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Theorem curf1cl 14002
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 2283 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
10 eqid 2283 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curf1 13999 . . 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 5539 . . . . . . . 8  |-  ( Base `  D )  e.  _V
136, 12eqeltri 2353 . . . . . . 7  |-  B  e. 
_V
1413mptex 5746 . . . . . 6  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  e.  _V
1513, 13mpt2ex 6198 . . . . . 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 6130 . . . . 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 15 . . . 4  |-  ( ph  ->  ( 1st `  K
)  =  ( y  e.  B  |->  ( X ( 1st `  F
) y ) ) )
1814, 15op2ndd 6131 . . . . 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 15 . . . 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 3804 . . 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 2318 . 2  |-  ( ph  ->  K  =  <. ( 1st `  K ) ,  ( 2nd `  K
) >. )
22 eqid 2283 . . . 4  |-  ( Base `  E )  =  (
Base `  E )
23 eqid 2283 . . . 4  |-  (  Hom  `  E )  =  (  Hom  `  E )
24 eqid 2283 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
25 eqid 2283 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
26 eqid 2283 . . . 4  |-  (comp `  D )  =  (comp `  D )
27 eqid 2283 . . . 4  |-  (comp `  E )  =  (comp `  E )
28 funcrcl 13737 . . . . . 6  |-  ( F  e.  ( ( C  X.c  D )  Func  E
)  ->  ( ( C  X.c  D )  e.  Cat  /\  E  e.  Cat )
)
295, 28syl 15 . . . . 5  |-  ( ph  ->  ( ( C  X.c  D
)  e.  Cat  /\  E  e.  Cat )
)
3029simprd 449 . . . 4  |-  ( ph  ->  E  e.  Cat )
31 eqid 2283 . . . . . . . . . 10  |-  ( C  X.c  D )  =  ( C  X.c  D )
3231, 2, 6xpcbas 13952 . . . . . . . . 9  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
33 relfunc 13736 . . . . . . . . . 10  |-  Rel  (
( C  X.c  D ) 
Func  E )
34 1st2ndbr 6169 . . . . . . . . . 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 644 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
3632, 22, 35funcf1 13740 . . . . . . . 8  |-  ( ph  ->  ( 1st `  F
) : ( A  X.  B ) --> (
Base `  E )
)
3736adantr 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  ( 1st `  F ) : ( A  X.  B
) --> ( Base `  E
) )
387adantr 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  X  e.  A )
39 simpr 447 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  y  e.  B )
4037, 38, 39fovrnd 5992 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  ( X ( 1st `  F
) y )  e.  ( Base `  E
) )
41 eqid 2283 . . . . . 6  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  =  ( y  e.  B  |->  ( X ( 1st `  F ) y ) )
4240, 41fmptd 5684 . . . . 5  |-  ( ph  ->  ( y  e.  B  |->  ( X ( 1st `  F ) y ) ) : B --> ( Base `  E ) )
4317feq1d 5379 . . . . 5  |-  ( ph  ->  ( ( 1st `  K
) : B --> ( Base `  E )  <->  ( y  e.  B  |->  ( X ( 1st `  F
) y ) ) : B --> ( Base `  E ) ) )
4442, 43mpbird 223 . . . 4  |-  ( ph  ->  ( 1st `  K
) : B --> ( Base `  E ) )
45 eqid 2283 . . . . . 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 5883 . . . . . . 7  |-  ( y (  Hom  `  D
) z )  e. 
