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Theorem yoneda 16120
Description: The Yoneda Lemma. There is a natural isomorphism between the functors  Z and  E, where  Z ( F ,  X ) is the natural transformations from Yon ( X )  =  Hom  (  -  ,  X ) to  F, and  E ( F ,  X )  =  F ( X ) is the evaluation functor. Here we need two universes to state the claim: the smaller universe  U is used for forming the functor category  Q  =  C op  ->  SetCat ( U ), which itself does not (necessarily) live in  U but instead is an element of the larger universe  V. (If  U is a Grothendieck universe, then it will be closed under this "presheaf" operation, and so we can set  U  =  V in this case.) (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y  |-  Y  =  (Yon `  C )
yoneda.b  |-  B  =  ( Base `  C
)
yoneda.1  |-  .1.  =  ( Id `  C )
yoneda.o  |-  O  =  (oppCat `  C )
yoneda.s  |-  S  =  ( SetCat `  U )
yoneda.t  |-  T  =  ( SetCat `  V )
yoneda.q  |-  Q  =  ( O FuncCat  S )
yoneda.h  |-  H  =  (HomF
`  Q )
yoneda.r  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
yoneda.e  |-  E  =  ( O evalF  S )
yoneda.z  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
yoneda.c  |-  ( ph  ->  C  e.  Cat )
yoneda.w  |-  ( ph  ->  V  e.  W )
yoneda.u  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
yoneda.v  |-  ( ph  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
yoneda.m  |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  |->  ( ( a `  x ) `
 (  .1.  `  x ) ) ) )
yoneda.i  |-  I  =  (  Iso  `  R
)
Assertion
Ref Expression
yoneda  |-  ( ph  ->  M  e.  ( Z I E ) )
Distinct variable groups:    f, a, x,  .1.    C, a, f, x    E, a, f    B, a, f, x    O, a, f, x    S, a, f, x    Q, a, f, x    T, f    ph, a, f, x    Y, a, f, x    Z, a, f, x
Allowed substitution hints:    R( x, f, a)    T( x, a)    U( x, f, a)    E( x)    H( x, f, a)    I( x, f, a)    M( x, f, a)    V( x, f, a)    W( x, f, a)

Proof of Theorem yoneda
Dummy variables  g 
y  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 yoneda.r . . 3  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
21fucbas 15817 . 2  |-  ( ( Q  X.c  O )  Func  T
)  =  ( Base `  R )
3 eqid 2420 . 2  |-  (Inv `  R )  =  (Inv
`  R )
4 yoneda.y . . . . . . 7  |-  Y  =  (Yon `  C )
5 yoneda.b . . . . . . 7  |-  B  =  ( Base `  C
)
6 yoneda.1 . . . . . . 7  |-  .1.  =  ( Id `  C )
7 yoneda.o . . . . . . 7  |-  O  =  (oppCat `  C )
8 yoneda.s . . . . . . 7  |-  S  =  ( SetCat `  U )
9 yoneda.t . . . . . . 7  |-  T  =  ( SetCat `  V )
10 yoneda.q . . . . . . 7  |-  Q  =  ( O FuncCat  S )
11 yoneda.h . . . . . . 7  |-  H  =  (HomF
`  Q )
12 yoneda.e . . . . . . 7  |-  E  =  ( O evalF  S )
13 yoneda.z . . . . . . 7  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
14 yoneda.c . . . . . . 7  |-  ( ph  ->  C  e.  Cat )
15 yoneda.w . . . . . . 7  |-  ( ph  ->  V  e.  W )
16 yoneda.u . . . . . . 7  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
17 yoneda.v . . . . . . 7  |-  ( ph  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
184, 5, 6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15, 16, 17yonedalem1 16109 . . . . . 6  |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O ) 
Func  T )  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
) )
1918simpld 460 . . . . 5  |-  ( ph  ->  Z  e.  ( ( Q  X.c  O )  Func  T
) )
20 funcrcl 15720 . . . . 5  |-  ( Z  e.  ( ( Q  X.c  O )  Func  T
)  ->  ( ( Q  X.c  O )  e.  Cat  /\  T  e.  Cat )
)
2119, 20syl 17 . . . 4  |-  ( ph  ->  ( ( Q  X.c  O
)  e.  Cat  /\  T  e.  Cat )
)
2221simpld 460 . . 3  |-  ( ph  ->  ( Q  X.c  O )  e.  Cat )
2321simprd 464 . . 3  |-  ( ph  ->  T  e.  Cat )
241, 22, 23fuccat 15827 . 2  |-  ( ph  ->  R  e.  Cat )
2518simprd 464 . 2  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  T
) )
26 yoneda.i . 2  |-  I  =  (  Iso  `  R
)
27 yoneda.m . . 3  |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  |->  ( ( a `  x ) `
 (  .1.  `  x ) ) ) )
28 eqid 2420 . . 3  |-  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( u  e.  ( ( 1st `  f
) `  x )  |->  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) x )  |->  ( ( ( x ( 2nd `  f ) y ) `  g
) `  u )
) ) ) )  =  ( f  e.  ( O  Func  S
) ,  x  e.  B  |->  ( u  e.  ( ( 1st `  f
) `  x )  |->  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) x )  |->  ( ( ( x ( 2nd `  f ) y ) `  g
) `  u )
) ) ) )
294, 5, 6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 27, 3, 28yonedainv 16118 . 2  |-  ( ph  ->  M ( Z (Inv
`  R ) E ) ( f  e.  ( O  Func  S
) ,  x  e.  B  |->  ( u  e.  ( ( 1st `  f
) `  x )  |->  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) x )  |->  ( ( ( x ( 2nd `  f ) y ) `  g
) `  u )
) ) ) ) )
302, 3, 24, 19, 25, 26, 29inviso1 15623 1  |-  ( ph  ->  M  e.  ( Z I E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867    u. cun 3431    C_ wss 3433   <.cop 3999    |-> cmpt 4475   ran crn 4846   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6796   2ndc2nd 6797  tpos ctpos 6971   Basecbs 15081   Hom chom 15161   Catccat 15522   Idccid 15523   Hom f chomf 15524  oppCatcoppc 15568  Invcinv 15602    Iso ciso 15603    Func cfunc 15711    o.func ccofu 15713   Nat cnat 15798   FuncCat cfuc 15799   SetCatcsetc 15922    X.c cxpc 16005    1stF c1stf 16006    2ndF c2ndf 16007   ⟨,⟩F cprf 16008   evalF cevlf 16046  HomFchof 16085  Yoncyon 16086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-tpos 6972  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-fz 11772  df-struct 15083  df-ndx 15084  df-slot 15085  df-base 15086  df-sets 15087  df-ress 15088  df-hom 15174  df-cco 15175  df-cat 15526  df-cid 15527  df-homf 15528  df-comf 15529  df-oppc 15569  df-sect 15604  df-inv 15605  df-iso 15606  df-ssc 15667  df-resc 15668  df-subc 15669  df-func 15715  df-cofu 15717  df-nat 15800  df-fuc 15801  df-setc 15923  df-xpc 16009  df-1stf 16010  df-2ndf 16011  df-prf 16012  df-evlf 16050  df-curf 16051  df-hof 16087  df-yon 16088
This theorem is referenced by: (None)
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