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Theorem curf1cl 14051
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 2316 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
10 eqid 2316 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10curf1 14048 . . 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 5577 . . . . . . . 8  |-  ( Base `  D )  e.  _V
136, 12eqeltri 2386 . . . . . . 7  |-  B  e. 
_V
1413mptex 5787 . . . . . 6  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  e.  _V
1513, 13mpt2ex 6240 . . . . . 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 6172 . . . . 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 6173 . . . . 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 3841 . . 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 2351 . 2  |-  ( ph  ->  K  =  <. ( 1st `  K ) ,  ( 2nd `  K
) >. )
22 eqid 2316 . . . 4  |-  ( Base `  E )  =  (
Base `  E )
23 eqid 2316 . . . 4  |-  (  Hom  `  E )  =  (  Hom  `  E )
24 eqid 2316 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
25 eqid 2316 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
26 eqid 2316 . . . 4  |-  (comp `  D )  =  (comp `  D )
27 eqid 2316 . . . 4  |-  (comp `  E )  =  (comp `  E )
28 funcrcl 13786 . . . . . 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 2316 . . . . . . . . . 10  |-  ( C  X.c  D )  =  ( C  X.c  D )
3231, 2, 6xpcbas 14001 . . . . . . . . 9  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
33 relfunc 13785 . . . . . . . . . 10  |-  Rel  (
( C  X.c  D ) 
Func  E )
34 1st2ndbr 6211 . . . . . . . . . 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 13789 . . . . . . . 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 6034 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  ( X ( 1st `  F
) y )  e.  ( Base `  E
) )
41 eqid 2316 . . . . . 6  |-  ( y  e.  B  |->  ( X ( 1st `  F
) y ) )  =  ( y  e.  B  |->  ( X ( 1st `  F ) y ) )
4240, 41fmptd 5722 . . . . 5  |-  ( ph  ->  ( y  e.  B  |->  ( X ( 1st `  F ) y ) ) : B --> ( Base `  E ) )
4317feq1d 5416 . . . . 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 2316 . . . . . 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 5925 . . . . . . 7  |-  ( y (  Hom  `  D
) z )  e. 
_V
4746mptex 5787 . . . . . 6  |-  ( g  e.  ( y (  Hom  `  D )
z )  |->  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g ) )  e.  _V
4845, 47fnmpt2i 6235 . . . . 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 5372 . . . . 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 2316 . . . . . . . . 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 4758 . . . . . . . . . 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 4758 . . . . . . . . . 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 13791 . . . . . . . 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 2316 . . . . . . . . . 10  |-  (  Hom  `  C )  =  (  Hom  `  C )
6231, 32, 61, 9, 51, 56, 59xpchom 14003 . . . . . . . . 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 14049 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  K ) `  y )  =  ( X ( 1st `  F
) y ) )
67 df-ov 5903 . . . . . . . . . . 11  |-  ( X ( 1st `  F
) y )  =  ( ( 1st `  F
) `  <. X , 
y >. )
6866, 67syl6req 2365 . . . . . . . . . 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 14049 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  K ) `  z )  =  ( X ( 1st `  F
) z ) )
70 df-ov 5903 . . . . . . . . . . 11  |-  ( X ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. X , 
z >. )
7169, 70syl6req 2365 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( 1st `  F ) `  <. X ,  z >.
)  =  ( ( 1st `  K ) `
 z ) )
7268, 71oveq12d 5918 . . . . . . . . 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 5424 . . . . . . . 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 13633 . . . . . . . 8  |-  ( ( ( ph  /\  (
y  e.  B  /\  z  e.  B )
)  /\  g  e.  ( y (  Hom  `  D ) z ) )  ->  ( ( Id `  C ) `  X )  e.  ( X (  Hom  `  C
) X ) )
76 op1stg 6174 . . . . . . . . . 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 6174 . . . . . . . . . 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 5918 . . . . . . . 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 2393 . . . . . . 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 6175 . . . . . . . . . 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 6175 . . . . . . . . . 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 5918 . . . . . . . 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 2393 . . . . . . 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 6034 . . . . . 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 2316 . . . . . 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 5722 . . . . 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 5917 . . . . . . 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 6012 . . . . . . . 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 2368 . . . . . 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 5416 . . . . 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 2316 . . . . . . . . 9  |-  ( Id
`  ( C  X.c  D
) )  =  ( Id `  ( C  X.c  D ) )
10131, 98, 99, 2, 6, 10, 24, 100, 38, 39xpcid 14012 . . . . . . . 8  |-  ( (
ph  /\  y  e.  B )  ->  (
( Id `  ( C  X.c  D ) ) `  <. X ,  y >.
