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Theorem curf1cl 14325
Description: The partially evaluated curry functor is a functor. (Contributed by Mario Carneiro, 13-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 )
Assertion
Ref Expression
curf1cl  |-  ( ph  ->  K  e.  ( D 
Func  E ) )

Proof of Theorem curf1cl
Dummy variables  g 
y  z  h  w 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 2436 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
10 eqid 2436 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curf1 14322 . . 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 5742 . . . . . . . 8  |-  ( Base `  D )  e.  _V
136, 12eqeltri 2506 . . . . . . 7  |-  B  e. 
_V
1413mptex 5966 . . . . . 6  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  e.  _V
1513, 13mpt2ex 6425 . . . . . 6  |-  ( 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 6357 . . . . 5  |-  ( 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 . . . 4  |-  ( ph  ->  ( 1st `  K
)  =  ( y  e.  B  |->  ( X ( 1st `  F
) y ) ) )
1814, 15op2ndd 6358 . . . . 5  |-  ( 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 ) ) ) >.  ->  ( 2nd `  K )  =  ( y  e.  B , 
z  e.  B  |->  ( g  e.  ( y (  Hom  `  D
) z )  |->  ( ( ( Id `  C ) `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) g ) ) ) )
1911, 18syl 16 . . . 4  |-  ( ph  ->  ( 2nd `  K
)  =  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y (  Hom  `  D )
z )  |->  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) ) ) )
2017, 19opeq12d 3992 . . 3  |-  ( ph  -> 
<. ( 1st `  K
) ,  ( 2nd `  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 ) ) ) >. )
2111, 20eqtr4d 2471 . 2  |-  ( ph  ->  K  =  <. ( 1st `  K ) ,  ( 2nd `  K
) >. )
22 eqid 2436 . . . 4  |-  ( Base `  E )  =  (
Base `  E )
23 eqid 2436 . . . 4  |-  (  Hom  `  E )  =  (  Hom  `  E )
24 eqid 2436 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
25 eqid 2436 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
26 eqid 2436 . . . 4  |-  (comp `  D )  =  (comp `  D )
27 eqid 2436 . . . 4  |-  (comp `  E )  =  (comp `  E )
28 funcrcl 14060 . . . . . 6  |-  ( F  e.  ( ( C  X.c  D )  Func  E
)  ->  ( ( C  X.c  D )  e.  Cat  /\  E  e.  Cat )
)
295, 28syl 16 . . . . 5  |-  ( ph  ->  ( ( C  X.c  D
)  e.  Cat  /\  E  e.  Cat )
)
3029simprd 450 . . . 4  |-  ( ph  ->  E  e.  Cat )
31 eqid 2436 . . . . . . . . . 10  |-  ( C  X.c  D )  =  ( C  X.c  D )
3231, 2, 6xpcbas 14275 . . . . . . . . 9  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
33 relfunc 14059 . . . . . . . . . 10  |-  Rel  (
( C  X.c  D ) 
Func  E )
34 1st2ndbr 6396 . . . . . . . . . 10  |-  ( ( Rel  ( ( C  X.c  D )  Func  E
)  /\  F  e.  ( ( C  X.c  D
)  Func  E )
)  ->  ( 1st `  F ) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
3533, 5, 34sylancr 645 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
3632, 22, 35funcf1 14063 . . . . . . . 8  |-  ( ph  ->  ( 1st `  F
) : ( A  X.  B ) --> (
Base `  E )
)
3736adantr 452 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  ( 1st `  F ) : ( A  X.  B
) --> ( Base `  E
) )
387adantr 452 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  X  e.  A )
39 simpr 448 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  y  e.  B )
4037, 38, 39fovrnd 6218 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  ( X ( 1st `  F
) y )  e.  ( Base `  E
) )
41 eqid 2436 . . . . . 6  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  =  ( y  e.  B  |->  ( X ( 1st `  F ) y ) )
4240, 41fmptd 5893 . . . . 5  |-  ( ph  ->  ( y  e.  B  |->  ( X ( 1st `  F ) y ) ) : B --> ( Base `  E ) )
4317feq1d 5580 . . . . 5  |-  ( ph  ->  ( ( 1st `  K
) : B --> ( Base `  E )  <->  ( y  e.  B  |->  ( X ( 1st `  F
) y ) ) : B --> ( Base `  E ) ) )
4442, 43mpbird 224 . . . 4  |-  ( ph  ->  ( 1st `  K
) : B --> ( Base `  E ) )
45 eqid 2436 . . . . . 6  |-  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y (  Hom  `  D )
z )  |->  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) ) )  =  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y (  Hom  `  D )
z )  |->  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) ) )
46 ovex 6106 . . . . . . 7  |-  ( y (  Hom  `  D
) z )  e. 
