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Theorem curf1cl 14280
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 2404 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
10 eqid 2404 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curf1 14277 . . 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 5701 . . . . . . . 8  |-  ( Base `  D )  e.  _V
136, 12eqeltri 2474 . . . . . . 7  |-  B  e. 
_V
1413mptex 5925 . . . . . 6  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  e.  _V
1513, 13mpt2ex 6384 . . . . . 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 6316 . . . . 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 6317 . . . . 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 3952 . . 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 2439 . 2  |-  ( ph  ->  K  =  <. ( 1st `  K ) ,  ( 2nd `  K
) >. )
22 eqid 2404 . . . 4  |-  ( Base `  E )  =  (
Base `  E )
23 eqid 2404 . . . 4  |-  (  Hom  `  E )  =  (  Hom  `  E )
24 eqid 2404 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
25 eqid 2404 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
26 eqid 2404 . . . 4  |-  (comp `  D )  =  (comp `  D )
27 eqid 2404 . . . 4  |-  (comp `  E )  =  (comp `  E )
28 funcrcl 14015 . . . . . 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 2404 . . . . . . . . . 10  |-  ( C  X.c  D )  =  ( C  X.c  D )
3231, 2, 6xpcbas 14230 . . . . . . . . 9  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
33 relfunc 14014 . . . . . . . . . 10  |-  Rel  (
( C  X.c  D ) 
Func  E )
34 1st2ndbr 6355 . . . . . . . . . 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 14018 . . . . . . . 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 6177 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  ( X ( 1st `  F
) y )  e.  ( Base `  E
) )
41 eqid 2404 . . . . . 6  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  =  ( y  e.  B  |->  ( X ( 1st `  F ) y ) )
4240, 41fmptd 5852 . . . . 5  |-  ( ph  ->  ( y  e.  B  |->  ( X ( 1st `  F ) y ) ) : B --> ( Base `  E ) )
4317feq1d 5539 . . . . 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 2404 . . . . . 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 6065 . . . . . . 7  |-  ( y (  Hom  `  D
) z )  e. 
_V
4746mptex 5925 . . . . . 6  |-  ( g  e.  ( y (  Hom  `  D )
z )  |->  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) )  e.  _V
4845, 47fnmpt2i 6379 . . . . 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 5495 . . . . 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 2404 . . . . . . . . 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 4869 . . . . . . . . . 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 4869 . . . . . . . . . 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 14020 . . . . . . . 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 2404 . . . . . . . . . 10  |-  (  Hom  `  C )  =  (  Hom  `  C )
6231, 32, 61, 9, 51, 56, 59xpchom 14232 . . . . . . . . 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 14278 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  K ) `  y )  =  ( X ( 1st `  F
) y ) )
67 df-ov 6043 . . . . . . . . . . 11  |-  ( X ( 1st `  F
) y )  =  ( ( 1st `  F
) `  <. X , 
y >. )
6866, 67syl6req 2453 . . . . . . . . . 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 14278 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  K ) `  z )  =  ( X ( 1st `  F
) z ) )
70 df-ov 6043 . . . . . . . . . . 11  |-  ( X ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. X , 
z >. )
7169, 70syl6req 2453 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  F ) `  <. X ,  z >.
)  =  ( ( 1st `  K ) `
 z ) )
7268, 71oveq12d 6058 . . . . . . . . 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 5547 . . . . . . . 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 13862 . . . . . . . 8  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( Id `  C ) `  X )  e.  ( X (  Hom  `  C
) X ) )
76 op1stg 6318 . . . . . . . . . 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 6318 . . . . . . . . . 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 6058 . . . . . . . 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 2481 . . . . . . 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 6319 . . . . . . . . . 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 6319 . . . . . . . . . 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 6058 . . . . . . . 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 2481 . . . . . . 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 6177 . . . . . 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 2404 . . . . . 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 5852 . . . . 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 6057 . . . . . . 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 6155 . . . . . . . 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 2456 . . . . . 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 5539 . . . . 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 2404 . . . . . . . . 9  |-  ( Id
`  ( C  X.c  D
) )  =  ( Id `  ( C  X.c  D ) )
10131, 98, 99, 2, 6, 10, 24, 100, 38, 39xpcid 14241 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B )  ->  (
( Id `  ( C  X.c  D ) ) `  <. X ,  y >.
)  =  <. (
( Id `  C
) `  X ) ,  ( ( Id
`  D ) `  y ) >. )
102101fveq2d 5691 . . . . . . 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 6043 . . . . . . 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 2454 . . . . . 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 14022 . . . . . 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 2438 . . . . 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 13862 . . . . . 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 14279 . . . . 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 14278 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  (
( 1st `  K
) `  y )  =  ( X ( 1st `  F ) y ) )
113112, 67syl6eq 2452 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
( 1st `  K
) `  y )  =  ( ( 1st `  F ) `  <. X ,  y >. )
)
114113fveq2d 5691 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  (
( Id `  E
) `  ( ( 1st `  K ) `  y ) )  =  ( ( Id `  E ) `  (
( 1st `  F
) `  <. X , 
y >. ) ) )
115108, 111, 1143eqtr4d 2446 . . . 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 2404 . . . . . . . . . 10  |-  (comp `  C )  =  (comp `  C )
120 eqid 2404 . . . . . . . . . 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 13862 . . . . . . . . . 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 14239 . . . . . . . . 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 13863 . . . . . . . . . 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 3950 . . . . . . . . 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 2436 . . . . . . . 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 5691 . . . . . . 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 6043 . . . . . . 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 2454 . . . . . 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 4869 . . . . . . . 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 4869 . . . . . . . . 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 14238 . . . . . . . 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 2481 . . . . . . 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 4869 . . . . . . . . 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 14238 . . . . . . . 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 2481 . . . . . . 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 14023 . . . . . 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 2438 . . . . 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 13865 . . . . . 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 14279 . . . . 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 14278 . . . . . . . . 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 2452 . . . . . . . 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 14278 . . . . . . . . 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 2452 . . . . . . . 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 3952 . . . . . . 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 14278 . . . . . . . 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 6043 . . . . . . . 8  |-  ( X ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. X ,  w >. )
159157, 158syl6eq 2452 . . . . . . 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 6058 . . . . . 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 14279 . . . . . . 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 6043 . . . . . . 7  |-  ( ( ( Id `  C
) `  X )
( <. X ,  z
>. ( 2nd `  F
) <. X ,  w >. ) h )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  h >. )
163161, 162syl6eq 2452 . . . . . 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 14279 . . . . . . 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 6043 . . . . . . 7  |-  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g )  =  ( ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) `  <. (
( Id `  C
) `  X ) ,  g >. )
166164, 165syl6eq 2452 . . . . . 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 6061 . . . . 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 2446 . . . 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 14017 . . 3  |-  ( ph  ->  ( 1st `  K
) ( D  Func  E ) ( 2nd `  K
) )
170 df-br 4173 . . 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 2478 1  |-  ( ph  ->  K  e.  ( D 
Func  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   Rel wrel 4842    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844   Idccid 13845    Func cfunc 14006    X.c cxpc 14220   curryF ccurf 14262
This theorem is referenced by:  curf2cl  14283  curfcl  14284
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-hom 13508  df-cco 13509  df-cat 13848  df-cid 13849  df-func 14010  df-xpc 14224  df-curf 14266
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