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Theorem yonedalem4b 15392
Description: Lemma for yoneda 15399. (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 )
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  ( Hom f  `  C ) 
C_  U )
15 yoneda.v . . . . 5  |-  ( ph  ->  ( ran  ( Hom f  `  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 15391 . . . 4  |-  ( ph  ->  ( ( F N X ) `  A
)  =  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) )
2120fveq1d 5859 . . 3  |-  ( ph  ->  ( ( ( F N X ) `  A ) `  P
)  =  ( ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) ) `  P
) )
2221fveq1d 5859 . 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 2461 . . 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 6300 . . . . . 6  |-  ( y ( Hom  `  C
) X )  e. 
_V
2625mptex 6122 . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  y  =  P )  ->  G  e.  ( P ( Hom  `  C ) X ) )
30 simpr 461 . . . . . . 7  |-  ( (
ph  /\  y  =  P )  ->  y  =  P )
3130oveq1d 6290 . . . . . 6  |-  ( (
ph  /\  y  =  P )  ->  (
y ( Hom  `  C
) X )  =  ( P ( Hom  `  C ) X ) )
3229, 31eleqtrrd 2551 . . . . 5  |-  ( (
ph  /\  y  =  P )  ->  G  e.  ( y ( Hom  `  C ) X ) )
33 fvex 5867 . . . . . 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 754 . . . . . . . 8  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  y  =  P )
3635oveq2d 6291 . . . . . . 7  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  ( X ( 2nd `  F
) y )  =  ( X ( 2nd `  F ) P ) )
37 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  g  =  G )
3836, 37fveq12d 5863 . . . . . 6  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  (
( X ( 2nd `  F ) y ) `
 g )  =  ( ( X ( 2nd `  F ) P ) `  G
) )
3938fveq1d 5859 . . . . 5  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  (
( ( X ( 2nd `  F ) y ) `  g
) `  A )  =  ( ( ( X ( 2nd `  F
) P ) `  G ) `  A
) )
4032, 34, 39fvmptdv2 5954 . . . 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 4529 . . . 4  |-  F/_ y
( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) )
42 nffvmpt1 5865 . . . . . 6  |-  F/_ y
( ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) y ) `  g ) `  A
) ) ) `  P )
43 nfcv 2622 . . . . . 6  |-  F/_ y G
4442, 43nffv 5864 . . . . 5  |-  F/_ y
( ( ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) `
 P ) `  G )
4544nfeq1 2637 . . . 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 5952 . . 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 2501 1  |-  ( ph  ->  ( ( ( ( F N X ) `
 A ) `  P ) `  G
)  =  ( ( ( X ( 2nd `  F ) P ) `
 G ) `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106    u. cun 3467    C_ wss 3469   <.cop 4026    |-> cmpt 4498   ran crn 4993   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   1stc1st 6772   2ndc2nd 6773  tpos ctpos 6944   Basecbs 14479   Hom chom 14555   Catccat 14908   Idccid 14909   Hom f chomf 14910  oppCatcoppc 14956    Func cfunc 15070    o.func ccofu 15072   FuncCat cfuc 15158   SetCatcsetc 15249    X.c cxpc 15284    1stF c1stf 15285    2ndF c2ndf 15286   ⟨,⟩F cprf 15287   evalF cevlf 15325  HomFchof 15364  Yoncyon 15365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280
This theorem is referenced by:  yonedalem4c  15393  yonedainv  15397
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