_V
4746mptex 5966 . . . . . 6  |-  ( g  e.  ( y (  Hom  `  D )
z )  |->  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) )  e.  _V
4845, 47fnmpt2i 6420 . . . . 5  |-  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y (  Hom  `  D )
z )  |->  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) ) )  Fn  ( B  X.  B )
4919fneq1d 5536 . . . . 5  |-  ( ph  ->  ( ( 2nd `  K
)  Fn  ( B  X.  B )  <->  ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y (  Hom  `  D ) z ) 
|->  ( ( ( Id
`  C ) `  X ) ( <. X ,  y >. ( 2nd `  F )
<. X ,  z >.
) g ) ) )  Fn  ( B  X.  B ) ) )
5048, 49mpbiri 225 . . . 4  |-  ( ph  ->  ( 2nd `  K
)  Fn  ( B  X.  B ) )
51 eqid 2436 . . . . . . . . 9  |-  (  Hom  `  ( C  X.c  D ) )  =  (  Hom  `  ( C  X.c  D ) )
5235ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( 1st `  F ) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
537ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  X  e.  A )
54 simplrl 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  y  e.  B )
55 opelxpi 4910 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  y  e.  B )  -> 
<. X ,  y >.  e.  ( A  X.  B
) )
5653, 54, 55syl2anc 643 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  <. X , 
y >.  e.  ( A  X.  B ) )
57 simplrr 738 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  z  e.  B )
58 opelxpi 4910 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  z  e.  B )  -> 
<. X ,  z >.  e.  ( A  X.  B
) )
5953, 57, 58syl2anc 643 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  <. X , 
z >.  e.  ( A  X.  B ) )
6032, 51, 23, 52, 56, 59funcf2 14065 . . . . . . . 8  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( <. X ,  y >. ( 2nd `  F ) <. X ,  z >. ) : ( <. X , 
y >. (  Hom  `  ( C  X.c  D ) ) <. X ,  z >. ) --> ( ( ( 1st `  F ) `  <. X ,  y >. )
(  Hom  `  E ) ( ( 1st `  F
) `  <. X , 
z >. ) ) )
61 eqid 2436 . . . . . . . . . 10  |-  (  Hom  `  C )  =  (  Hom  `  C )
6231, 32, 61, 9, 51, 56, 59xpchom 14277 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( <. X ,  y >. (  Hom  `  ( C  X.c  D
) ) <. X , 
z >. )  =  ( ( ( 1st `  <. X ,  y >. )
(  Hom  `  C ) ( 1st `  <. X ,  z >. )
)  X.  ( ( 2nd `  <. X , 
y >. ) (  Hom  `  D ) ( 2nd `  <. X ,  z
>. ) ) ) )
633ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  C  e.  Cat )
644ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  D  e.  Cat )
655ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  F  e.  ( ( C  X.c  D
)  Func  E )
)
661, 2, 63, 64, 65, 6, 53, 8, 54curf11 14323 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  K ) `  y )  =  ( X ( 1st `  F
) y ) )
67 df-ov 6084 . . . . . . . . . . 11  |-  ( X ( 1st `  F
) y )  =  ( ( 1st `  F
) `  <. X , 
y >. )
6866, 67syl6req 2485 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  F ) `  <. X ,  y >.
)  =  ( ( 1st `  K ) `
 y ) )
691, 2, 63, 64, 65, 6, 53, 8, 57curf11 14323 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  K ) `  z )  =  ( X ( 1st `  F
) z ) )
70 df-ov 6084 . . . . . . . . . . 11  |-  ( X ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. X , 
z >. )
7169, 70syl6req 2485 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  F ) `  <. X ,  z >.
)  =  ( ( 1st `  K ) `
 z ) )
7268, 71oveq12d 6099 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( (
( 1st `  F
) `  <. X , 
y >. ) (  Hom  `  E ) ( ( 1st `  F ) `
 <. X ,  z
>. ) )  =  ( ( ( 1st `  K
) `  y )
(  Hom  `  E ) ( ( 1st `  K
) `  z )
) )
7362, 72feq23d 5588 . . . . . . . 8  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( <. X ,  y >.
