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Theorem curf2 14289
Description: Value of the curry functor at a morphism. (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
)
Assertion
Ref Expression
curf2  |-  ( ph  ->  L  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
Distinct variable groups:    z, C    z, F    z, H    z, L    z, E    z, G    z, I    ph, z    z, B   
z, D    z, X    z, K    z, Y
Allowed substitution hint:    A( z)

Proof of Theorem curf2
Dummy variables  x  y  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curf2.l . 2  |-  L  =  ( ( X ( 2nd `  G ) Y ) `  K
)
2 curf2.g . . . . 5  |-  G  =  ( <. C ,  D >. curryF  F
)
3 curf2.a . . . . 5  |-  A  =  ( Base `  C
)
4 curf2.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
5 curf2.d . . . . 5  |-  ( ph  ->  D  e.  Cat )
6 curf2.f . . . . 5  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
7 curf2.b . . . . 5  |-  B  =  ( Base `  D
)
8 eqid 2412 . . . . 5  |-  (  Hom  `  D )  =  (  Hom  `  D )
9 eqid 2412 . . . . 5  |-  ( Id
`  C )  =  ( Id `  C
)
10 curf2.h . . . . 5  |-  H  =  (  Hom  `  C
)
11 curf2.i . . . . 5  |-  I  =  ( Id `  D
)
122, 3, 4, 5, 6, 7, 8, 9, 10, 11curfval 14283 . . . 4  |-  ( ph  ->  G  =  <. (
x  e.  A  |->  <.
( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y (  Hom  `  D )
z )  |->  ( ( ( Id `  C
) `  x )
( <. x ,  y
>. ( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. ) ,  ( x  e.  A , 
y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) ) ) >. )
13 fvex 5709 . . . . . . 7  |-  ( Base `  C )  e.  _V
143, 13eqeltri 2482 . . . . . 6  |-  A  e. 
_V
1514mptex 5933 . . . . 5  |-  ( x  e.  A  |->  <. (
y  e.  B  |->  ( x ( 1st `  F
) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y (  Hom  `  D ) z ) 
|->  ( ( ( Id
`  C ) `  x ) ( <.
x ,  y >.
( 2nd `  F
) <. x ,  z
>. ) g ) ) ) >. )  e.  _V
1614, 14mpt2ex 6392 . . . . 5  |-  ( x  e.  A ,  y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) ) )  e.  _V
1715, 16op2ndd 6325 . . . 4  |-  ( G  =  <. ( x  e.  A  |->  <. ( y  e.  B  |->  ( x ( 1st `  F ) y ) ) ,  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y (  Hom  `  D
) z )  |->  ( ( ( Id `  C ) `  x
) ( <. x ,  y >. ( 2nd `  F ) <.
x ,  z >.
) g ) ) ) >. ) ,  ( x  e.  A , 
y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) ) ) >.  ->  ( 2nd `  G )  =  ( x  e.  A ,  y  e.  A  |->  ( g  e.  ( x H y ) 
|->  ( z  e.  B  |->  ( g ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) ) ) )
1812, 17syl 16 . . 3  |-  ( ph  ->  ( 2nd `  G
)  =  ( x  e.  A ,  y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) ) ) )
19 curf2.x . . . 4  |-  ( ph  ->  X  e.  A )
20 curf2.y . . . . 5  |-  ( ph  ->  Y  e.  A )
2120adantr 452 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  A )
22 ovex 6073 . . . . . 6  |-  ( x H y )  e. 
_V
2322mptex 5933 . . . . 5  |-  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) )  e.  _V
2423a1i 11 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( g  e.  ( x H y ) 
|->  ( z  e.  B  |->  ( g ( <.
x ,  z >.
( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) )  e.  _V )
25 curf2.k . . . . . . 7  |-  ( ph  ->  K  e.  ( X H Y ) )
2625adantr 452 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  K  e.  ( X H Y ) )
27 simprl 733 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
28 simprr 734 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
2927, 28oveq12d 6066 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x H y )  =  ( X H Y ) )
3026, 29eleqtrrd 2489 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  K  e.  ( x H y ) )
31 fvex 5709 . . . . . . . 8  |-  ( Base `  D )  e.  _V
327, 31eqeltri 2482 . . . . . . 7  |-  B  e. 
_V
3332mptex 5933 . . . . . 6  |-  ( z  e.  B  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) )  e.  _V
3433a1i 11 . . . . 5  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  (
z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) )  e.  _V )
35 simplrl 737 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  x  =  X )
3635opeq1d 3958 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  <. x ,  z >.  =  <. X ,  z >. )
37 simplrr 738 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  y  =  Y )
3837opeq1d 3958 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  <. y ,  z >.  =  <. Y ,  z >. )
3936, 38oveq12d 6066 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  ( <. x ,  z >.
( 2nd `  F
) <. y ,  z
>. )  =  ( <. X ,  z >.
( 2nd `  F
) <. Y ,  z
>. ) )
40 simpr 448 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  g  =  K )
41 eqidd 2413 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  (
I `  z )  =  ( I `  z ) )
4239, 40, 41oveq123d 6069 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  (
g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) )  =  ( K ( <. X ,  z >. ( 2nd `  F )
<. Y ,  z >.
) ( I `  z ) ) )
4342mpteq2dv 4264 . . . . 5  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  g  =  K )  ->  (
z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) )  =  ( z  e.  B  |->  ( K (
<. X ,  z >.
( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
4430, 34, 43fvmptdv2 5785 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( X ( 2nd `  G ) Y )  =  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z >. ( 2nd `  F ) <.
y ,  z >.
) ( I `  z ) ) ) )  ->  ( ( X ( 2nd `  G
) Y ) `  K )  =  ( z  e.  B  |->  ( K ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) ) )
4519, 21, 24, 44ovmpt2dv 6173 . . 3  |-  ( ph  ->  ( ( 2nd `  G
)  =  ( x  e.  A ,  y  e.  A  |->  ( g  e.  ( x H y )  |->  ( z  e.  B  |->  ( g ( <. x ,  z
>. ( 2nd `  F
) <. y ,  z
>. ) ( I `  z ) ) ) ) )  ->  (
( X ( 2nd `  G ) Y ) `
 K )  =  ( z  e.  B  |->  ( K ( <. X ,  z >. ( 2nd `  F )
<. Y ,  z >.
) ( I `  z ) ) ) ) )
4618, 45mpd 15 . 2  |-  ( ph  ->  ( ( X ( 2nd `  G ) Y ) `  K
)  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
471, 46syl5eq 2456 1  |-  ( ph  ->  L  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924   <.cop 3785    e. cmpt 4234   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   1stc1st 6314   2ndc2nd 6315   Basecbs 13432    Hom chom 13503   Catccat 13852   Idccid 13853    Func cfunc 14014    X.c cxpc 14228   curryF ccurf 14270
This theorem is referenced by:  curf2val  14290  curf2cl  14291  curfcl  14292  diag2  14305  curf2ndf  14307
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-curf 14274
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