)  =  <. (
( Id `  C
) `  X ) ,  ( ( Id
`  D ) `  y ) >. )
102101fveq2d 5567 . . . . . . 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 5903 . . . . . . 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 2366 . . . . . 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 13793 . . . . . 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 2350 . . . . 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 13633 . . . . . 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 14050 . . . . 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 14049 . . . . . . 7  |-  ( (
ph  /\  y  e.  B )  ->  (
( 1st `  K
) `  y )  =  ( X ( 1st `  F ) y ) )
113112, 67syl6eq 2364 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
( 1st `  K
) `  y )  =  ( ( 1st `  F ) `  <. X ,  y >. )
)
114113fveq2d 5567 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  (
( Id `  E
) `  ( ( 1st `  K ) `  y ) )  =  ( ( Id `  E ) `  (
( 1st `  F
) `  <. X , 
y >. ) ) )
115108, 111, 1143eqtr4d 2358 . . . 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 2316 . . . . . . . . . 10  |-  (comp `  C )  =  (comp `  C )
120 eqid 2316 . . . . . . . . . 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 13633 . . . . . . . . . 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 14010 . . . . . . . . 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 13634 . . . . . . . . . 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 3839 . . . . . . . . 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 2348 . . . . . . . 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 5567 . . . . . . 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 5903 . . . . . . 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 2366 . . . . . 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 4758 . . . . . . . 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 4758 . . . . . . . . 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 14009 . . . . . . . 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 2393 . . . . . . 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 4758 . . . . . . . . 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 14009 . . . . . . . 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 2393 . . . . . . 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 13794 . . . . . 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 2350 . . . . 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 13636 . . . . . 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 14050 . . . . 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 14049 . . . . . . . . 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 2364 . . . . . . . 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 14049 . . . . . . . . 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 2364 . . . . . . . 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 3841 . . . . . . 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 14049 . . . . . . . 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 5903 . . . . . . . 8  |-  ( X ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. X ,  w >. )
159157, 158syl6eq 2364 . . . . . . 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 5918 . . . . . 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 14050 . . . . . . 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 5903 . . . . . . 7  |-  ( ( ( Id `  C
) `  X )
( <. X ,  z
>. ( 2nd `  F
) <. X ,  w >. ) h )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  h >. )
163161, 162syl6eq 2364 . . . . . 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 14050 . . . . . . 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 5903 . . . . . . 7  |-  ( ( ( Id `  C
) `  X )
( <. X ,  y
>. ( 2nd `  F
) <. X ,  z
>. ) g )  =  ( ( <. X , 
y >. ( 2nd `  F
) <. X ,  z
>. ) `  <. (
( Id `  C
) `  X ) ,  g >. )
166164, 165syl6eq 2364 . . . . . 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 5921 . . . . 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 2358 . . . 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 13788 . . 3  |-  ( ph  ->  ( 1st `  K
) ( D  Func  E ) ( 2nd `  K
) )
170 df-br 4061 . . 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 2390 1  |-  ( ph  ->  K  e.  ( D 
Func  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   _Vcvv 2822   <.cop 3677   class class class wbr 4060    e. cmpt 4114    X. cxp 4724   Rel wrel 4731    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   1stc1st 6162   2ndc2nd 6163   Basecbs 13195    Hom chom 13266  compcco 13267   Catccat 13615   Idccid 13616    Func cfunc 13777    X.c cxpc 13991   curryF ccurf 14033
This theorem is referenced by:  curf2cl  14054  curfcl  14055
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-hom 13279  df-cco 13280  df-cat 13619  df-cid 13620  df-func 13781  df-xpc 13995  df-curf 14037
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