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Theorem curf1 14314
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 14313 . . 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 6088 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
x ( 1st `  F
) y )  =  ( X ( 1st `  F ) y ) )
1312mpteq2dv 4288 . . . 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 3982 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  <. x ,  y >.  =  <. X ,  y >. )
1614opeq1d 3982 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  <. x ,  z >.  =  <. X ,  z >. )
1715, 16oveq12d 6091 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  ( <. x ,  y >.
( 2nd `  F
) <. x ,  z
>. )  =  ( <. X ,  y >.
( 2nd `  F
) <. X ,  z
>. ) )
1814fveq2d 5724 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
19 eqidd 2436 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  g  =  g )
2017, 18, 19oveq123d 6094 . . . . . 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 4288 . . . . 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 6130 . . . 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 3984 . . 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 4419 . . . 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 5802 . 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 2479 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 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809    e. cmpt 4258   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340   Basecbs 13461    Hom chom 13532   Catccat 13881   Idccid 13882    Func cfunc 14043    X.c cxpc 14257   curryF ccurf 14299
This theorem is referenced by:  curf11  14315  curf12  14316  curf1cl  14317  curf2ndf  14336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-curf 14303
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