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Theorem curf12 15029
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 15027 . . 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 5696 . . . . . 6  |-  ( Base `  D )  e.  _V
136, 12eqeltri 2508 . . . . 5  |-  B  e. 
_V
1413mptex 5943 . . . 4  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  e.  _V
1513, 13mpt2ex 6645 . . . 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 6583 . . 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 465 . . 3  |-  ( (
ph  /\  y  =  Y )  ->  Z  e.  B )
21 ovex 6111 . . . . 5  |-  ( y J z )  e. 
_V
2221mptex 5943 . . . 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 465 . . . . 5  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  ->  H  e.  ( Y J Z ) )
26 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  -> 
y  =  Y )
27 simprr 756 . . . . . 6  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  -> 
z  =  Z )
2826, 27oveq12d 6104 . . . . 5  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  -> 
( y J z )  =  ( Y J Z ) )
2925, 28eleqtrrd 2515 . . . 4  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  ->  H  e.  ( y J z ) )
30 ovex 6111 . . . . 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 759 . . . . . . 7  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  y  =  Y )
3332opeq2d 4061 . . . . . 6  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  <. X , 
y >.  =  <. X ,  Y >. )
34 simplrr 760 . . . . . . 7  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  z  =  Z )
3534opeq2d 4061 . . . . . 6  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  <. X , 
z >.  =  <. X ,  Z >. )
3633, 35oveq12d 6104 . . . . 5  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  ( <. X ,  y >.
( 2nd `  F
) <. X ,  z
>. )  =  ( <. X ,  Y >. ( 2nd `  F )
<. X ,  Z >. ) )
37 eqidd 2439 . . . . 5  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  (  .1.  `  X )  =  (  .1.  `  X
) )
38 simpr 461 . . . . 5  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  g  =  H )
3936, 37, 38oveq123d 6107 . . . 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 5782 . . 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 6218 . 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 1369    e. wcel 1756   _Vcvv 2967   <.cop 3878    e. cmpt 4345   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   1stc1st 6570   2ndc2nd 6571   Basecbs 14166   Hom chom 14241   Catccat 14594   Idccid 14595    Func cfunc 14756    X.c cxpc 14970   curryF ccurf 15012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-curf 15016
This theorem is referenced by:  curf1cl  15030  curf2cl  15033  uncfcurf  15041  diag12  15046  yon12  15067
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