprr 734 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
62 eqid 2435 . . . . . . . . . . . . 13  |-  (  Hom  `  C )  =  (  Hom  `  C )
63 simprl 733 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
642, 21, 60, 61, 62, 63yon11 14353 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  (
( 1st `  Y
) `  y )
) `  x )  =  ( x (  Hom  `  C )
y ) )
6564fveq2d 5724 . . . . . . . . . . 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 2467 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( F `  ( ( 1st `  ( ( 1st `  Y ) `  y
) ) `  x
) )  =  y )
6859, 67chvarv 1969 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  x  e.  ( Base `  C )
) )  ->  ( F `  ( ( 1st `  ( ( 1st `  Y ) `  x
) ) `  x
) )  =  x )
6949, 68sylan2 461 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( F `  ( ( 1st `  ( ( 1st `  Y ) `  x
) ) `  x
) )  =  x )
7069, 67eqeq12d 2449 . . . . . . 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 211 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  x  =  y ) )
7271ralrimivva 2790 . . . . 5  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  x  =  y ) )
73 dff13 5996 . . . . 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 646 . . . 4  |-  ( ph  ->  ( 1st `  Y
) : ( Base `  C ) -1-1-> ( O 
Func  S ) )
75 f1f1orn 5677 . . . 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 5589 . . . . . 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 13512 . . . . 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 5659 . . . 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 202 . 2  |-  ( ph  ->  ( 1st `  Y
) : ( Base `  C ) -1-1-onto-> ( Base `  E
) )
84 eqid 2435 . . 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 14254 . 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 889 1  |-  ( ph  ->  Y  e.  ( C I E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    i^i cin 3311    C_ wss 3312   <.cop 3809   class class class wbr 4204   ran crn 4871   Rel wrel 4875    Fn wfn 5441   -->wf 5442   -1-1->wf1 5443   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   Basecbs 13461   ↾s cress 13462    Hom chom 13532   Catccat 13881    Homf chomf 13883  oppCatcoppc 13929    Iso ciso 13964    Func cfunc 14043   Full cful 14091   Faith cfth 14092   FuncCat cfuc 14131   SetCatcsetc 14222  CatCatccatc 14241  Yoncyon 14338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-hom 13545  df-cco 13546  df-cat 13885  df-cid 13886  df-homf 13887  df-comf 13888  df-oppc 13930  df-sect 13965  df-inv 13966  df-iso 13967  df-ssc 14002  df-resc 14003  df-subc 14004  df-func 14047  df-idfu 14048  df-cofu 14049  df-full 14093  df-fth 14094  df-nat 14132  df-fuc 14133  df-setc 14223  df-catc 14242  df-xpc 14261  df-1stf 14262  df-2ndf 14263  df-prf 14264  df-evlf 14302  df-curf 14303  df-hof 14339  df-yon 14340
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