_V
4746mptex 5746 . . . . . 6  |-  ( g  e.  ( y (  Hom  `  D )
z )  |->  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) )  e.  _V
4845, 47fnmpt2i 6193 . . . . 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 5335 . . . . 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 224 . . . 4  |-  ( ph  ->  ( 2nd `  K
)  Fn  ( B  X.  B ) )
51 eqid 2283 . . . . . . . . 9  |-  (  Hom  `  ( C  X.c  D ) )  =  (  Hom  `  ( C  X.c  D ) )
5235ad2antrr 706 . . . . . . . . 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 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  X  e.  A )
54 simplrl 736 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  y  e.  B )
55 opelxpi 4721 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  y  e.  B )  -> 
<. X ,  y >.  e.  ( A  X.  B
) )
5653, 54, 55syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  <. X , 
y >.  e.  ( A  X.  B ) )
57 simplrr 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  z  e.  B )
58 opelxpi 4721 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  z  e.  B )  -> 
<. X ,  z >.  e.  ( A  X.  B
) )
5953, 57, 58syl2anc 642 . . . . . . . . 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 13742 . . . . . . . 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 2283 . . . . . . . . . 10  |-  (  Hom  `  C )  =  (  Hom  `  C )
6231, 32, 61, 9, 51, 56, 59xpchom 13954 . . . . . . . . 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 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  C  e.  Cat )
644ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  D  e.  Cat )
655ad2antrr 706 . . . . . . . . . . . 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 14000 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  K ) `  y )  =  ( X ( 1st `  F
) y ) )
67 df-ov 5861 . . . . . . . . . . 11  |-  ( X ( 1st `  F
) y )  =  ( ( 1st `  F
) `  <. X , 
y >. )
6866, 67syl6req 2332 . . . . . . . . . 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 14000 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  K ) `  z )  =  ( X ( 1st `  F
) z ) )
70 df-ov 5861 . . . . . . . . . . 11  |-  ( X ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. X , 
z >. )
7169, 70syl6req 2332 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  F ) `  <. X ,  z >.
)  =  ( ( 1st `  K ) `
 z ) )
7268, 71oveq12d 5876 . . . . . . . . 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 5386 . . . . . . . 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 201 . . . . . . 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 13584 . . . . . . . 8  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( Id `  C ) `  X )  e.  ( X (  Hom  `  C
) X ) )
76 op1stg 6132 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  y  e.  B )  ->  ( 1st `  <. X ,  y >. )  =  X )
7753, 54, 76syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( 1st ` 
<. X ,  y >.
)  =  X )
78 op1stg 6132 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  z  e.  B )  ->  ( 1st `  <. X ,  z >. )  =  X )
7953, 57, 78syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( 1st ` 
<. X ,  z >.
)  =  X )
8077, 79oveq12d 5876 . . . . . . . 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 2360 . . . . . . 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 447 . . . . . . . 8  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  g  e.  ( y (  Hom  `  D ) z ) )
83 op2ndg 6133 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  y  e.  B )  ->  ( 2nd `  <. X ,  y >. )  =  y )
8453, 54, 83syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( 2nd ` 
<. X ,  y >.
)  =  y )
85 op2ndg 6133 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  z  e.  B )  ->  ( 2nd `  <. X ,  z >. )  =  z )
8653, 57, 85syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( 2nd ` 
<. X ,  z >.
)  =  z )
8784, 86oveq12d 5876 . . . . . . . 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 2360 . . . . . . 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 5992 . . . . . 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 2283 . . . . . 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 5684 . . . . 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 5875 . . . . . . 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 5970 . . . . . . . 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 1266 . . . . . . 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 2335 . . . . . 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 5379 . . . . 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 223 . . . 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 451 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  B )  ->  C  e.  Cat )
994adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  Cat )
100 eqid 2283 . . . . . . . . 9  |-  ( Id
`  ( C  X.c  D
) )  =  ( Id `  ( C  X.c  D ) )
10131, 98, 99, 2, 6, 10, 24, 100, 38, 39xpcid 13963 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B )  ->  (
( Id `  ( C  X.c  D ) ) `  <. X ,  y >.