( 2nd `  F
) <. X ,  z
>. ) : ( <. X ,  y >. (  Hom  `  ( C  X.c  D ) ) <. X ,  z >. ) --> ( ( ( 1st `  F ) `  <. X ,  y >. )
(  Hom  `  E ) ( ( 1st `  F
) `  <. X , 
z >. ) )  <->  ( <. X ,  y >. ( 2nd `  F ) <. X ,  z >. ) : ( ( ( 1st `  <. X , 
y >. ) (  Hom  `  C ) ( 1st `  <. X ,  z
>. ) )  X.  (
( 2nd `  <. X ,  y >. )
(  Hom  `  D ) ( 2nd `  <. X ,  z >. )
) ) --> ( ( ( 1st `  K
) `  y )
(  Hom  `  E ) ( ( 1st `  K
) `  z )
) ) )
7460, 73mpbid 202 . . . . . . 7  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( <. X ,  y >. ( 2nd `  F ) <. X ,  z >. ) : ( ( ( 1st `  <. X , 
y >. ) (  Hom  `  C ) ( 1st `  <. X ,  z
>. ) )  X.  (
( 2nd `  <. X ,  y >. )
(  Hom  `  D ) ( 2nd `  <. X ,  z >. )
) ) --> ( ( ( 1st `  K
) `  y )
(  Hom  `  E ) ( ( 1st `  K
) `  z )
) )
752, 61, 10, 63, 53catidcl 13907 . . . . . . . 8  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( Id `  C ) `  X )  e.  ( X (  Hom  `  C
) X ) )
76 op1stg 6359 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  y  e.  B )  ->  ( 1st `  <. X ,  y >. )  =  X )
7753, 54, 76syl2anc 643 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( 1st ` 
<. X ,  y >.
)  =  X )
78 op1stg 6359 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  z  e.  B )  ->  ( 1st `  <. X ,  z >. )  =  X )
7953, 57, 78syl2anc 643 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( 1st ` 
<. X ,  z >.
)  =  X )
8077, 79oveq12d 6099 . . . . . . . 8  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  <. X ,  y
>. ) (  Hom  `  C
) ( 1st `  <. X ,  z >. )
)  =  ( X (  Hom  `  C
) X ) )
8175, 80eleqtrrd 2513 . . . . . . 7  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( Id `  C ) `  X )  e.  ( ( 1st `  <. X ,  y >. )
(  Hom  `  C ) ( 1st `  <. X ,  z >. )
) )
82 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  g  e.  ( y (  Hom  `  D ) z ) )
83 op2ndg 6360 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  y  e.  B )  ->  ( 2nd `  <. X ,  y >. )  =  y )
8453, 54, 83syl2anc 643 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( 2nd ` 
<. X ,  y >.
)  =  y )
85 op2ndg 6360 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  z  e.  B )  ->  ( 2nd `  <. X ,  z >. )  =  z )
8653, 57, 85syl2anc 643 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( 2nd ` 
<. X ,  z >.
)  =  z )
8784, 86oveq12d 6099 . . . . . . . 8  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 2nd `  <. X ,  y
>. ) (  Hom  `  D
) ( 2nd `  <. X ,  z >. )
)  =  ( y (  Hom  `  D
) z ) )
8882, 87eleqtrrd 2513 . . . . . . 7  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  g  e.  ( ( 2nd `  <. X ,  y >. )
(  Hom  `  D ) ( 2nd `  <. X ,  z >. )
) )
8974, 81, 88fovrnd 6218 . . . . . 6  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( (
( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g )  e.  ( ( ( 1st `  K ) `  y
) (  Hom  `  E
) ( ( 1st `  K ) `  z
) ) )
90 eqid 2436 . . . . . 6  |-  ( g  e.  ( y (  Hom  `  D )
z )  |->  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) )  =  ( g  e.  ( y (  Hom  `  D ) z ) 
|->  ( ( ( Id
`  C ) `  X ) ( <. X ,  y >. ( 2nd `  F )
<. X ,  z >.
