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Theorem curf2val 15139
Description: Value of a component of the curry functor natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)
Hypotheses
Ref Expression
curf2.g  |-  G  =  ( <. C ,  D >. curryF  F
)
curf2.a  |-  A  =  ( Base `  C
)
curf2.c  |-  ( ph  ->  C  e.  Cat )
curf2.d  |-  ( ph  ->  D  e.  Cat )
curf2.f  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
curf2.b  |-  B  =  ( Base `  D
)
curf2.h  |-  H  =  ( Hom  `  C
)
curf2.i  |-  I  =  ( Id `  D
)
curf2.x  |-  ( ph  ->  X  e.  A )
curf2.y  |-  ( ph  ->  Y  e.  A )
curf2.k  |-  ( ph  ->  K  e.  ( X H Y ) )
curf2.l  |-  L  =  ( ( X ( 2nd `  G ) Y ) `  K
)
curf2.z  |-  ( ph  ->  Z  e.  B )
Assertion
Ref Expression
curf2val  |-  ( ph  ->  ( L `  Z
)  =  ( K ( <. X ,  Z >. ( 2nd `  F
) <. Y ,  Z >. ) ( I `  Z ) ) )

Proof of Theorem curf2val
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 curf2.g . . 3  |-  G  =  ( <. C ,  D >. curryF  F
)
2 curf2.a . . 3  |-  A  =  ( Base `  C
)
3 curf2.c . . 3  |-  ( ph  ->  C  e.  Cat )
4 curf2.d . . 3  |-  ( ph  ->  D  e.  Cat )
5 curf2.f . . 3  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
6 curf2.b . . 3  |-  B  =  ( Base `  D
)
7 curf2.h . . 3  |-  H  =  ( Hom  `  C
)
8 curf2.i . . 3  |-  I  =  ( Id `  D
)
9 curf2.x . . 3  |-  ( ph  ->  X  e.  A )
10 curf2.y . . 3  |-  ( ph  ->  Y  e.  A )
11 curf2.k . . 3  |-  ( ph  ->  K  e.  ( X H Y ) )
12 curf2.l . . 3  |-  L  =  ( ( X ( 2nd `  G ) Y ) `  K
)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12curf2 15138 . 2  |-  ( ph  ->  L  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
14 simpr 461 . . . . 5  |-  ( (
ph  /\  z  =  Z )  ->  z  =  Z )
1514opeq2d 4161 . . . 4  |-  ( (
ph  /\  z  =  Z )  ->  <. X , 
z >.  =  <. X ,  Z >. )
1614opeq2d 4161 . . . 4  |-  ( (
ph  /\  z  =  Z )  ->  <. Y , 
z >.  =  <. Y ,  Z >. )
1715, 16oveq12d 6205 . . 3  |-  ( (
ph  /\  z  =  Z )  ->  ( <. X ,  z >.
( 2nd `  F
) <. Y ,  z
>. )  =  ( <. X ,  Z >. ( 2nd `  F )
<. Y ,  Z >. ) )
18 eqidd 2452 . . 3  |-  ( (
ph  /\  z  =  Z )  ->  K  =  K )
1914fveq2d 5790 . . 3  |-  ( (
ph  /\  z  =  Z )  ->  (
I `  z )  =  ( I `  Z ) )
2017, 18, 19oveq123d 6208 . 2  |-  ( (
ph  /\  z  =  Z )  ->  ( K ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) )  =  ( K ( <. X ,  Z >. ( 2nd `  F )
<. Y ,  Z >. ) ( I `  Z
) ) )
21 curf2.z . 2  |-  ( ph  ->  Z  e.  B )
22 ovex 6212 . . 3  |-  ( K ( <. X ,  Z >. ( 2nd `  F
) <. Y ,  Z >. ) ( I `  Z ) )  e. 
_V
2322a1i 11 . 2  |-  ( ph  ->  ( K ( <. X ,  Z >. ( 2nd `  F )
<. Y ,  Z >. ) ( I `  Z
) )  e.  _V )
2413, 20, 21, 23fvmptd 5875 1  |-  ( ph  ->  ( L `  Z
)  =  ( K ( <. X ,  Z >. ( 2nd `  F
) <. Y ,  Z >. ) ( I `  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3065   <.cop 3978   ` cfv 5513  (class class class)co 6187   2ndc2nd 6673   Basecbs 14273   Hom chom 14348   Catccat 14701   Idccid 14702    Func cfunc 14863    X.c cxpc 15077   curryF ccurf 15119
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 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
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 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-1st 6674  df-2nd 6675  df-curf 15123
This theorem is referenced by:  curf2cl  15140  curfcl  15141  uncfcurf  15148  yon2  15175
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