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Theorem yoniso 15095
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 14772 . . . 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 14965 . . . . . . 7  |-  ( ph  ->  B  =  ( V  i^i  Cat ) )
7 inss2 3571 . . . . . . 7  |-  ( V  i^i  Cat )  C_  Cat
86, 7syl6eqss 3406 . . . . . 6  |-  ( ph  ->  B  C_  Cat )
9 yoniso.c . . . . . 6  |-  ( ph  ->  C  e.  B )
108, 9sseldd 3357 . . . . 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 15072 . . . 4  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
17 1st2nd 6620 . . . 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 15094 . . . . 5  |-  ( ph  ->  Y  e.  ( ( C Full  Q )  i^i  ( C Faith  Q ) ) )
2018, 19eqeltrrd 2518 . . . 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 14661 . . . . . . . 8  |-  ( C  e.  Cat  ->  O  e.  Cat )
2410, 23syl 16 . . . . . . 7  |-  ( ph  ->  O  e.  Cat )
2512setccat 14953 . . . . . . . 8  |-  ( U  e.  W  ->  S  e.  Cat )
2614, 25syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  Cat )
2713, 24, 26fuccat 14880 . . . . . 6  |-  ( ph  ->  Q  e.  Cat )
28 fvex 5701 . . . . . . . 8  |-  ( 1st `  Y )  e.  _V
2928rnex 6512 . . . . . . 7  |-  ran  ( 1st `  Y )  e. 
_V
3029a1i 11 . . . . . 6  |-  ( ph  ->  ran  ( 1st `  Y
)  e.  _V )
3113fucbas 14870 . . . . . . . . 9  |-  ( O 
Func  S )  =  (
Base `  Q )
32 1st2ndbr 6623 . . . . . . . . . 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 14776 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) : ( Base `  C ) --> ( O 
Func  S ) )
35 ffn 5559 . . . . . . . 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 5566 . . . . . . 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 14850 . . . . 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 4293 . . . . 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 4293 . . . . 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 2517 . 2  |-  ( ph  ->  Y  e.  ( ( C Full  E )  i^i  ( C Faith  E ) ) )
45 fveq2 5691 . . . . . . . . 9  |-  ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  ( 1st `  ( ( 1st `  Y
) `  x )
)  =  ( 1st `  ( ( 1st `  Y
) `  y )
) )
4645fveq1d 5693 . . . . . . . 8  |-  ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  ( ( 1st `  ( ( 1st `  Y ) `  x
) ) `  x
)  =  ( ( 1st `  ( ( 1st `  Y ) `
 y ) ) `
 x ) )
4746fveq2d 5695 . . . . . . 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 2503 . . . . . . . . . . . . 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 5691 . . . . . . . . . . . . . . 15  |-  ( y  =  x  ->  (
( 1st `  Y
) `  y )  =  ( ( 1st `  Y ) `  x
) )
5453fveq2d 5695 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  ( 1st `  ( ( 1st `  Y ) `  y
) )  =  ( 1st `  ( ( 1st `  Y ) `
 x ) ) )
5554fveq1d 5693 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  (
( 1st `  (
( 1st `  Y
) `  y )
) `  x )  =  ( ( 1st `  ( ( 1st `  Y
) `  x )
) `  x )
)
5655fveq2d 5695 . . . . . . . . . . . 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 2457 . . . . . . . . . . 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 756 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
62 eqid 2443 . . . . . . . . . . . . 13  |-  ( Hom  `  C )  =  ( Hom  `  C )
63 simprl 755 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
642, 21, 60, 61, 62, 63yon11 15074 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  (
( 1st `  Y
) `  y )
) `  x )  =  ( x ( Hom  `  C )
y ) )
6564fveq2d 5695 . . . . . . . . . . 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 2475 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( F `  ( ( 1st `  ( ( 1st `  Y ) `  y
) ) `  x
) )  =  y )
6859, 67chvarv 1958 . . . . . . . . 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 2457 . . . . . . 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 2808 . . . . 5  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  x  =  y ) )
73 dff13 5971 . . . . 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 5652 . . . 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 5565 . . . . . 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 14229 . . . . 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 5634 . . . 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 14975 . 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 913 1  |-  ( ph  ->  Y  e.  ( C I E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972    i^i cin 3327    C_ wss 3328   <.cop 3883   class class class wbr 4292   ran crn 4841   Rel wrel 4845    Fn wfn 5413   -->wf 5414   -1-1->wf1 5415   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   Basecbs 14174   ↾s cress 14175   Hom chom 14249   Catccat 14602   Hom f chomf 14604  oppCatcoppc 14650    Iso ciso 14685    Func cfunc 14764   Full cful 14812   Faith cfth 14813   FuncCat cfuc 14852   SetCatcsetc 14943  CatCatccatc 14962  Yoncyon 15059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-hom 14262  df-cco 14263  df-cat 14606  df-cid 14607  df-homf 14608  df-comf 14609  df-oppc 14651  df-sect 14686  df-inv 14687  df-iso 14688  df-ssc 14723  df-resc 14724  df-subc 14725  df-func 14768  df-idfu 14769  df-cofu 14770  df-full 14814  df-fth 14815  df-nat 14853  df-fuc 14854  df-setc 14944  df-catc 14963  df-xpc 14982  df-1stf 14983  df-2ndf 14984  df-prf 14985  df-evlf 15023  df-curf 15024  df-hof 15060  df-yon 15061
This theorem is referenced by: (None)
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