MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  yonedalem4b Structured version   Unicode version

Theorem yonedalem4b 14378
Description: Lemma for yoneda 14385. (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 14377 . . . 4  |-  ( ph  ->  ( ( F N X ) `  A
)  =  ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) )
2120fveq1d 5733 . . 3  |-  ( ph  ->  ( ( ( F N X ) `  A ) `  P
)  =  ( ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) ) `  P
) )
2221fveq1d 5733 . 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 2439 . . 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 6109 . . . . . 6  |-  ( y (  Hom  `  C
) X )  e. 
_V
2625mptex 5969 . . . . 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 453 . . . . . 6  |-  ( (
ph  /\  y  =  P )  ->  G  e.  ( P (  Hom  `  C ) X ) )
30 simpr 449 . . . . . . 7  |-  ( (
ph  /\  y  =  P )  ->  y  =  P )
3130oveq1d 6099 . . . . . 6  |-  ( (
ph  /\  y  =  P )  ->  (
y (  Hom  `  C
) X )  =  ( P (  Hom  `  C ) X ) )
3229, 31eleqtrrd 2515 . . . . 5  |-  ( (
ph  /\  y  =  P )  ->  G  e.  ( y (  Hom  `  C ) X ) )
33 fvex 5745 . . . . . 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 733 . . . . . . . 8  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  y  =  P )
3635oveq2d 6100 . . . . . . 7  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  ( X ( 2nd `  F
) y )  =  ( X ( 2nd `  F ) P ) )
37 simpr 449 . . . . . . 7  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  g  =  G )
3836, 37fveq12d 5737 . . . . . 6  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  (
( X ( 2nd `  F ) y ) `
 g )  =  ( ( X ( 2nd `  F ) P ) `  G
) )
3938fveq1d 5733 . . . . 5  |-  ( ( ( ph  /\  y  =  P )  /\  g  =  G )  ->  (
( ( X ( 2nd `  F ) y ) `  g
) `  A )  =  ( ( ( X ( 2nd `  F
) P ) `  G ) `  A
) )
4032, 34, 39fvmptdv2 5821 . . . 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 4301 . . . 4  |-  F/_ y
( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) )
42 nffvmpt1 5739 . . . . . 6  |-  F/_ y
( ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) y ) `  g ) `  A
) ) ) `  P )
43 nfcv 2574 . . . . . 6  |-  F/_ y G
4442, 43nffv 5738 . . . . 5  |-  F/_ y
( ( ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) `
 P ) `  G )
4544nfeq1 2583 . . . 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 5819 . . 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 2470 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 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    u. cun 3320    C_ wss 3322   <.cop 3819    e. cmpt 4269   ran crn 4882   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351  tpos ctpos 6481   Basecbs 13474    Hom chom 13545   Catccat 13894   Idccid 13895    Homf chomf 13896  oppCatcoppc 13942    Func cfunc 14056    o.func ccofu 14058   FuncCat cfuc 14144   SetCatcsetc 14235    X.c cxpc 14270    1stF c1stf 14271    2ndF c2ndf 14272   ⟨,⟩F cprf 14273   evalF cevlf 14311  HomFchof 14350  Yoncyon 14351
This theorem is referenced by:  yonedalem4c  14379  yonedainv  14383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089
  Copyright terms: Public domain W3C validator