) g ) )
9189, 90fmptd 5893 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( g  e.  ( y (  Hom  `  D
) z )  |->  ( ( ( Id `  C ) `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) g ) ) : ( y (  Hom  `  D )
z ) --> ( ( ( 1st `  K
) `  y )
(  Hom  `  E ) ( ( 1st `  K
) `  z )
) )
9219oveqd 6098 . . . . . . 7  |-  ( ph  ->  ( y ( 2nd `  K ) z )  =  ( y ( y  e.  B , 
z  e.  B  |->  ( g  e.  ( y (  Hom  `  D
) z )  |->  ( ( ( Id `  C ) `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) g ) ) ) z ) )
9345ovmpt4g 6196 . . . . . . . 8  |-  ( ( y  e.  B  /\  z  e.  B  /\  ( g  e.  ( y (  Hom  `  D
) z )  |->  ( ( ( Id `  C ) `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) g ) )  e.  _V )  -> 
( y ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y (  Hom  `  D )
z )  |->  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) ) ) z )  =  ( g  e.  ( y (  Hom  `  D
) z )  |->  ( ( ( Id `  C ) `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) g ) ) )
9447, 93mp3an3 1268 . . . . . . 7  |-  ( ( y  e.  B  /\  z  e.  B )  ->  ( y ( y  e.  B ,  z  e.  B  |->  ( g  e.  ( y (  Hom  `  D )
z )  |->  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) ) ) z )  =  ( g  e.  ( y (  Hom  `  D
) z )  |->  ( ( ( Id `  C ) `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) g ) ) )
9592, 94sylan9eq 2488 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( 2nd `  K ) z )  =  ( g  e.  ( y (  Hom  `  D ) z ) 
|->  ( ( ( Id
`  C ) `  X ) ( <. X ,  y >. ( 2nd `  F )
<. X ,  z >.
) g ) ) )
9695feq1d 5580 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( ( y ( 2nd `  K ) z ) : ( y (  Hom  `  D
) z ) --> ( ( ( 1st `  K
) `  y )
(  Hom  `  E ) ( ( 1st `  K
) `  z )
)  <->  ( g  e.  ( y (  Hom  `  D ) z ) 
|->  ( ( ( Id
`  C ) `  X ) ( <. X ,  y >. ( 2nd `  F )
<. X ,  z >.
) g ) ) : ( y (  Hom  `  D )
z ) --> ( ( ( 1st `  K
) `  y )
(  Hom  `  E ) ( ( 1st `  K
) `  z )
) ) )
9791, 96mpbird 224 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( 2nd `  K ) z ) : ( y (  Hom  `  D )
z ) --> ( ( ( 1st `  K
) `  y )
(  Hom  `  E ) ( ( 1st `  K
) `  z )
) )
983adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  B )  ->  C  e.  Cat )
994adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  Cat )
100 eqid 2436 . . . . . . . . 9  |-  ( Id
`  ( C  X.c  D
) )  =  ( Id `  ( C  X.c  D ) )
10131, 98, 99, 2, 6, 10, 24, 100, 38, 39xpcid 14286 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B )  ->  (
( Id `  ( C  X.c  D ) ) `  <. X ,  y >.
)  =  <. (
( Id `  C
) `  X ) ,  ( ( Id
`  D ) `  y ) >. )
102101fveq2d 5732 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  (
( <. X ,  y
>. ( 2nd `  F
) <. X ,  y
>. ) `  ( ( Id `  ( C  X.c  D ) ) `  <. X ,  y >.
) )  =  ( ( <. X ,  y
>. ( 2nd `  F
) <. X ,  y
>. ) `  <. (
( Id `  C
) `  X ) ,  ( ( Id
`  D ) `  y ) >. )
)
103 df-ov 6084 . . . . . . 7  |-  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  y
>. ) ( ( Id
`  D ) `  y ) )  =  ( ( <. X , 
y >. ( 2nd `  F
) <. X ,  y
>. ) `  <. (
( Id `  C
) `  X ) ,  ( ( Id
`  D ) `  y ) >. )
104102, 103syl6eqr 2486 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
( <. X ,  y
>. ( 2nd `  F
) <. X ,  y
>. ) `  ( ( Id `  ( C  X.c  D ) ) `  <. X ,  y >.
