MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  yonedalem4a Structured version   Unicode version

Theorem yonedalem4a 15207
Description: Lemma for yoneda 15215. (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 5806 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
5 simprr 756 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  x  =  X ) )  ->  x  =  X )
64, 5fveq12d 5808 . . . 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 6219 . . . . . 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 5806 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  ( 2nd `  f )  =  ( 2nd `  F
) )
11 eqidd 2455 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  y  =  y )
1210, 7, 11oveq123d 6224 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  (
x ( 2nd `  f
) y )  =  ( X ( 2nd `  F ) y ) )
1312fveq1d 5804 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  (
( x ( 2nd `  f ) y ) `
 g )  =  ( ( X ( 2nd `  F ) y ) `  g
) )
1413fveq1d 5804 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  x  =  X )
)  /\  y  e.  B )  ->  (
( ( x ( 2nd `  f ) y ) `  g
) `  u )  =  ( ( ( X ( 2nd `  F
) y ) `  g ) `  u
) )
158, 14mpteq12dv 4481 . . . . 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 4489 . . . 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 4481 . . 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 5812 . . . . 5  |-  ( ( 1st `  F ) `
 X )  e. 
_V
2120mptex 6060 . . . 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 6331 . 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 5806 . . . 4  |-  ( (
ph  /\  u  =  A )  ->  (
( ( X ( 2nd `  F ) y ) `  g
) `  u )  =  ( ( ( X ( 2nd `  F
) y ) `  g ) `  A
) )
2625mpteq2dv 4490 . . 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 4490 . 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 5812 . . . . 5  |-  ( Base `  C )  e.  _V
3129, 30eqeltri 2538 . . . 4  |-  B  e. 
_V
3231mptex 6060 . . 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 5891 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 1370    e. wcel 1758   _Vcvv 3078    u. cun 3437    C_ wss 3439   <.cop 3994    |-> cmpt 4461   ran crn 4952   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689  tpos ctpos 6857   Basecbs 14295   Hom chom 14371   Catccat 14724   Idccid 14725   Hom f chomf 14726  oppCatcoppc 14772    Func cfunc 14886    o.func ccofu 14888   FuncCat cfuc 14974   SetCatcsetc 15065    X.c cxpc 15100    1stF c1stf 15101    2ndF c2ndf 15102   ⟨,⟩F cprf 15103   evalF cevlf 15141  HomFchof 15180  Yoncyon 15181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208
This theorem is referenced by:  yonedalem4b  15208  yonedalem4c  15209  yonffthlem  15214
  Copyright terms: Public domain W3C validator