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Theorem curf1 14015
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 14014 . . 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 2297 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  B  =  B )
12 simpr 447 . . . . . 6  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
1312oveq1d 5889 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  (
x ( 1st `  F
) y )  =  ( X ( 1st `  F ) y ) )
1411, 13mpteq12dv 4114 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  (
y  e.  B  |->  ( x ( 1st `  F
) y ) )  =  ( y  e.  B  |->  ( X ( 1st `  F ) y ) ) )
15 eqidd 2297 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  J  =  J )
1615oveqd 5891 . . . . . 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 3818 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  <. x ,  y >.  =  <. X ,  y >. )
1917opeq1d 3818 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  <. x ,  z >.  =  <. X ,  z >. )
2018, 19oveq12d 5892 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  ( <. x ,  y >.
( 2nd `  F
) <. x ,  z
>. )  =  ( <. X ,  y >.
( 2nd `  F
) <. X ,  z
>. ) )
21 eqidd 2297 . . . . . . . 8  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  .1.  =  .1.  )
2221, 17fveq12d 5547 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
23 eqidd 2297 . . . . . . 7  |-  ( ( ( ph  /\  x  =  X )  /\  y  e.  B  /\  z  e.  B )  ->  g  =  g )
2420, 22, 23oveq123d 5895 . . . . . 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 4114 . . . . 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 5928 . . . 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 3820 . . 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 4253 . . . 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 5622 . 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 2340 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   Basecbs 13164    Hom chom 13235   Catccat 13582   Idccid 13583    Func cfunc 13744    X.c cxpc 13958   curryF ccurf 14000
This theorem is referenced by:  curf11  14016  curf12  14017  curf1cl  14018  curf2ndf  14037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-curf 14004
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