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Theorem yoniso 14309
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 13986 . . . 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 14179 . . . . . . 7  |-  ( ph  ->  B  =  ( V  i^i  Cat ) )
7 inss2 3505 . . . . . . 7  |-  ( V  i^i  Cat )  C_  Cat
86, 7syl6eqss 3341 . . . . . 6  |-  ( ph  ->  B  C_  Cat )
9 yoniso.c . . . . . 6  |-  ( ph  ->  C  e.  B )
108, 9sseldd 3292 . . . . 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 14286 . . . 4  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
17 1st2nd 6332 . . . 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 14308 . . . . 5  |-  ( ph  ->  Y  e.  ( ( C Full  Q )  i^i  ( C Faith  Q ) ) )
2018, 19eqeltrrd 2462 . . . 4  |-  ( ph  -> 
<. ( 1st `  Y
) ,  ( 2nd `  Y ) >.  e.  ( ( C Full  Q )  i^i  ( C Faith  Q
) ) )
21 eqid 2387 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
22 yoniso.e . . . . . 6  |-  E  =  ( Qs  ran  ( 1st `  Y
) )
2311oppccat 13875 . . . . . . . 8  |-  ( C  e.  Cat  ->  O  e.  Cat )
2410, 23syl 16 . . . . . . 7  |-  ( ph  ->  O  e.  Cat )
2512setccat 14167 . . . . . . . 8  |-  ( U  e.  W  ->  S  e.  Cat )
2614, 25syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  Cat )
2713, 24, 26fuccat 14094 . . . . . 6  |-  ( ph  ->  Q  e.  Cat )
28 fvex 5682 . . . . . . . 8  |-  ( 1st `  Y )  e.  _V
2928rnex 5073 . . . . . . 7  |-  ran  ( 1st `  Y )  e. 
_V
3029a1i 11 . . . . . 6  |-  ( ph  ->  ran  ( 1st `  Y
)  e.  _V )
3113fucbas 14084 . . . . . . . . 9  |-  ( O 
Func  S )  =  (
Base `  Q )
32 1st2ndbr 6335 . . . . . . . . . 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 13990 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) : ( Base `  C ) --> ( O 
Func  S ) )
35 ffn 5531 . . . . . . . 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 5538 . . . . . . 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 14064 . . . . 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 4154 . . . . 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 4154 . . . . 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 2461 . 2  |-  ( ph  ->  Y  e.  ( ( C Full  E )  i^i  ( C Faith  E ) ) )
45 fveq2 5668 . . . . . . . . 9  |-  ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  ( 1st `  ( ( 1st `  Y
) `  x )
)  =  ( 1st `  ( ( 1st `  Y
) `  y )
) )
4645fveq1d 5670 . . . . . . . 8  |-  ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  ( ( 1st `  ( ( 1st `  Y ) `  x
) ) `  x
)  =  ( ( 1st `  ( ( 1st `  Y ) `
 y ) ) `
 x ) )
4746fveq2d 5672 . . . . . . 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 2447 . . . . . . . . . . . . 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 5668 . . . . . . . . . . . . . . 15  |-  ( y  =  x  ->  (
( 1st `  Y
) `  y )  =  ( ( 1st `  Y ) `  x
) )
5453fveq2d 5672 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  ( 1st `  ( ( 1st `  Y ) `  y
) )  =  ( 1st `  ( ( 1st `  Y ) `
 x ) ) )
5554fveq1d 5670 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  (
( 1st `  (
( 1st `  Y
) `  y )
) `  x )  =  ( ( 1st `  ( ( 1st `  Y
) `  x )
) `  x )
)
5655fveq2d 5672 . . . . . . . . . . . 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 2401 . . . . . . . . . . 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 2387 . . . . . . . . . . . . 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 14288 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  (
( 1st `  Y
) `  y )
) `  x )  =  ( x (  Hom  `  C )
y ) )
6564fveq2d 5672 . . . . . . . . . . 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 2419 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( F `  ( ( 1st `  ( ( 1st `  Y ) `  y
) ) `  x
) )  =  y )
6859, 67chvarv 2047 . . . . . . . . 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 2401 . . . . . . 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 2741 . . . . 5  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) ( ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  y
)  ->  x  =  y ) )
73 dff13 5943 . . . . 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 5625 . . . 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 5537 . . . . . 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 13447 . . . . 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 5607 . . . 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 2387 . . 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 14189 . 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 1649    e. wcel 1717   A.wral 2649   _Vcvv 2899    i^i cin 3262    C_ wss 3263   <.cop 3760   class class class wbr 4153   ran crn 4819   Rel wrel 4823    Fn wfn 5389   -->wf 5390   -1-1->wf1 5391   -1-1-onto->wf1o 5393   ` cfv 5394  (class class class)co 6020   1stc1st 6286   2ndc2nd 6287   Basecbs 13396   ↾s cress 13397    Hom chom 13467   Catccat 13816    Homf chomf 13818  oppCatcoppc 13864    Iso ciso 13899    Func cfunc 13978   Full cful 14026   Faith cfth 14027   FuncCat cfuc 14066   SetCatcsetc 14157  CatCatccatc 14176  Yoncyon 14273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-hom 13480  df-cco 13481  df-cat 13820  df-cid 13821  df-homf 13822  df-comf 13823  df-oppc 13865  df-sect 13900  df-inv 13901  df-iso 13902  df-ssc 13937  df-resc 13938  df-subc 13939  df-func 13982  df-idfu 13983  df-cofu 13984  df-full 14028  df-fth 14029  df-nat 14067  df-fuc 14068  df-setc 14158  df-catc 14177  df-xpc 14196  df-1stf 14197  df-2ndf 14198  df-prf 14199  df-evlf 14237  df-curf 14238  df-hof 14274  df-yon 14275
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