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Theorem curf12 15343
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 15341 . . 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 5867 . . . . . 6  |-  ( Base `  D )  e.  _V
136, 12eqeltri 2544 . . . . 5  |-  B  e. 
_V
1413mptex 6122 . . . 4  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  e.  _V
1513, 13mpt2ex 6850 . . . 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 6785 . . 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 6300 . . . . 5  |-  ( y J z )  e. 
_V
2221mptex 6122 . . . 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 6293 . . . . 5  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  -> 
( y J z )  =  ( Y J Z ) )
2925, 28eleqtrrd 2551 . . . 4  |-  ( (
ph  /\  ( y  =  Y  /\  z  =  Z ) )  ->  H  e.  ( y J z ) )
30 ovex 6300 . . . . 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 4213 . . . . . 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 4213 . . . . . 6  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  <. X , 
z >.  =  <. X ,  Z >. )
3633, 35oveq12d 6293 . . . . 5  |-  ( ( ( ph  /\  (
y  =  Y  /\  z  =  Z )
)  /\  g  =  H )  ->  ( <. X ,  y >.
( 2nd `  F
) <. X ,  z
>. )  =  ( <. X ,  Y >. ( 2nd `  F )
<. X ,  Z >. ) )
37 eqidd 2461 . . . . 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 6296 . . . 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 5954 . . 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 6410 . 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 1374    e. wcel 1762   _Vcvv 3106   <.cop 4026    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   1stc1st 6772   2ndc2nd 6773   Basecbs 14479   Hom chom 14555   Catccat 14908   Idccid 14909    Func cfunc 15070    X.c cxpc 15284   curryF ccurf 15326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-curf 15330
This theorem is referenced by:  curf1cl  15344  curf2cl  15347  uncfcurf  15355  diag12  15360  yon12  15381
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