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Theorem yonedalem4b 14336
Description: Lemma for yoneda 14343. (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  (  Homf  `  C ) 
C_  U )
yoneda.v  |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U
)  C_  V )
yonedalem21.f  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
yonedalem21.x  |-  ( ph  ->  X  e.  B )
yonedalem4.n  |-  N  =  ( 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 )
) ) ) )
yonedalem4.p  |-  ( ph  ->  A  e.  ( ( 1st `  F ) `
 X ) )
yonedalem4b.p  |-  ( ph  ->  P  e.  B )
yonedalem4b.g  |-  ( ph  ->  G  e.  ( P (  Hom  `  C
) X ) )
Assertion
Ref Expression
yonedalem4b  |-  ( ph  ->  ( ( ( ( F N X ) `
 A ) `  P ) `  G
)  =  ( ( ( X ( 2nd `  F ) P ) `
 G ) `  A ) )
Distinct variable groups:    f, g, x, y,  .1.    u, g, A, y    u, f, C, g, x, y   
f, E, g, u, y    f, F, g, u, x, y    B, f, g, u, x, y   
f, G, g, x, y    f, O, g, u, x, y    S, f, g, u, x, y    Q, f, g, u, x    T, f, g, u, y    P, f, g, x, y    ph, f, g, u, x, y    u, R    f, Y, g, u, x, y   
f, Z, g, u, x, y    f, X, g, u, x, y
Allowed substitution hints:    A( x, f)    P( u)    Q( y)    R( x, y, f, g)    T( x)    U( x, y, u, f, g)    .1. ( u)    E( x)    G( u)    H( x, y, u, f, g)    N( x, y, u, f, g)    V( x, y, u, f, g)    W( x, y, u, f, g)

Proof of Theorem yonedalem4b
StepHypRef Expression
1 yoneda.y . . . . 5  |-  Y  =  (Yon `  C )
2 yoneda.b . . . . 5  |-  B  =  ( Base `  C
)
3 yoneda.1 . . . . 5  |-  .1.  =  ( Id `  C )
4 yoneda.o . . . . 5  |-  O  =  (oppCat `  C )
5 yoneda.s . . . . 5  |-  S  =  ( SetCat `  U )
6 yoneda.t . . . . 5  |-  T  =  ( SetCat `  V )
7 yoneda.q . . . . 5  |-  Q  =  ( O FuncCat  S )
8 yoneda.h . . . . 5  |-  H  =  (HomF
`  Q )
9 yoneda.r . . . . 5  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
10 yoneda.e . . . . 5  |-  E  =  ( O evalF  S )
11 yoneda.z . . . . 5  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
12 yoneda.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
13 yoneda.w . . . . 5  |-  ( ph  ->  V  e.  W )
14 yoneda.u . . . . 5  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
15 yoneda.v . . . . 5  |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U
)  C_  V )
16 yonedalem21.f . . . . 5  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
17 yonedalem21.x . . . . 5  |-  ( ph  ->  X  e.  B )
18 yonedalem4.n . . . . 5  |-  N  =  ( 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 )
) ) ) )
19 yonedalem4.p . . . . 5  |-  ( ph  ->  A  e.  ( ( 1st `  F ) `
 X ) )
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19yonedalem4a 14335 . . . 4  |-  ( ph  ->  ( ( F N X ) `  A
)  =  ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) )
2120fveq1d 5697 . . 3  |-  ( ph  ->  ( ( ( F N X ) `  A ) `  P
)  =  ( ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) ) `  P
) )
2221fveq1d 5697 . 2  |-  ( ph  ->  ( ( ( ( F N X ) `
 A ) `  P ) `  G
)  =  ( ( ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) ) `  P
) `  G )
)
23 eqidd 2413 . . 3  |-  ( ph  ->  ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) )  =  ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) ) )
24 yonedalem4b.p . . . 4  |-  ( ph  ->  P  e.  B )
25 ovex 6073 . . . . . 6  |-  ( y (  Hom  `  C
) X )  e. 
_V
2625mptex 5933 . . . . 5  |-  ( g  e.  ( y (  Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) )  e. 
