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Theorem yonedalem4a 15758
Description: Lemma for yoneda 15766. (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 ) )
Assertion
Ref Expression
yonedalem4a  |-  ( ph  ->  ( ( F N X ) `  A
)  =  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 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, O, g, u, x, y    S, f, g, u, x, y    Q, f, g, u, x    T, f, g, u, 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)    Q( y)    R( x, y, f, g)    T( x)    U( x, y, u, f, g)    .1. ( u)    E( x)    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 yonedalem4a
StepHypRef Expression
1 yonedalem4.n . . . 4  |-  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 )
) ) ) )
21a1i 11 . . 3  |-  ( ph  ->  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 )
) ) ) ) )
3 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
f  =  F )
43fveq2d 5807 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
5 simprr 756 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  ->  x  =  X )
64, 5fveq12d 5809 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
( ( 1st `  f
) `  x )  =  ( ( 1st `  F ) `  X
) )
7 simplrr 761 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  x  =  X )
87oveq2d 6248 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  (
y ( Hom  `  C
) x )  =  ( y ( Hom  `  C ) X ) )
9 simplrl 760 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  f  =  F )
109fveq2d 5807 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  ( 2nd `  f )  =  ( 2nd `  F
) )
11 eqidd 2401 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  y  =  y )
1210, 7, 11oveq123d 6253 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  (
x ( 2nd `  f
) y )  =  ( X ( 2nd `  F ) y ) )
1312fveq1d 5805 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  (
( x ( 2nd `  f ) y ) `
 g )  =  ( ( X ( 2nd `  F ) y ) `  g
) )
1413fveq1d 5805 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  (
( ( x ( 2nd `  f ) y ) `  g
) `  u )  =  ( ( ( X ( 2nd `  F
) y ) `  g ) `  u
) )
158, 14mpteq12dv 4470 . . . . 5  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  (
g  e.  ( y ( Hom  `  C
) x )  |->  ( ( ( x ( 2nd `  f ) y ) `  g
) `  u )
)  =  ( g  e.  ( y ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  u ) ) )
1615mpteq2dva 4478 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) x )  |->  ( ( ( x ( 2nd `  f ) y ) `  g
) `  u )
) )  =  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  u )
) ) )
176, 16mpteq12dv 4470 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
( u  e.  ( ( 1st `  f
) `  x )  |->  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) x )  |->  ( ( ( x ( 2nd `  f ) y ) `  g
) `  u )
) ) )  =  ( u  e.  ( ( 1st `  F
) `  X )  |->  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  u )
) ) ) )
18 yonedalem21.f . . 3  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
19 yonedalem21.x . . 3  |-  ( ph  ->  X  e.  B )
20 fvex 5813 . . . . 5  |-  ( ( 1st `  F ) `
 X )  e. 
_V
2120mptex 6078 . . . 4  |-  ( u  e.  ( ( 1st `  F ) `  X
)  |->  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) y ) `  g ) `  u
) ) ) )  e.  _V
2221a1i 11 . . 3  |-  ( ph  ->  ( u  e.  ( ( 1st `  F
) `  X )  |->  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  u )
) ) )  e. 
_V )
232, 17, 18, 19, 22ovmpt2d 6365 . 2  |-  ( ph  ->  ( F N X )  =  ( u  e.  ( ( 1st `  F ) `  X
)  |->  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) y ) `  g ) `  u
) ) ) ) )
24 simpr 459 . . . . 5  |-  ( (
ph  /\  u  =  A )  ->  u  =  A )
2524fveq2d 5807 . . . 4  |-  ( (
ph  /\  u  =  A )  ->  (
( ( X ( 2nd `  F ) y ) `  g
) `  u )  =  ( ( ( X ( 2nd `  F
) y ) `  g ) `  A
) )
2625mpteq2dv 4479 . . 3  |-  ( (
ph  /\  u  =  A )  ->  (
g  e.  ( y ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  u )
)  =  ( g  e.  ( y ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) )
2726mpteq2dv 4479 . 2  |-  ( (
ph  /\  u  =  A )  ->  (
y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  u )
) )  =  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) ) )
28 yonedalem4.p . 2  |-  ( ph  ->  A  e.  ( ( 1st `  F ) `
 X ) )
29 yoneda.b . . . . 5  |-  B  =  ( Base `  C
)
30 fvex 5813 . . . . 5  |-  ( Base `  C )  e.  _V
3129, 30eqeltri 2484 . . . 4  |-  B  e. 
_V
3231mptex 6078 . . 3  |-  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) )  e.  _V
3332a1i 11 . 2  |-  ( ph  ->  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
) )  e.  _V )
3423, 27, 28, 33fvmptd 5892 1  |-  ( ph  ->  ( ( F N X ) `  A
)  =  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840   _Vcvv 3056    u. cun 3409    C_ wss 3411   <.cop 3975    |-> cmpt 4450   ran crn 4941   ` cfv 5523  (class class class)co 6232    |-> cmpt2 6234   1stc1st 6734   2ndc2nd 6735  tpos ctpos 6909   Basecbs 14731   Hom chom 14810   Catccat 15168   Idccid 15169   Hom f chomf 15170  oppCatcoppc 15214    Func cfunc 15357    o.func ccofu 15359   FuncCat cfuc 15445   SetCatcsetc 15568    X.c cxpc 15651    1stF c1stf 15652    2ndF c2ndf 15653   ⟨,⟩F cprf 15654   evalF cevlf 15692  HomFchof 15731  Yoncyon 15732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237
This theorem is referenced by:  yonedalem4b  15759  yonedalem4c  15760  yonffthlem  15765
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