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Theorem curf2 14102
Description: Value of the curry functor at a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
Hypotheses
Ref Expression
curf2.g  |-  G  =  ( <. C ,  D >. curryF  F
)
curf2.a  |-  A  =  ( Base `  C
)
curf2.c  |-  ( ph  ->  C  e.  Cat )
curf2.d  |-  ( ph  ->  D  e.  Cat )
curf2.f  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
curf2.b  |-  B  =  ( Base `  D
)
curf2.h  |-  H  =  (  Hom  `  C
)
curf2.i  |-  I  =  ( Id `  D
)
curf2.x  |-  ( ph  ->  X  e.  A )
curf2.y  |-  ( ph  ->  Y  e.  A )
curf2.k  |-  ( ph  ->  K  e.  ( X H Y ) )
curf2.l  |-  L  =  ( ( X ( 2nd `  G ) Y ) `  K
)
Assertion
Ref Expression
curf2  |-  ( ph  ->  L  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
Distinct variable groups:    z, C    z, F    z, H    z, L    z, E    z, G    z, I    ph, z    z, B   
z, D    z, X    z, K    z, Y
Allowed substitution hint:    A( z)

Proof of Theorem curf2
Dummy variables  x  y  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curf2.l . 2  |-  L  =  ( ( X ( 2nd `  G ) Y ) `  K
)
2 curf2.g . . . . 5  |-  G  =  ( <. C ,  D >. curryF  F
)
3 curf2.a . . . . 5  |-  A  =  ( Base `  C
)
4 curf2.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
5 curf2.d . . . . 5  |-  ( ph  ->  D  e.  Cat )
6 curf2.f . . . . 5  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
7 curf2.b . . . . 5  |-  B  =  ( Base `  D
)
8 eqid 2358 . . . . 5  |-  (  Hom  `  D )  =  (  Hom  `  D )
9 eqid 2358 . . . . 5  |-  ( Id
`  C )  =  ( Id `  C
)
10 curf2.h . . . . 5  |-  H  =  (  Hom  `  C
)
11 curf2.i . . . . 5  |-  I  =  ( Id `  D
)
122, 3, 4, 5, 6, 7, 8, 9, 10, 11curfval 14096 . . . 4  |-  ( ph  ->  G  =  <. (
x  e.  A  |->  <.
( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y (  Hom  `  D )
z )  |->  ( ( ( Id `  C
) `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  A , 
y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) ) ) >. )
13 fvex 5622 . . . . . . 7  |-  ( Base `  C )  e.  _V
143, 13eqeltri 2428 . . . . . 6  |-  A  e. 
_V
1514mptex 5832 . . . . 5  |-  ( x  e.  A  |->  <. (
y  e.  B  |->  ( x ( 1st `  F
) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y (  Hom  `  D ) z ) 
|->  ( ( ( Id
`  C ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. )  e.  _V
1614, 14mpt2ex 6285 . . . . 5  |-  ( x  e.  A ,  y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) ) )  e.  _V
1715, 16op2ndd 6218 . . . 4  |-  ( G  =  <. ( x  e.  A  |->  <. ( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y (  Hom  `  D
) z )  |->  ( ( ( Id `  C ) `  x
) ( <. x ,  y >. ( 2nd `  F ) <.
x ,  z >.
) g ) ) ) >. ) ,  ( x  e.  A , 
y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) ) ) >.  ->  ( 2nd `  G )  =  ( x  e.  A ,  y  e.  A  |->  ( g  e.  ( x H y ) 
|->  ( z  e.  B  |->  ( g ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) ) ) )
1812, 17syl 15 . . 3  |-  ( ph  ->  ( 2nd `  G
)  =  ( x  e.  A ,  y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) ) ) )
19 curf2.x . . . 4  |-  ( ph  ->  X  e.  A )
20 curf2.y . . . . 5  |-  ( ph  ->  Y  e.  A )
2120adantr 451 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  A )
22 ovex 5970 . . . . . 6  |-  ( x H y )  e. 
_V
2322mptex 5832 . . . . 5  |-  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) )  e.  _V
2423a1i 10 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( g  e.  ( x H y ) 
|->  ( z  e.  B  |->  ( g ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) )  e.  _V )
25 curf2.k . . . . . . 7  |-  ( ph  ->  K  e.  ( X H Y ) )
2625adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  K  e.  ( X H Y ) )
27 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
28 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
2927, 28oveq12d 5963 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x H y )  =  ( X H Y ) )
3026, 29eleqtrrd 2435 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  K  e.  ( x H y ) )
31 fvex 5622 . . . . . . . 8  |-  ( Base `  D )  e.  _V
327, 31eqeltri 2428 . . . . . . 7  |-  B  e. 
_V
3332mptex 5832 . . . . . 6  |-  ( z  e.  B  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) )  e.  _V
3433a1i 10 . . . . 5  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  (
z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) )  e.  _V )
35 eqidd 2359 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  B  =  B )
36 simplrl 736 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  x  =  X )
3736opeq1d 3883 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  <. x ,  z >.  =  <. X ,  z >. )
38 simplrr 737 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  y  =  Y )
3938opeq1d 3883 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  <. y ,  z >.  =  <. Y ,  z >. )
4037, 39oveq12d 5963 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  ( <. x ,  z >.
( 2nd `  F
) <. y ,  z
>. )  =  ( <. X ,  z >.
( 2nd `  F
) <. Y ,  z
>. ) )
41 simpr 447 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  g  =  K )
42 eqidd 2359 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  (
I `  z )  =  ( I `  z ) )
4340, 41, 42oveq123d 5966 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  (
g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) )  =  ( K ( <. X ,  z >. ( 2nd `  F )
<. Y ,  z >.
) ( I `  z ) ) )
4435, 43mpteq12dv 4179 . . . . 5  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  (
z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) )  =  ( z  e.  B  |->  ( K (
<. X ,  z >.
( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
4530, 34, 44fvmptdv2 5696 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( X ( 2nd `  G ) Y )  =  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) )  ->  ( ( X ( 2nd `  G
) Y ) `  K )  =  ( z  e.  B  |->  ( K ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) ) )
4619, 21, 24, 45ovmpt2dv 6067 . . 3  |-  ( ph  ->  ( ( 2nd `  G
)  =  ( x  e.  A ,  y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) ) )  ->  (
( X ( 2nd `  G ) Y ) `
 K )  =  ( z  e.  B  |->  ( K ( <. X ,  z >. ( 2nd `  F )
<. Y ,  z >.
) ( I `  z ) ) ) ) )
4718, 46mpd 14 . 2  |-  ( ph  ->  ( ( X ( 2nd `  G ) Y ) `  K
)  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
481, 47syl5eq 2402 1  |-  ( ph  ->  L  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864   <.cop 3719    e. cmpt 4158   ` cfv 5337  (class class class)co 5945    e. cmpt2 5947   1stc1st 6207   2ndc2nd 6208   Basecbs 13245    Hom chom 13316   Catccat 13665   Idccid 13666    Func cfunc 13827    X.c cxpc 14041   curryF ccurf 14083
This theorem is referenced by:  curf2val  14103  curf2cl  14104  curfcl  14105  diag2  14118  curf2ndf  14120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-curf 14087
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