) )  =  ( ( ( Id `  C ) `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  y
>. ) ( ( Id
`  D ) `  y ) ) )
10535adantr 452 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  ( 1st `  F ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  F ) )
1067, 55sylan 458 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  <. X , 
y >.  e.  ( A  X.  B ) )
10732, 100, 25, 105, 106funcid 14067 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
( <. X ,  y
>. ( 2nd `  F
) <. X ,  y
>. ) `  ( ( Id `  ( C  X.c  D ) ) `  <. X ,  y >.
) )  =  ( ( Id `  E
) `  ( ( 1st `  F ) `  <. X ,  y >.
) ) )
108104, 107eqtr3d 2470 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  (
( ( Id `  C ) `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  y
>. ) ( ( Id
`  D ) `  y ) )  =  ( ( Id `  E ) `  (
( 1st `  F
) `  <. X , 
y >. ) ) )
1095adantr 452 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  F  e.  ( ( C  X.c  D
)  Func  E )
)
1106, 9, 24, 99, 39catidcl 13907 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
( Id `  D
) `  y )  e.  ( y (  Hom  `  D ) y ) )
1111, 2, 98, 99, 109, 6, 38, 8, 39, 9, 10, 39, 110curf12 14324 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  (
( y ( 2nd `  K ) y ) `
 ( ( Id
`  D ) `  y ) )  =  ( ( ( Id
`  C ) `  X ) ( <. X ,  y >. ( 2nd `  F )
<. X ,  y >.
) ( ( Id
`  D ) `  y ) ) )
1121, 2, 98, 99, 109, 6, 38, 8, 39curf11 14323 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  (
( 1st `  K
) `  y )  =  ( X ( 1st `  F ) y ) )
113112, 67syl6eq 2484 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
( 1st `  K
) `  y )  =  ( ( 1st `  F ) `  <. X ,  y >. )
)
114113fveq2d 5732 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  (
( Id `  E
) `  ( ( 1st `  K ) `  y ) )  =  ( ( Id `  E ) `  (
( 1st `  F
) `  <. X , 
y >. ) ) )
115108, 111, 1143eqtr4d 2478 . . . 4  |-  ( (
ph  /\  y  e.  B )  ->  (
( y ( 2nd `  K ) y ) `
 ( ( Id
`  D ) `  y ) )  =  ( ( Id `  E ) `  (
( 1st `  K
) `  y )
) )
11673ad2ant1 978 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  X  e.  A
)
117 simp21 990 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  y  e.  B
)
118 simp22 991 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  z  e.  B
)
119 eqid 2436 . . . . . . . . . 10  |-  (comp `  C )  =  (comp `  C )
120 eqid 2436 . . . . . . . . . 10  |-  (comp `  ( C  X.c  D )
)  =  (comp `  ( C  X.c  D )
)
121 simp23 992 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  w  e.  B
)
12233ad2ant1 978 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  C  e.  Cat )
1232, 61, 10, 122, 116catidcl 13907 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( Id
`  C ) `  X )  e.  ( X (  Hom  `  C
) X ) )
124 simp3l 985 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  g  e.  ( y (  Hom  `  D
) z ) )
125 simp3r 986 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  h  e.  ( z (  Hom  `  D
) w ) )
12631, 2, 6, 61, 9, 116, 117, 116, 118, 119, 26, 120, 116, 121, 123, 124, 123, 125xpcco2 14284 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( <. (
( Id `  C
) `  X ) ,  h >. ( <. <. X , 
y >. ,  <. X , 
z >. >. (comp `  ( C  X.c  D ) ) <. X ,  w >. )
<. ( ( Id `  C ) `  X
) ,  g >.
)  =  <. (
( ( Id `  C ) `  X
) ( <. X ,  X >. (comp `  C
) X ) ( ( Id `  C
) `  X )
) ,  ( h ( <. y ,  z
>. (comp `  D )
w ) g )
>. )
1272, 61, 10, 122, 116, 119, 116, 123catlid 13908 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( ( Id `  C ) `
 X ) (
<. X ,  X >. (comp `  C ) X ) ( ( Id `  C ) `  X
) )  =  ( ( Id `  C
) `  X )
)
128127opeq1d 3990 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  <. ( ( ( Id `  C ) `
 X ) (
<. X ,  X >. (comp `  C ) X ) ( ( Id `  C ) `  X
) ) ,  ( h ( <. y ,  z >. (comp `  D ) w ) g ) >.  =  <. ( ( Id `  C
) `  X ) ,  ( h (
<. y ,  z >.
