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Theorem curf1 13999
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 13998 . . 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 eqidd 2284 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  B  =  B )
12 simpr 447 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
1312oveq1d 5873 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
x ( 1st `  F
) y )  =  ( X ( 1st `  F ) y ) )
1411, 13mpteq12dv 4098 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  (
y  e.  B  |->  ( x ( 1st `  F
) y ) )  =  ( y  e.  B  |->  ( X ( 1st `  F ) y ) ) )
15 eqidd 2284 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  J  =  J )
1615oveqd 5875 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  (
y J z )  =  ( y J z ) )
17 simp1r 980 . . . . . . . . 9  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  x  =  X )
1817opeq1d 3802 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  <. x ,  y >.  =  <. X ,  y >. )
1917opeq1d 3802 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  <. x ,  z >.  =  <. X ,  z >. )
2018, 19oveq12d 5876 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  ( <. x ,  y >.
( 2nd `  F
) <. x ,  z
>. )  =  ( <. X ,  y >.
( 2nd `  F
) <. X ,  z
>. ) )
21 eqidd 2284 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  .1.  =  .1.  )
2221, 17fveq12d 5531 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
23 eqidd 2284 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  g  =  g )
2420, 22, 23oveq123d 5879 . . . . . 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 ) )
2516, 24mpteq12dv 4098 . . . . 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 ) ) )
2625mpt2eq3dva 5912 . . . 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 ) ) ) )
2714, 26opeq12d 3804 . . 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 ) ) ) >. )
28 curf1.x . . 3  |-  ( ph  ->  X  e.  A )
29 opex 4237 . . . 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
3029a1i 10 . . 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 )
3110, 27, 28, 30fvmptd 5606 . 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 ) ) ) >. )
321, 31syl5eq 2327 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   Basecbs 13148    Hom chom 13219   Catccat 13566   Idccid 13567    Func cfunc 13728    X.c cxpc 13942   curryF ccurf 13984
This theorem is referenced by:  curf11  14000  curf12  14001  curf1cl  14002  curf2ndf  14021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-curf 13988
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