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Theorem curf2val 15701
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 15700 . 2  |-  ( ph  ->  L  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
14 simpr 459 . . . . 5  |-  ( (
ph  /\  z  =  Z )  ->  z  =  Z )
1514opeq2d 4210 . . . 4  |-  ( (
ph  /\  z  =  Z )  ->  <. X , 
z >.  =  <. X ,  Z >. )
1614opeq2d 4210 . . . 4  |-  ( (
ph  /\  z  =  Z )  ->  <. Y , 
z >.  =  <. Y ,  Z >. )
1715, 16oveq12d 6288 . . 3  |-  ( (
ph  /\  z  =  Z )  ->  ( <. X ,  z >.
( 2nd `  F
) <. Y ,  z
>. )  =  ( <. X ,  Z >. ( 2nd `  F )
<. Y ,  Z >. ) )
18 eqidd 2455 . . 3  |-  ( (
ph  /\  z  =  Z )  ->  K  =  K )
1914fveq2d 5852 . . 3  |-  ( (
ph  /\  z  =  Z )  ->  (
I `  z )  =  ( I `  Z ) )
2017, 18, 19oveq123d 6291 . 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 6298 . . 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 5936 1  |-  ( ph  ->  ( L `  Z
)  =  ( K ( <. X ,  Z >. ( 2nd `  F
) <. Y ,  Z >. ) ( I `  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   <.cop 4022   ` cfv 5570  (class class class)co 6270   2ndc2nd 6772   Basecbs 14719   Hom chom 14798   Catccat 15156   Idccid 15157    Func cfunc 15345    X.c cxpc 15639   curryF ccurf 15681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-curf 15685
This theorem is referenced by:  curf2cl  15702  curfcl  15703  uncfcurf  15710  yon2  15737
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