(comp `  D )
w ) g )
>. )
129126, 128eqtrd 2468 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( <. (
( Id `  C
) `  X ) ,  h >. ( <. <. X , 
y >. ,  <. X , 
z >. >. (comp `  ( C  X.c  D ) ) <. X ,  w >. )
<. ( ( Id `  C ) `  X
) ,  g >.
)  =  <. (
( Id `  C
) `  X ) ,  ( h (
<. y ,  z >.
(comp `  D )
w ) g )
>. )
130129fveq2d 5732 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( <. X ,  y >. ( 2nd `  F )
<. X ,  w >. ) `
 ( <. (
( Id `  C
) `  X ) ,  h >. ( <. <. X , 
y >. ,  <. X , 
z >. >. (comp `  ( C  X.c  D ) ) <. X ,  w >. )
<. ( ( Id `  C ) `  X
) ,  g >.
) )  =  ( ( <. X ,  y
>. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  ( h (
<. y ,  z >.
(comp `  D )
w ) g )
>. ) )
131 df-ov 6084 . . . . . . 7  |-  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  w >. ) ( h (
<. y ,  z >.
(comp `  D )
w ) g ) )  =  ( (
<. X ,  y >.
( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  ( h (
<. y ,  z >.
(comp `  D )
w ) g )
>. )
132130, 131syl6eqr 2486 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( <. X ,  y >. ( 2nd `  F )
<. X ,  w >. ) `
 ( <. (
( Id `  C
) `  X ) ,  h >. ( <. <. X , 
y >. ,  <. X , 
z >. >. (comp `  ( C  X.c  D ) ) <. X ,  w >. )
<. ( ( Id `  C ) `  X
) ,  g >.
) )  =  ( ( ( Id `  C ) `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  w >. ) ( h (
<. y ,  z >.
(comp `  D )
w ) g ) ) )
133353ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( 1st `  F
) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
134116, 117, 55syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  <. X ,  y
>.  e.  ( A  X.  B ) )
135116, 118, 58syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  <. X ,  z
>.  e.  ( A  X.  B ) )
136 opelxpi 4910 . . . . . . . 8  |-  ( ( X  e.  A  /\  w  e.  B )  -> 
<. X ,  w >.  e.  ( A  X.  B
) )
137116, 121, 136syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  <. X ,  w >.  e.  ( A  X.  B ) )
138 opelxpi 4910 . . . . . . . . 9  |-  ( ( ( ( Id `  C ) `  X
)  e.  ( X (  Hom  `  C
) X )  /\  g  e.  ( y
(  Hom  `  D ) z ) )  ->  <. ( ( Id `  C ) `  X
) ,  g >.  e.  ( ( X (  Hom  `  C ) X )  X.  (
y (  Hom  `  D
) z ) ) )
139123, 124, 138syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  <. ( ( Id
`  C ) `  X ) ,  g
>.  e.  ( ( X (  Hom  `  C
) X )  X.  ( y (  Hom  `  D ) z ) ) )
14031, 2, 6, 61, 9, 116, 117, 116, 118, 51xpchom2 14283 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( <. X , 
y >. (  Hom  `  ( C  X.c  D ) ) <. X ,  z >. )  =  ( ( X (  Hom  `  C
) X )  X.  ( y (  Hom  `  D ) z ) ) )
141139, 140eleqtrrd 2513 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  <. ( ( Id
`  C ) `  X ) ,  g
>.  e.  ( <. X , 
y >. (  Hom  `  ( C  X.c  D ) ) <. X ,  z >. ) )
142 opelxpi 4910 . . . . . . . . 9  |-  ( ( ( ( Id `  C ) `  X
)  e.  ( X (  Hom  `  C
) X )  /\  h  e.  ( z
(  Hom  `  D ) w ) )  ->  <. ( ( Id `  C ) `  X
) ,  h >.  e.  ( ( X (  Hom  `  C ) X )  X.  (
z (  Hom  `  D
) w ) ) )
143123, 125, 142syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  <. ( ( Id
`  C ) `  X ) ,  h >.  e.  ( ( X (  Hom  `  C
) X )  X.  ( z (  Hom  `  D ) w ) ) )
14431, 2, 6, 61, 9, 116, 118, 116, 121, 51xpchom2 14283 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( <. X , 
z >. (  Hom  `  ( C  X.c  D ) ) <. X ,  w >. )  =  ( ( X (  Hom  `  C
) X )  X.  ( z (  Hom  `  D ) w ) ) )
145143, 144eleqtrrd 2513 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  <. ( ( Id
`  C ) `  X ) ,  h >.  e.  ( <. X , 
z >. (  Hom  `  ( C  X.c  D ) ) <. X ,  w >. ) )
14632, 51, 120, 27, 133, 134, 135, 137, 141, 145funcco 14068 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( <. X ,  y >. ( 2nd `  F )
<. X ,  w >. ) `
 ( <. (
( Id `  C
) `  X ) ,  h >. ( <. <. X , 
y >. ,  <. X , 
z >. >. (comp `  ( C  X.c  D ) ) <. X ,  w >. )
<. ( ( Id `  C ) `  X
) ,  g >.
