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Theorem curf2 15372
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 2443 . . . . 5  |-  ( Hom  `  D )  =  ( Hom  `  D )
9 eqid 2443 . . . . 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 15366 . . . 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 5866 . . . . . . 7  |-  ( Base `  C )  e.  _V
143, 13eqeltri 2527 . . . . . 6  |-  A  e. 
_V
1514mptex 6128 . . . . 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 6862 . . . . 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 6796 . . . 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 465 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  A )
22 ovex 6309 . . . . . 6  |-  ( x H y )  e. 
_V
2322mptex 6128 . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  K  e.  ( X H Y ) )
27 simprl 756 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
28 simprr 757 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
2927, 28oveq12d 6299 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x H y )  =  ( X H Y ) )
3026, 29eleqtrrd 2534 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  K  e.  ( x H y ) )
31 fvex 5866 . . . . . . . 8  |-  ( Base `  D )  e.  _V
327, 31eqeltri 2527 . . . . . . 7  |-  B  e. 
_V
3332mptex 6128 . . . . . 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 761 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  x  =  X )
3635opeq1d 4208 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  <. x ,  z >.  =  <. X ,  z >. )
37 simplrr 762 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  y  =  Y )
3837opeq1d 4208 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  <. y ,  z >.  =  <. Y ,  z >. )
3936, 38oveq12d 6299 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  ( <. x ,  z >.
( 2nd `  F
) <. y ,  z
>. )  =  ( <. X ,  z >.
( 2nd `  F
) <. Y ,  z
>. ) )
40 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  g  =  K )
41 eqidd 2444 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  (
I `  z )  =  ( I `  z ) )
4239, 40, 41oveq123d 6302 . . . . . 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 4524 . . . . 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 5954 . . . 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 6420 . . 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 2496 1  |-  ( ph  ->  L  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095   <.cop 4020    |-> cmpt 4495   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   1stc1st 6783   2ndc2nd 6784   Basecbs 14509   Hom chom 14585   Catccat 14938   Idccid 14939    Func cfunc 15097    X.c cxpc 15311   curryF ccurf 15353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-curf 15357
This theorem is referenced by:  curf2val  15373  curf2cl  15374  curfcl  15375  diag2  15388  curf2ndf  15390
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