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Theorem yoneda 15413
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 15190 . 2  |-  ( ( Q  X.c  O )  Func  T
)  =  ( Base `  R )
3 eqid 2467 . 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 15402 . . . . . 6  |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O ) 
Func  T )  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
) )
1918simpld 459 . . . . 5  |-  ( ph  ->  Z  e.  ( ( Q  X.c  O )  Func  T
) )
20 funcrcl 15093 . . . . 5  |-  ( Z  e.  ( ( Q  X.c  O )  Func  T
)  ->  ( ( Q  X.c  O )  e.  Cat  /\  T  e.  Cat )
)
2119, 20syl 16 . . . 4  |-  ( ph  ->  ( ( Q  X.c  O
)  e.  Cat  /\  T  e.  Cat )
)
2221simpld 459 . . 3  |-  ( ph  ->  ( Q  X.c  O )  e.  Cat )
2321simprd 463 . . 3  |-  ( ph  ->  T  e.  Cat )
241, 22, 23fuccat 15200 . 2  |-  ( ph  ->  R  e.  Cat )
2518simprd 463 . 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 2467 . . 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 15411 . 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 15024 1  |-  ( ph  ->  M  e.  ( Z I E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    u. cun 3474    C_ wss 3476   <.cop 4033    |-> cmpt 4505   ran crn 5000   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   1stc1st 6783   2ndc2nd 6784  tpos ctpos 6955   Basecbs 14493   Hom chom 14569   Catccat 14922   Idccid 14923   Hom f chomf 14924  oppCatcoppc 14970  Invcinv 15004    Iso ciso 15005    Func cfunc 15084    o.func ccofu 15086   Nat cnat 15171   FuncCat cfuc 15172   SetCatcsetc 15263    X.c cxpc 15298    1stF c1stf 15299    2ndF c2ndf 15300   ⟨,⟩F cprf 15301   evalF cevlf 15339  HomFchof 15378  Yoncyon 15379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6956  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-fz 11674  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-hom 14582  df-cco 14583  df-cat 14926  df-cid 14927  df-homf 14928  df-comf 14929  df-oppc 14971  df-sect 15006  df-inv 15007  df-iso 15008  df-ssc 15043  df-resc 15044  df-subc 15045  df-func 15088  df-cofu 15090  df-nat 15173  df-fuc 15174  df-setc 15264  df-xpc 15302  df-1stf 15303  df-2ndf 15304  df-prf 15305  df-evlf 15343  df-curf 15344  df-hof 15380  df-yon 15381
This theorem is referenced by: (None)
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