) )  =  ( ( ( <. X , 
z >. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  h >. ) ( <.
( ( 1st `  F
) `  <. X , 
y >. ) ,  ( ( 1st `  F
) `  <. X , 
z >. ) >. (comp `  E ) ( ( 1st `  F ) `
 <. X ,  w >. ) ) ( (
<. X ,  y >.
( 2nd `  F
) <. X ,  z
>. ) `  <. (
( Id `  C
) `  X ) ,  g >. )
) )
147132, 146eqtr3d 2470 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( ( Id `  C ) `
 X ) (
<. X ,  y >.
( 2nd `  F
) <. X ,  w >. ) ( h (
<. y ,  z >.
(comp `  D )
w ) g ) )  =  ( ( ( <. X ,  z
>. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  h >. ) ( <.
( ( 1st `  F
) `  <. X , 
y >. ) ,  ( ( 1st `  F
) `  <. X , 
z >. ) >. (comp `  E ) ( ( 1st `  F ) `
 <. X ,  w >. ) ) ( (
<. X ,  y >.
( 2nd `  F
) <. X ,  z
>. ) `  <. (
( Id `  C
) `  X ) ,  g >. )
) )
14843ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  D  e.  Cat )
14953ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  F  e.  ( ( C  X.c  D ) 
Func  E ) )
1506, 9, 26, 148, 117, 118, 121, 124, 125catcocl 13910 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( h (
<. y ,  z >.
(comp `  D )
w ) g )  e.  ( y (  Hom  `  D )
w ) )
1511, 2, 122, 148, 149, 6, 116, 8, 117, 9, 10, 121, 150curf12 14324 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( y ( 2nd `  K
) w ) `  ( h ( <.
y ,  z >.
(comp `  D )
w ) g ) )  =  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  w >. ) ( h (
<. y ,  z >.
(comp `  D )
w ) g ) ) )
1521, 2, 122, 148, 149, 6, 116, 8, 117curf11 14323 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( 1st `  K ) `  y
)  =  ( X ( 1st `  F
) y ) )
153152, 67syl6eq 2484 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( 1st `  K ) `  y
)  =  ( ( 1st `  F ) `
 <. X ,  y
>. ) )
1541, 2, 122, 148, 149, 6, 116, 8, 118curf11 14323 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( 1st `  K ) `  z
)  =  ( X ( 1st `  F
) z ) )
155154, 70syl6eq 2484 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( 1st `  K ) `  z
)  =  ( ( 1st `  F ) `
 <. X ,  z
>. ) )
156153, 155opeq12d 3992 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  <. ( ( 1st `  K ) `  y
) ,  ( ( 1st `  K ) `
 z ) >.  =  <. ( ( 1st `  F ) `  <. X ,  y >. ) ,  ( ( 1st `  F ) `  <. X ,  z >. ) >. )
1571, 2, 122, 148, 149, 6, 116, 8, 121curf11 14323 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( 1st `  K ) `  w
)  =  ( X ( 1st `  F
) w ) )
158 df-ov 6084 . . . . . . . 8  |-  ( X ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. X ,  w >. )
159157, 158syl6eq 2484 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( 1st `  K ) `  w
)  =  ( ( 1st `  F ) `
 <. X ,  w >. ) )
160156, 159oveq12d 6099 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( <. (
( 1st `  K
) `  y ) ,  ( ( 1st `  K ) `  z
) >. (comp `  E
) ( ( 1st `  K ) `  w
) )  =  (
<. ( ( 1st `  F
) `  <. X , 
y >. ) ,  ( ( 1st `  F
) `  <. X , 
z >. ) >. (comp `  E ) ( ( 1st `  F ) `
 <. X ,  w >. ) ) )
1611, 2, 122, 148, 149, 6, 116, 8, 118, 9, 10, 121, 125curf12 14324 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( z ( 2nd `  K
