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Theorem curf2 14331
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 2438 . . . . 5  |-  (  Hom  `  D )  =  (  Hom  `  D )
9 eqid 2438 . . . . 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 14325 . . . 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 5745 . . . . . . 7  |-  ( Base `  C )  e.  _V
143, 13eqeltri 2508 . . . . . 6  |-  A  e. 
_V
1514mptex 5969 . . . . 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 6428 . . . . 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 6361 . . . 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 16 . . 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 453 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  A )
22 ovex 6109 . . . . . 6  |-  ( x H y )  e. 
_V
2322mptex 5969 . . . . 5  |-  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) )  e.  _V
2423a1i 11 . . . 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 453 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  K  e.  ( X H Y ) )
27 simprl 734 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
28 simprr 735 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
2927, 28oveq12d 6102 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x H y )  =  ( X H Y ) )
3026, 29eleqtrrd 2515 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  K  e.  ( x H y ) )
31 fvex 5745 . . . . . . . 8  |-  ( Base `  D )  e.  _V
327, 31eqeltri 2508 . . . . . . 7  |-  B  e. 
_V
3332mptex 5969 . . . . . 6  |-  ( z  e.  B  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) )  e.  _V
3433a1i 11 . . . . 5  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  (
z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) )  e.  _V )
35 simplrl 738 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  x  =  X )
3635opeq1d 3992 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  <. x ,  z >.  =  <. X ,  z >. )
37 simplrr 739 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  y  =  Y )
3837opeq1d 3992 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  <. y ,  z >.  =  <. Y ,  z >. )
3936, 38oveq12d 6102 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  ( <. x ,  z >.
( 2nd `  F
) <. y ,  z
>. )  =  ( <. X ,  z >.
( 2nd `  F
) <. Y ,  z
>. ) )
40 simpr 449 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  g  =  K )
41 eqidd 2439 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  (
I `  z )  =  ( I `  z ) )
4239, 40, 41oveq123d 6105 . . . . . 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 ) ) )
4342mpteq2dv 4299 . . . . 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 ) ) ) )
4430, 34, 43fvmptdv2 5821 . . . 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 ) ) ) ) )
4519, 21, 24, 44ovmpt2dv 6209 . . 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 ) ) ) ) )
4618, 45mpd 15 . 2  |-  ( ph  ->  ( ( X ( 2nd `  G ) Y ) `  K
)  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
471, 46syl5eq 2482 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 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819    e. cmpt 4269   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351   Basecbs 13474    Hom chom 13545   Catccat 13894   Idccid 13895    Func cfunc 14056    X.c cxpc 14270   curryF ccurf 14312
This theorem is referenced by:  curf2val  14332  curf2cl  14333  curfcl  14334  diag2  14347  curf2ndf  14349
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  ax-un 4704
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-pw 3803  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  df-1st 6352  df-2nd 6353  df-curf 14316
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