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Theorem curf1 14250
Description: Value of the object part of the curry functor. (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 )
curf1.j  |-  J  =  (  Hom  `  D
)
curf1.1  |-  .1.  =  ( Id `  C )
Assertion
Ref Expression
curf1  |-  ( 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 ) ) ) >. )
Distinct variable groups:    y, g,
z,  .1.    y, A    B, g, y, z    C, g, y, z    D, g, y, z    ph, g,
y, z    g, E, y, z    g, J    g, K, y, z    g, X, y, z    g, F, y, z
Allowed substitution hints:    A( z, g)    G( y, z, g)    J( y, z)

Proof of Theorem curf1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 curf1.k . 2  |-  K  =  ( ( 1st `  G
) `  X )
2 curfval.g . . . 4  |-  G  =  ( <. C ,  D >. curryF  F
)
3 curfval.a . . . 4  |-  A  =  ( Base `  C
)
4 curfval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
5 curfval.d . . . 4  |-  ( ph  ->  D  e.  Cat )
6 curfval.f . . . 4  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
7 curfval.b . . . 4  |-  B  =  ( Base `  D
)
8 curf1.j . . . 4  |-  J  =  (  Hom  `  D
)
9 curf1.1 . . . 4  |-  .1.  =  ( Id `  C )
102, 3, 4, 5, 6, 7, 8, 9curf1fval 14249 . . 3  |-  ( ph  ->  ( 1st `  G
)  =  ( x  e.  A  |->  <. (
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 ) ) ) >. ) )
11 simpr 448 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
1211oveq1d 6036 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
x ( 1st `  F
) y )  =  ( X ( 1st `  F ) y ) )
1312mpteq2dv 4238 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  (
y  e.  B  |->  ( x ( 1st `  F
) y ) )  =  ( y  e.  B  |->  ( X ( 1st `  F ) y ) ) )
14 simp1r 982 . . . . . . . . 9  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  x  =  X )
1514opeq1d 3933 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  <. x ,  y >.  =  <. X ,  y >. )
1614opeq1d 3933 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  <. x ,  z >.  =  <. X ,  z >. )
1715, 16oveq12d 6039 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  ( <. x ,  y >.
( 2nd `  F
) <. x ,  z
>. )  =  ( <. X ,  y >.
( 2nd `  F
) <. X ,  z
>. ) )
1814fveq2d 5673 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
19 eqidd 2389 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  g  =  g )
2017, 18, 19oveq123d 6042 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  (
(  .1.  `  x
) ( <. x ,  y >. ( 2nd `  F ) <.
x ,  z >.
) g )  =  ( (  .1.  `  X ) ( <. X ,  y >. ( 2nd `  F )
<. X ,  z >.
) g ) )
2120mpteq2dv 4238 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  (
g  e.  ( y J z )  |->  ( (  .1.  `  x
) ( <. x ,  y >. ( 2nd `  F ) <.
x ,  z >.
) g ) )  =  ( g  e.  ( y J z )  |->  ( (  .1.  `  X ) ( <. X ,  y >. ( 2nd `  F )
<. X ,  z >.
) g ) ) )
2221mpt2eq3dva 6078 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  (
y  e.  B , 
z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  x
) ( <. x ,  y >. ( 2nd `  F ) <.
x ,  z >.
) g ) ) )  =  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y J z )  |->  ( (  .1.  `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) ) ) )
2313, 22opeq12d 3935 . . 3  |-  ( (
ph  /\  x  =  X )  ->  <. (
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 ) ) ) >.  =  <. ( 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 ) ) ) >. )
24 curf1.x . . 3  |-  ( ph  ->  X  e.  A )
25 opex 4369 . . . 4  |-  <. (
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 ) ) ) >.  e.  _V
2625a1i 11 . . 3  |-  ( ph  -> 
<. ( 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 ) ) ) >.  e.  _V )
2710, 23, 24, 26fvmptd 5750 . 2  |-  ( ph  ->  ( ( 1st `  G
) `  X )  =  <. ( 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 ) ) ) >. )
281, 27syl5eq 2432 1  |-  ( 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 ) ) ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2900   <.cop 3761    e. cmpt 4208   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   1stc1st 6287   2ndc2nd 6288   Basecbs 13397    Hom chom 13468   Catccat 13817   Idccid 13818    Func cfunc 13979    X.c cxpc 14193   curryF ccurf 14235
This theorem is referenced by:  curf11  14251  curf12  14252  curf1cl  14253  curf2ndf  14272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-curf 14239
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