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Theorem yoneda 15213
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 14990 . 2  |-  ( ( Q  X.c  O )  Func  T
)  =  ( Base `  R )
3 eqid 2454 . 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 15202 . . . . . 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 14893 . . . . 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 15000 . 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 2454 . . 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 15211 . 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 14824 1  |-  ( ph  ->  M  e.  ( Z I E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    u. cun 3435    C_ wss 3437   <.cop 3992    |-> cmpt 4459   ran crn 4950   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   1stc1st 6686   2ndc2nd 6687  tpos ctpos 6855   Basecbs 14293   Hom chom 14369   Catccat 14722   Idccid 14723   Hom f chomf 14724  oppCatcoppc 14770  Invcinv 14804    Iso ciso 14805    Func cfunc 14884    o.func ccofu 14886   Nat cnat 14971   FuncCat cfuc 14972   SetCatcsetc 15063    X.c cxpc 15098    1stF c1stf 15099    2ndF c2ndf 15100   ⟨,⟩F cprf 15101   evalF cevlf 15139  HomFchof 15178  Yoncyon 15179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-tpos 6856  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-ixp 7375  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-hom 14382  df-cco 14383  df-cat 14726  df-cid 14727  df-homf 14728  df-comf 14729  df-oppc 14771  df-sect 14806  df-inv 14807  df-iso 14808  df-ssc 14843  df-resc 14844  df-subc 14845  df-func 14888  df-cofu 14890  df-nat 14973  df-fuc 14974  df-setc 15064  df-xpc 15102  df-1stf 15103  df-2ndf 15104  df-prf 15105  df-evlf 15143  df-curf 15144  df-hof 15180  df-yon 15181
This theorem is referenced by: (None)
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