_V
2726a1i 11 . . . 4  |-  ( (
ph  /\  y  =  P )  ->  (
g  e.  ( y (  Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
)  e.  _V )
28 yonedalem4b.g . . . . . . 7  |-  ( ph  ->  G  e.  ( P (  Hom  `  C
) X ) )
2928adantr 452 . . . . . 6  |-  ( (
ph  /\  y  =  P )  ->  G  e.  ( P (  Hom  `  C ) X ) )
30 simpr 448 . . . . . . 7  |-  ( (
ph  /\  y  =  P )  ->  y  =  P )
3130oveq1d 6063 . . . . . 6  |-  ( (
ph  /\  y  =  P )  ->  (
y (  Hom  `  C
) X )  =  ( P (  Hom  `  C ) X ) )
3229, 31eleqtrrd 2489 . . . . 5  |-  ( (
ph  /\  y  =  P )  ->  G  e.  ( y (  Hom  `  C ) X ) )
33 fvex 5709 . . . . . 6  |-  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A )  e.  _V
3433a1i 11 . . . . 5  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  (
( ( X ( 2nd `  F ) y ) `  g
) `  A )  e.  _V )
35 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  y  =  P )
3635oveq2d 6064 . . . . . . 7  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  ( X ( 2nd `  F
) y )  =  ( X ( 2nd `  F ) P ) )
37 simpr 448 . . . . . . 7  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  g  =  G )
3836, 37fveq12d 5701 . . . . . 6  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  (
( X ( 2nd `  F ) y ) `
 g )  =  ( ( X ( 2nd `  F ) P ) `  G
) )
3938fveq1d 5697 . . . . 5  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  (
( ( X ( 2nd `  F ) y ) `  g
) `  A )  =  ( ( ( X ( 2nd `  F
) P ) `  G ) `  A
) )
4032, 34, 39fvmptdv2 5785 . . . 4  |-  ( (
ph  /\  y  =  P )  ->  (
( ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) y ) `  g ) `  A
) ) ) `  P )  =  ( g  e.  ( y (  Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
)  ->  ( (
( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) ) `  P
) `  G )  =  ( ( ( X ( 2nd `  F
) P ) `  G ) `  A
) ) )
41 nfmpt1 4266 . . . 4  |-  F/_ y
( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) )
42 nffvmpt1 5703 . . . . . 6  |-  F/_ y
( ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) y ) `  g ) `  A
) ) ) `  P )
43 nfcv 2548 . . . . . 6  |-  F/_ y G
4442, 43nffv 5702 . . . . 5  |-  F/_ y
( ( ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) `
 P ) `  G )
4544nfeq1 2557 . . . 4  |-  F/ y ( ( ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) `
 P ) `  G )  =  ( ( ( X ( 2nd `  F ) P ) `  G
) `  A )
4624, 27, 40, 41, 45fvmptdf 5783 . . 3  |-  ( ph  ->  ( ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) y ) `  g ) `  A
) ) )  =  ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) )  ->  (
( ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) y ) `  g ) `  A
) ) ) `  P ) `  G
)  =  ( ( ( X ( 2nd `  F ) P ) `
 G ) `  A ) ) )
4723, 46mpd 15 . 2  |-  ( ph  ->  ( ( ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) `
 P ) `  G )  =  ( ( ( X ( 2nd `  F ) P ) `  G
) `  A )
)
4822, 47eqtrd 2444 1  |-  ( ph  ->  ( ( ( ( F N X ) `
 A ) `  P ) `  G
)  =  ( ( ( X ( 2nd `  F ) P ) `
 G ) `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924    u. cun 3286    C_ wss 3288   <.cop 3785    e. cmpt 4234   ran crn 4846   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   1stc1st 6314   2ndc2nd 6315  tpos ctpos 6445   Basecbs 13432    Hom chom 13503   Catccat 13852   Idccid 13853    Homf chomf 13854  oppCatcoppc 13900    Func cfunc 14014    o.func ccofu 14016   FuncCat cfuc 14102   SetCatcsetc 14193    X.c cxpc 14228    1stF c1stf 14229    2ndF c2ndf 14230   ⟨,⟩F cprf 14231   evalF cevlf 14269  HomFchof 14308  Yoncyon 14309
This theorem is referenced by:  yonedalem4c  14337  yonedainv  14341
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053
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