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Theorem curf12 14329
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 14327 . . 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 5745 . . . . . 6  |-  ( Base `  D )  e.  _V
136, 12eqeltri 2508 . . . . 5  |-  B  e. 
_V
1413mptex 5969 . . . 4  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  e.  _V
1513, 13mpt2ex 6428 . . . 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 6361 . . 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 453 . . 3  |-  ( (
ph  /\  y  =  Y )  ->  Z  e.  B )
21 ovex 6109 . . . . 5  |-  ( y J z )  e. 
_V
2221mptex 5969 . . . 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 453 . . . . 5  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  ->  H  e.  ( Y J Z ) )
26 simprl 734 . . . . . 6  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  -> 
y  =  Y )
27 simprr 735 . . . . . 6  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  -> 
z  =  Z )
2826, 27oveq12d 6102 . . . . 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 6109 . . . . 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 738 . . . . . . 7  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  y  =  Y )
3332opeq2d 3993 . . . . . 6  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  <. X , 
y >.  =  <. X ,  Y >. )
34 simplrr 739 . . . . . . 7  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  z  =  Z )
3534opeq2d 3993 . . . . . 6  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  <. X , 
z >.  =  <. X ,  Z >. )
3633, 35oveq12d 6102 . . . . 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 449 . . . . 5  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  g  =  H )
3936, 37, 38oveq123d 6105 . . . 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 5821 . . 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 6209 . 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 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:  curf1cl  14330  curf2cl  14333  uncfcurf  14341  diag12  14346  yon12  14367
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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