) w ) `  h )  =  ( ( ( Id `  C ) `  X
) ( <. X , 
z >. ( 2nd `  F
) <. X ,  w >. ) h ) )
162 df-ov 6084 . . . . . . 7  |-  ( ( ( Id `  C
) `  X )
( <. X ,  z
>. ( 2nd `  F
) <. X ,  w >. ) h )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  h >. )
163161, 162syl6eq 2484 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( z ( 2nd `  K
) w ) `  h )  =  ( ( <. X ,  z
>. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  h >. ) )
1641, 2, 122, 148, 149, 6, 116, 8, 117, 9, 10, 118, 124curf12 14324 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( y ( 2nd `  K
) z ) `  g )  =  ( ( ( Id `  C ) `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) g ) )
165 df-ov 6084 . . . . . . 7  |-  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g )  =  ( ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) `  <. (
( Id `  C
) `  X ) ,  g >. )
166164, 165syl6eq 2484 . . . . . 6  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( y ( 2nd `  K
) z ) `  g )  =  ( ( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) `  <. (
( Id `  C
) `  X ) ,  g >. )
)
167160, 163, 166oveq123d 6102 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( ( z ( 2nd `  K
) w ) `  h ) ( <.
( ( 1st `  K
) `  y ) ,  ( ( 1st `  K ) `  z
) >. (comp `  E
) ( ( 1st `  K ) `  w
) ) ( ( y ( 2nd `  K
) z ) `  g ) )  =  ( ( ( <. X ,  z >. ( 2nd `  F )
<. X ,  w >. ) `
 <. ( ( Id
`  C ) `  X ) ,  h >. ) ( <. (
( 1st `  F
) `  <. X , 
y >. ) ,  ( ( 1st `  F
) `  <. X , 
z >. ) >. (comp `  E ) ( ( 1st `  F ) `
 <. X ,  w >. ) ) ( (
<. X ,  y >.
( 2nd `  F
) <. X ,  z
>. ) `  <. (
( Id `  C
) `  X ) ,  g >. )
) )
168147, 151, 1673eqtr4d 2478 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  z  e.  B  /\  w  e.  B )  /\  (
g  e.  ( y (  Hom  `  D
) z )  /\  h  e.  ( z
(  Hom  `  D ) w ) ) )  ->  ( ( y ( 2nd `  K
) w ) `  ( h ( <.
y ,  z >.
(comp `  D )
w ) g ) )  =  ( ( ( z ( 2nd `  K ) w ) `
 h ) (
<. ( ( 1st `  K
) `  y ) ,  ( ( 1st `  K ) `  z
) >. (comp `  E
) ( ( 1st `  K ) `  w
) ) ( ( y ( 2nd `  K
) z ) `  g ) ) )
1696, 22, 9, 23, 24, 25, 26, 27, 4, 30, 44, 50, 97, 115, 168isfuncd 14062 . . 3  |-  ( ph  ->  ( 1st `  K
) ( D  Func  E ) ( 2nd `  K
) )
170 df-br 4213 . . 3  |-  ( ( 1st `  K ) ( D  Func  E
) ( 2nd `  K
)  <->  <. ( 1st `  K
) ,  ( 2nd `  K ) >.  e.  ( D  Func  E )
)
171169, 170sylib 189 . 2  |-  ( ph  -> 
<. ( 1st `  K
) ,  ( 2nd `  K ) >.  e.  ( D  Func  E )
)
17221, 171eqeltrd 2510 1  |-  ( ph  ->  K  e.  ( D 
Func  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817   class class class wbr 4212    e. cmpt 4266    X. cxp 4876   Rel wrel 4883    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348   Basecbs 13469    Hom chom 13540  compcco 13541   Catccat 13889   Idccid 13890    Func cfunc 14051    X.c cxpc 14265   curryF ccurf 14307
This theorem is referenced by:  curf2cl  14328  curfcl  14329
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-hom 13553  df-cco 13554  df-cat 13893  df-cid 13894  df-func 14055  df-xpc 14269  df-curf 14311
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