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Theorem yonedalem4a 15077
Description: Lemma for yoneda 15085. (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 5690 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
5 simprr 756 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  ->  x  =  X )
64, 5fveq12d 5692 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
( ( 1st `  f
) `  x )  =  ( ( 1st `  F ) `  X
) )
7 simplrr 760 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  x  =  X )
87oveq2d 6102 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  (
y ( Hom  `  C
) x )  =  ( y ( Hom  `  C ) X ) )
9 simplrl 759 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  f  =  F )
109fveq2d 5690 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  ( 2nd `  f )  =  ( 2nd `  F
) )
11 eqidd 2439 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  y  =  y )
1210, 7, 11oveq123d 6107 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  (
x ( 2nd `  f
) y )  =  ( X ( 2nd `  F ) y ) )
1312fveq1d 5688 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  (
( x ( 2nd `  f ) y ) `
 g )  =  ( ( X ( 2nd `  F ) y ) `  g
) )
1413fveq1d 5688 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  (
( ( x ( 2nd `  f ) y ) `  g
) `  u )  =  ( ( ( X ( 2nd `  F
) y ) `  g ) `  u
) )
158, 14mpteq12dv 4365 . . . . 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 4373 . . . 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 4365 . . 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 5696 . . . . 5  |-  ( ( 1st `  F ) `
 X )  e. 
_V
2120mptex 5943 . . . 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 6213 . 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 461 . . . . 5  |-  ( (
ph  /\  u  =  A )  ->  u  =  A )
2524fveq2d 5690 . . . 4  |-  ( (
ph  /\  u  =  A )  ->  (
( ( X ( 2nd `  F ) y ) `  g
) `  u )  =  ( ( ( X ( 2nd `  F
) y ) `  g ) `  A
) )
2625mpteq2dv 4374 . . 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 4374 . 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 5696 . . . . 5  |-  ( Base `  C )  e.  _V
3129, 30eqeltri 2508 . . . 4  |-  B  e. 
_V
3231mptex 5943 . . 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 5774 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 369    = wceq 1369    e. wcel 1756   _Vcvv 2967    u. cun 3321    C_ wss 3323   <.cop 3878    e. cmpt 4345   ran crn 4836   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   1stc1st 6570   2ndc2nd 6571  tpos ctpos 6739   Basecbs 14166   Hom chom 14241   Catccat 14594   Idccid 14595   Hom f chomf 14596  oppCatcoppc 14642    Func cfunc 14756    o.func ccofu 14758   FuncCat cfuc 14844   SetCatcsetc 14935    X.c cxpc 14970    1stF c1stf 14971    2ndF c2ndf 14972   ⟨,⟩F cprf 14973   evalF cevlf 15011  HomFchof 15050  Yoncyon 15051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091
This theorem is referenced by:  yonedalem4b  15078  yonedalem4c  15079  yonffthlem  15084
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