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Theorem yonedalem4b 15869
Description: Lemma for yoneda 15876. (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 15868 . . . 4  |-  ( ph  ->  ( ( F N X ) `  A
)  =  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) )
2120fveq1d 5851 . . 3  |-  ( ph  ->  ( ( ( F N X ) `  A ) `  P
)  =  ( ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) ) `  P
) )
2221fveq1d 5851 . 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 2403 . . 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 6306 . . . . . 6  |-  ( y ( Hom  `  C
) X )  e. 
_V
2625mptex 6124 . . . . 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 463 . . . . . 6  |-  ( (
ph  /\  y  =  P )  ->  G  e.  ( P ( Hom  `  C ) X ) )
30 simpr 459 . . . . . . 7  |-  ( (
ph  /\  y  =  P )  ->  y  =  P )
3130oveq1d 6293 . . . . . 6  |-  ( (
ph  /\  y  =  P )  ->  (
y ( Hom  `  C
) X )  =  ( P ( Hom  `  C ) X ) )
3229, 31eleqtrrd 2493 . . . . 5  |-  ( (
ph  /\  y  =  P )  ->  G  e.  ( y ( Hom  `  C ) X ) )
33 fvex 5859 . . . . . 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 6294 . . . . . . 7  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  ( X ( 2nd `  F
) y )  =  ( X ( 2nd `  F ) P ) )
37 simpr 459 . . . . . . 7  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  g  =  G )
3836, 37fveq12d 5855 . . . . . 6  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  (
( X ( 2nd `  F ) y ) `
 g )  =  ( ( X ( 2nd `  F ) P ) `  G
) )
3938fveq1d 5851 . . . . 5  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  (
( ( X ( 2nd `  F ) y ) `  g
) `  A )  =  ( ( ( X ( 2nd `  F
) P ) `  G ) `  A
) )
4032, 34, 39fvmptdv2 5947 . . . 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 4484 . . . 4  |-  F/_ y
( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) )
42 nffvmpt1 5857 . . . . . 6  |-  F/_ y
( ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) y ) `  g ) `  A
) ) ) `  P )
43 nfcv 2564 . . . . . 6  |-  F/_ y G
4442, 43nffv 5856 . . . . 5  |-  F/_ y
( ( ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) `
 P ) `  G )
4544nfeq1 2579 . . . 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 5945 . . 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 2443 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 367    = wceq 1405    e. wcel 1842   _Vcvv 3059    u. cun 3412    C_ wss 3414   <.cop 3978    |-> cmpt 4453   ran crn 4824   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   1stc1st 6782   2ndc2nd 6783  tpos ctpos 6957   Basecbs 14841   Hom chom 14920   Catccat 15278   Idccid 15279   Hom f chomf 15280  oppCatcoppc 15324    Func cfunc 15467    o.func ccofu 15469   FuncCat cfuc 15555   SetCatcsetc 15678    X.c cxpc 15761    1stF c1stf 15762    2ndF c2ndf 15763   ⟨,⟩F cprf 15764   evalF cevlf 15802  HomFchof 15841  Yoncyon 15842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283
This theorem is referenced by:  yonedalem4c  15870  yonedainv  15874
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