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Theorem curf12 15020
Description: The partially evaluated curry functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
curfval.g  |-  G  =  ( <. C ,  D >. curryF  F
)
curfval.a  |-  A  =  ( Base `  C
)
curfval.c  |-  ( ph  ->  C  e.  Cat )
curfval.d  |-  ( ph  ->  D  e.  Cat )
curfval.f  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
curfval.b  |-  B  =  ( Base `  D
)
curf1.x  |-  ( ph  ->  X  e.  A )
curf1.k  |-  K  =  ( ( 1st `  G
) `  X )
curf11.y  |-  ( ph  ->  Y  e.  B )
curf12.j  |-  J  =  ( Hom  `  D
)
curf12.1  |-  .1.  =  ( Id `  C )
curf12.y  |-  ( ph  ->  Z  e.  B )
curf12.g  |-  ( ph  ->  H  e.  ( Y J Z ) )
Assertion
Ref Expression
curf12  |-  ( ph  ->  ( ( Y ( 2nd `  K ) Z ) `  H
)  =  ( (  .1.  `  X )
( <. X ,  Y >. ( 2nd `  F
) <. X ,  Z >. ) H ) )

Proof of Theorem curf12
Dummy variables  g 
y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curfval.g . . . 4  |-  G  =  ( <. C ,  D >. curryF  F
)
2 curfval.a . . . 4  |-  A  =  ( Base `  C
)
3 curfval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
4 curfval.d . . . 4  |-  ( ph  ->  D  e.  Cat )
5 curfval.f . . . 4  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
6 curfval.b . . . 4  |-  B  =  ( Base `  D
)
7 curf1.x . . . 4  |-  ( ph  ->  X  e.  A )
8 curf1.k . . . 4  |-  K  =  ( ( 1st `  G
) `  X )
9 curf12.j . . . 4  |-  J  =  ( Hom  `  D
)
10 curf12.1 . . . 4  |-  .1.  =  ( Id `  C )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curf1 15018 . . 3  |-  ( ph  ->  K  =  <. (
y  e.  B  |->  ( X ( 1st `  F
) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  X ) ( <. X ,  y >. ( 2nd `  F )
<. X ,  z >.
) g ) ) ) >. )
12 fvex 5689 . . . . . 6  |-  ( Base `  D )  e.  _V
136, 12eqeltri 2503 . . . . 5  |-  B  e. 
_V
1413mptex 5935 . . . 4  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  e.  _V
1513, 13mpt2ex 6639 . . . 4  |-  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) ) )  e.  _V
1614, 15op2ndd 6577 . . 3  |-  ( K  =  <. ( y  e.  B  |->  ( X ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z ) 
|->  ( (  .1.  `  X ) ( <. X ,  y >. ( 2nd `  F )
<. X ,  z >.
) g ) ) ) >.  ->  ( 2nd `  K )  =  ( y  e.  B , 
z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) g ) ) ) )
1711, 16syl 16 . 2  |-  ( ph  ->  ( 2nd `  K
)  =  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) ) ) )
18 curf11.y . . 3  |-  ( ph  ->  Y  e.  B )
19 curf12.y . . . 4  |-  ( ph  ->  Z  e.  B )
2019adantr 462 . . 3  |-  ( (
ph  /\  y  =  Y )  ->  Z  e.  B )
21 ovex 6105 . . . . 5  |-  ( y J z )  e. 
_V
2221mptex 5935 . . . 4  |-  ( g  e.  ( y J z )  |->  ( (  .1.  `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) )  e.  _V
2322a1i 11 . . 3  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  -> 
( g  e.  ( y J z ) 
|->  ( (  .1.  `  X ) ( <. X ,  y >. ( 2nd `  F )
<. X ,  z >.
) g ) )  e.  _V )
24 curf12.g . . . . . 6  |-  ( ph  ->  H  e.  ( Y J Z ) )
2524adantr 462 . . . . 5  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  ->  H  e.  ( Y J Z ) )
26 simprl 748 . . . . . 6  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  -> 
y  =  Y )
27 simprr 749 . . . . . 6  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  -> 
z  =  Z )
2826, 27oveq12d 6098 . . . . 5  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  -> 
( y J z )  =  ( Y J Z ) )
2925, 28eleqtrrd 2510 . . . 4  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  ->  H  e.  ( y J z ) )
30 ovex 6105 . . . . 5  |-  ( (  .1.  `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g )  e. 
_V
3130a1i 11 . . . 4  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  (
(  .1.  `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) g )  e. 
_V )
32 simplrl 752 . . . . . . 7  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  y  =  Y )
3332opeq2d 4054 . . . . . 6  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  <. X , 
y >.  =  <. X ,  Y >. )
34 simplrr 753 . . . . . . 7  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  z  =  Z )
3534opeq2d 4054 . . . . . 6  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  <. X , 
z >.  =  <. X ,  Z >. )
3633, 35oveq12d 6098 . . . . 5  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  ( <. X ,  y >.
( 2nd `  F
) <. X ,  z
>. )  =  ( <. X ,  Y >. ( 2nd `  F )
<. X ,  Z >. ) )
37 eqidd 2434 . . . . 5  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  (  .1.  `  X )  =  (  .1.  `  X
) )
38 simpr 458 . . . . 5  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  g  =  H )
3936, 37, 38oveq123d 6101 . . . 4  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  (
(  .1.  `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) g )  =  ( (  .1.  `  X ) ( <. X ,  Y >. ( 2nd `  F )
<. X ,  Z >. ) H ) )
4029, 31, 39fvmptdv2 5775 . . 3  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  -> 
( ( Y ( 2nd `  K ) Z )  =  ( g  e.  ( y J z )  |->  ( (  .1.  `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) g ) )  ->  ( ( Y ( 2nd `  K
) Z ) `  H )  =  ( (  .1.  `  X
) ( <. X ,  Y >. ( 2nd `  F
) <. X ,  Z >. ) H ) ) )
4118, 20, 23, 40ovmpt2dv 6212 . 2  |-  ( ph  ->  ( ( 2nd `  K
)  =  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) ) )  ->  ( ( Y ( 2nd `  K
) Z ) `  H )  =  ( (  .1.  `  X
) ( <. X ,  Y >. ( 2nd `  F
) <. X ,  Z >. ) H ) ) )
4217, 41mpd 15 1  |-  ( ph  ->  ( ( Y ( 2nd `  K ) Z ) `  H
)  =  ( (  .1.  `  X )
( <. X ,  Y >. ( 2nd `  F
) <. X ,  Z >. ) H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   _Vcvv 2962   <.cop 3871    e. cmpt 4338   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   1stc1st 6564   2ndc2nd 6565   Basecbs 14157   Hom chom 14232   Catccat 14585   Idccid 14586    Func cfunc 14747    X.c cxpc 14961   curryF ccurf 15003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-curf 15007
This theorem is referenced by:  curf1cl  15021  curf2cl  15024  uncfcurf  15032  diag12  15037  yon12  15058
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