)  =  <. (
( Id `  C
) `  X ) ,  ( ( Id
`  D ) `  y ) >. )
102101fveq2d 5529 . . . . . . 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 5861 . . . . . . 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 2333 . . . . . 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 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  ( 1st `  F ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  F ) )
1067, 55sylan 457 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  <. X , 
y >.  e.  ( A  X.  B ) )
10732, 100, 25, 105, 106funcid 13744 . . . . . 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 2317 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  (
( ( Id `  C ) `  X
) ( <. X , 
y >. ( 2nd `  F
) <. X ,  y
>. ) ( ( Id
`  D ) `  y ) )  =  ( ( Id `  E ) `  (
( 1st `  F
) `  <. X , 
y >. ) ) )
1095adantr 451 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  F  e.  ( ( C  X.c  D
)  Func  E )
)
1106, 9, 24, 99, 39catidcl 13584 . . . . . 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 14001 . . . . 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 14000 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  (
( 1st `  K
) `  y )  =  ( X ( 1st `  F ) y ) )
113112, 67syl6eq 2331 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
( 1st `  K
) `  y )  =  ( ( 1st `  F ) `  <. X ,  y >. )
)
114113fveq2d 5529 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  (
( Id `  E
) `  ( ( 1st `  K ) `  y ) )  =  ( ( Id `  E ) `  (
( 1st `  F
) `  <. X , 
y >. ) ) )
115108, 111, 1143eqtr4d 2325 . . . 4  |-  ( (
ph  /\  y  e.  B )  ->  (
( y ( 2nd `  K ) y ) `
 ( ( Id
`  D ) `  y ) )  =  ( ( Id `  E ) `  (
( 1st `  K
) `  y )
) )
11673ad2ant1 976 . . . . . . . . . 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 988 . . . . . . . . . 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 989 . . . . . . . . . 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 2283 . . . . . . . . . 10  |-  (comp `  C )  =  (comp `  C )
120 eqid 2283 . . . . . . . . . 10  |-  (comp `  ( C  X.c  D )
)  =  (comp `  ( C  X.c  D )
)
121 simp23 990 . . . . . . . . . 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 976 . . . . . . . . . . 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 13584 . . . . . . . . . 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 983 . . . . . . . . . 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 984 . . . . . . . . . 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 13961 . . . . . . . . 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 13585 . . . . . . . . . 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 3802 . . . . . . . . 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 2315 . . . . . . . 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 5529 . . . . . . 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 5861 . . . . . . 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 2333 . . . . . 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 976 . . . . . . 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 642 . . . . . . 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 642 . . . . . . 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 4721 . . . . . . . 8  |-  ( ( X  e.  A  /\  w  e.  B )  -> 
<. X ,  w >.  e.  ( A  X.  B
) )
137116, 121, 136syl2anc 642 . . . . . . 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 4721 . . . . . . . . 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 642 . . . . . . . 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 13960 . . . . . . . 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 2360 . . . . . . 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 4721 . . . . . . . . 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 642 . . . . . . . 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 13960 . . . . . . . 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 2360 . . . . . . 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 13745 . . . . . 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 2317 . . . . 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 976 . . . . . 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 976 . . . . . 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 13587 . . . . . 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 14001 . . . . 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 14000 . . . . . . . . 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 2331 . . . . . . . 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 14000 . . . . . . . . 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 2331 . . . . . . . 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 3804 . . . . . . 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 14000 . . . . . . . 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 5861 . . . . . . . 8  |-  ( X ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. X ,  w >. )
159157, 158syl6eq 2331 . . . . . . 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 5876 . . . . . 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 14001 . . . . . . 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 5861 . . . . . . 7  |-  ( ( ( Id `  C
) `  X )
( <. X ,  z
>. ( 2nd `  F
) <. X ,  w >. ) h )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  h >. )
163161, 162syl6eq 2331 . . . . . 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 14001 . . . . . . 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 5861 . . . . . . 7  |-  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g )  =  ( ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) `  <. (
( Id `  C
) `  X ) ,  g >. )
166164, 165syl6eq 2331 . . . . . 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 5879 . . . . 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 2325 . . . 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 13739 . . 3  |-  ( ph  ->  ( 1st `  K
) ( D  Func  E ) ( 2nd `  K
) )
170 df-br 4024 . . 3  |-  ( ( 1st `  K ) ( D  Func  E
) ( 2nd `  K
)  <->  <. ( 1st `  K
) ,  ( 2nd `  K ) >.  e.  ( D  Func  E )
)
171169, 170sylib 188 . 2  |-  ( ph  -> 
<. ( 1st `  K
) ,  ( 2nd `  K ) >.  e.  ( D  Func  E )
)
17221, 171eqeltrd 2357 1  |-  ( ph  ->  K  e.  ( D 
Func  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   Rel wrel 4694    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567    Func cfunc 13728    X.c cxpc 13942   curryF ccurf 13984
This theorem is referenced by:  curf2cl  14005  curfcl  14006
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  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-hom 13232  df-cco 13233  df-cat 13570  df-cid 13571  df-func 13732  df-xpc 13946  df-curf 13988
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