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Theorem curf12 15139
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 15137 . . 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 5799 . . . . . 6  |-  ( Base `  D )  e.  _V
136, 12eqeltri 2535 . . . . 5  |-  B  e. 
_V
1413mptex 6047 . . . 4  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  e.  _V
1513, 13mpt2ex 6750 . . . 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 6688 . . 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 6215 . . . . 5  |-  ( y J z )  e. 
_V
2221mptex 6047 . . . 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 6208 . . . . 5  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  -> 
( y J z )  =  ( Y J Z ) )
2925, 28eleqtrrd 2542 . . . 4  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  ->  H  e.  ( y J z ) )
30 ovex 6215 . . . . 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 4164 . . . . . 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 4164 . . . . . 6  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  <. X , 
z >.  =  <. X ,  Z >. )
3633, 35oveq12d 6208 . . . . 5  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  ( <. X ,  y >.
( 2nd `  F
) <. X ,  z
>. )  =  ( <. X ,  Y >. ( 2nd `  F )
<. X ,  Z >. ) )
37 eqidd 2452 . . . . 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 6211 . . . 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 5886 . . 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 6323 . 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 1370    e. wcel 1758   _Vcvv 3068   <.cop 3981    |-> cmpt 4448   ` cfv 5516  (class class class)co 6190    |-> cmpt2 6192   1stc1st 6675   2ndc2nd 6676   Basecbs 14276   Hom chom 14351   Catccat 14704   Idccid 14705    Func cfunc 14866    X.c cxpc 15080   curryF ccurf 15122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-curf 15126
This theorem is referenced by:  curf1cl  15140  curf2cl  15143  uncfcurf  15151  diag12  15156  yon12  15177
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