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Theorem curf2cl 14005
Description: The curry functor at a morphism is a natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)
Hypotheses
Ref Expression
curf2.g  |-  G  =  ( <. C ,  D >. curryF  F
)
curf2.a  |-  A  =  ( Base `  C
)
curf2.c  |-  ( ph  ->  C  e.  Cat )
curf2.d  |-  ( ph  ->  D  e.  Cat )
curf2.f  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
curf2.b  |-  B  =  ( Base `  D
)
curf2.h  |-  H  =  (  Hom  `  C
)
curf2.i  |-  I  =  ( Id `  D
)
curf2.x  |-  ( ph  ->  X  e.  A )
curf2.y  |-  ( ph  ->  Y  e.  A )
curf2.k  |-  ( ph  ->  K  e.  ( X H Y ) )
curf2.l  |-  L  =  ( ( X ( 2nd `  G ) Y ) `  K
)
curf2.n  |-  N  =  ( D Nat  E )
Assertion
Ref Expression
curf2cl  |-  ( ph  ->  L  e.  ( ( ( 1st `  G
) `  X ) N ( ( 1st `  G ) `  Y
) ) )

Proof of Theorem curf2cl
Dummy variables  z  w  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 curf2.g . . . 4  |-  G  =  ( <. C ,  D >. curryF  F
)
2 curf2.a . . . 4  |-  A  =  ( Base `  C
)
3 curf2.c . . . 4  |-  ( ph  ->  C  e.  Cat )
4 curf2.d . . . 4  |-  ( ph  ->  D  e.  Cat )
5 curf2.f . . . 4  |-  ( ph  ->  F  e.  ( ( C  X.c  D )  Func  E
) )
6 curf2.b . . . 4  |-  B  =  ( Base `  D
)
7 curf2.h . . . 4  |-  H  =  (  Hom  `  C
)
8 curf2.i . . . 4  |-  I  =  ( Id `  D
)
9 curf2.x . . . 4  |-  ( ph  ->  X  e.  A )
10 curf2.y . . . 4  |-  ( ph  ->  Y  e.  A )
11 curf2.k . . . 4  |-  ( ph  ->  K  e.  ( X H Y ) )
12 curf2.l . . . 4  |-  L  =  ( ( X ( 2nd `  G ) Y ) `  K
)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12curf2 14003 . . 3  |-  ( ph  ->  L  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
14 eqid 2283 . . . . . . . . . 10  |-  ( C  X.c  D )  =  ( C  X.c  D )
1514, 2, 6xpcbas 13952 . . . . . . . . 9  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
16 eqid 2283 . . . . . . . . 9  |-  (  Hom  `  ( C  X.c  D ) )  =  (  Hom  `  ( C  X.c  D ) )
17 eqid 2283 . . . . . . . . 9  |-  (  Hom  `  E )  =  (  Hom  `  E )
18 relfunc 13736 . . . . . . . . . . 11  |-  Rel  (
( C  X.c  D ) 
Func  E )
19 1st2ndbr 6169 . . . . . . . . . . 11  |-  ( ( 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
) )
2018, 5, 19sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
2120adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  ( 1st `  F ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  F ) )
229adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  B )  ->  X  e.  A )
23 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  B )  ->  z  e.  B )
24 opelxpi 4721 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  z  e.  B )  -> 
<. X ,  z >.  e.  ( A  X.  B
) )
2522, 23, 24syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  <. X , 
z >.  e.  ( A  X.  B ) )
2610adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  B )  ->  Y  e.  A )
27 opelxpi 4721 . . . . . . . . . 10  |-  ( ( Y  e.  A  /\  z  e.  B )  -> 
<. Y ,  z >.  e.  ( A  X.  B
) )
2826, 23, 27syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  <. Y , 
z >.  e.  ( A  X.  B ) )
2915, 16, 17, 21, 25, 28funcf2 13742 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  ( <. X ,  z >.
( 2nd `  F
) <. Y ,  z
>. ) : ( <. X ,  z >. (  Hom  `  ( C  X.c  D ) ) <. Y ,  z >. ) --> ( ( ( 1st `  F ) `  <. X ,  z >. )
(  Hom  `  E ) ( ( 1st `  F
) `  <. Y , 
z >. ) ) )
30 eqid 2283 . . . . . . . . . 10  |-  (  Hom  `  D )  =  (  Hom  `  D )
3114, 2, 6, 7, 30, 22, 23, 26, 23, 16xpchom2 13960 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  ( <. X ,  z >.
(  Hom  `  ( C  X.c  D ) ) <. Y ,  z >. )  =  ( ( X H Y )  X.  ( z (  Hom  `  D ) z ) ) )
3231feq2d 5380 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) : ( <. X ,  z >. (  Hom  `  ( C  X.c  D ) ) <. Y ,  z >. ) --> ( ( ( 1st `  F ) `  <. X ,  z >. )
(  Hom  `  E ) ( ( 1st `  F
) `  <. Y , 
z >. ) )  <->  ( <. X ,  z >. ( 2nd `  F ) <. Y ,  z >. ) : ( ( X H Y )  X.  ( z (  Hom  `  D ) z ) ) --> ( ( ( 1st `  F ) `
 <. X ,  z
>. ) (  Hom  `  E
) ( ( 1st `  F ) `  <. Y ,  z >. )
) ) )
3329, 32mpbid 201 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  ( <. X ,  z >.
