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Theorem curf11 15051
Description: Value of the double evaluated 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 )
curf11.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
curf11  |-  ( ph  ->  ( ( 1st `  K
) `  Y )  =  ( X ( 1st `  F ) Y ) )

Proof of Theorem curf11
Dummy variables  g 
y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curfval.g . . . 4  |-  G  =  ( <. C ,  D >. curryF  F
)
2 curfval.a . . . 4  |-  A  =  ( Base `  C
)
3 curfval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
4 curfval.d . . . 4  |-  ( ph  ->  D  e.  Cat )
5 curfval.f . . . 4  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
6 curfval.b . . . 4  |-  B  =  ( Base `  D
)
7 curf1.x . . . 4  |-  ( ph  ->  X  e.  A )
8 curf1.k . . . 4  |-  K  =  ( ( 1st `  G
) `  X )
9 eqid 2443 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
10 eqid 2443 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curf1 15050 . . 3  |-  ( ph  ->  K  =  <. (
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 ) ) ) >. )
12 fvex 5716 . . . . . 6  |-  ( Base `  D )  e.  _V
136, 12eqeltri 2513 . . . . 5  |-  B  e. 
_V
1413mptex 5963 . . . 4  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  e.  _V
1513, 13mpt2ex 6665 . . . 4  |-  ( 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, 15op1std 6602 . . 3  |-  ( K  =  <. ( 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 ) ) ) >.  ->  ( 1st `  K )  =  ( y  e.  B  |->  ( X ( 1st `  F
) y ) ) )
1711, 16syl 16 . 2  |-  ( ph  ->  ( 1st `  K
)  =  ( y  e.  B  |->  ( X ( 1st `  F
) y ) ) )
18 simpr 461 . . 3  |-  ( (
ph  /\  y  =  Y )  ->  y  =  Y )
1918oveq2d 6122 . 2  |-  ( (
ph  /\  y  =  Y )  ->  ( X ( 1st `  F
) y )  =  ( X ( 1st `  F ) Y ) )
20 curf11.y . 2  |-  ( ph  ->  Y  e.  B )
21 ovex 6131 . . 3  |-  ( X ( 1st `  F
) Y )  e. 
_V
2221a1i 11 . 2  |-  ( ph  ->  ( X ( 1st `  F ) Y )  e.  _V )
2317, 19, 20, 22fvmptd 5794 1  |-  ( ph  ->  ( ( 1st `  K
) `  Y )  =  ( X ( 1st `  F ) Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2987   <.cop 3898    e. cmpt 4365   ` cfv 5433  (class class class)co 6106    e. cmpt2 6108   1stc1st 6590   2ndc2nd 6591   Basecbs 14189   Hom chom 14264   Catccat 14617   Idccid 14618    Func cfunc 14779    X.c cxpc 14993   curryF ccurf 15035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-1st 6592  df-2nd 6593  df-curf 15039
This theorem is referenced by:  curf1cl  15053  curf2cl  15056  curfcl  15057  uncfcurf  15064  diag11  15068  yon11  15089
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