( 2nd `  F
) <. Y ,  z
>. ) : ( ( X H Y )  X.  ( z (  Hom  `  D )
z ) ) --> ( ( ( 1st `  F
) `  <. X , 
z >. ) (  Hom  `  E ) ( ( 1st `  F ) `
 <. Y ,  z
>. ) ) )
3411adantr 451 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  K  e.  ( X H Y ) )
354adantr 451 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  D  e.  Cat )
366, 30, 8, 35, 23catidcl 13584 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
I `  z )  e.  ( z (  Hom  `  D ) z ) )
3733, 34, 36fovrnd 5992 . . . . . 6  |-  ( (
ph  /\  z  e.  B )  ->  ( K ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) )  e.  ( ( ( 1st `  F ) `  <. X ,  z >. )
(  Hom  `  E ) ( ( 1st `  F
) `  <. Y , 
z >. ) ) )
383adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  C  e.  Cat )
395adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  F  e.  ( ( C  X.c  D
)  Func  E )
)
40 eqid 2283 . . . . . . . . 9  |-  ( ( 1st `  G ) `
 X )  =  ( ( 1st `  G
) `  X )
411, 2, 38, 35, 39, 6, 22, 40, 23curf11 14000 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  z )  =  ( X ( 1st `  F ) z ) )
42 df-ov 5861 . . . . . . . 8  |-  ( X ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. X , 
z >. )
4341, 42syl6eq 2331 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  z )  =  ( ( 1st `  F ) `  <. X ,  z >. )
)
44 eqid 2283 . . . . . . . . 9  |-  ( ( 1st `  G ) `
 Y )  =  ( ( 1st `  G
) `  Y )
451, 2, 38, 35, 39, 6, 26, 44, 23curf11 14000 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  z )  =  ( Y ( 1st `  F ) z ) )
46 df-ov 5861 . . . . . . . 8  |-  ( Y ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. Y , 
z >. )
4745, 46syl6eq 2331 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  z )  =  ( ( 1st `  F ) `  <. Y ,  z >. )
)
4843, 47oveq12d 5876 . . . . . 6  |-  ( (
ph  /\  z  e.  B )  ->  (
( ( 1st `  (
( 1st `  G
) `  X )
) `  z )
(  Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
)  =  ( ( ( 1st `  F
) `  <. X , 
z >. ) (  Hom  `  E ) ( ( 1st `  F ) `
 <. Y ,  z
>. ) ) )
4937, 48eleqtrrd 2360 . . . . 5  |-  ( (
ph  /\  z  e.  B )  ->  ( K ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) )  e.  ( ( ( 1st `  ( ( 1st `  G
) `  X )
) `  z )
(  Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
) )
5049ralrimiva 2626 . . . 4  |-  ( ph  ->  A. z  e.  B  ( K ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) )  e.  ( ( ( 1st `  ( ( 1st `  G
) `  X )
) `  z )
(  Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
) )
51 fvex 5539 . . . . . 6  |-  ( Base `  D )  e.  _V
526, 51eqeltri 2353 . . . . 5  |-  B  e. 
_V
53 mptelixpg 6853 . . . . 5  |-  ( B  e.  _V  ->  (
( z  e.  B  |->  ( K ( <. X ,  z >. ( 2nd `  F )
<. Y ,  z >.
) ( I `  z ) ) )  e.  X_ z  e.  B  ( ( ( 1st `  ( ( 1st `  G
) `  X )
) `  z )
(  Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
)  <->  A. z  e.  B  ( K ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) )  e.  ( ( ( 1st `  ( ( 1st `  G
) `  X )
) `  z )
(  Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
) ) )
5452, 53ax-mp 8 . . . 4  |-  ( ( z  e.  B  |->  ( K ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) )  e.  X_ z  e.  B  ( ( ( 1st `  ( ( 1st `  G
) `  X )
) `  z )
(  Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
)  <->  A. z  e.  B  ( K ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) )  e.  ( ( ( 1st `  ( ( 1st `  G
) `  X )
) `  z )
(  Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
) )
5550, 54sylibr 203 . . 3  |-  ( ph  ->  ( z  e.  B  |->  ( K ( <. X ,  z >. ( 2nd `  F )
<. Y ,  z >.
) ( I `  z ) ) )  e.  X_ z  e.  B  ( ( ( 1st `  ( ( 1st `  G
) `  X )
) `  z )
(  Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
) )
5613, 55eqeltrd 2357 . 2  |-  ( ph  ->  L  e.  X_ z  e.  B  ( (
( 1st `  (
( 1st `  G
) `  X )
) `  z )
(  Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
) )
57 eqid 2283 . . . . . . . . . 10  |-  ( Id
`  C )  =  ( Id `  C
)
583adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  C  e.  Cat )
599adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  X  e.  A )
60 eqid 2283 . . . . . . . . . 10  |-  (comp `  C )  =  (comp `  C )
6110adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  Y  e.  A )
6211adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  K  e.  ( X H Y ) )
632, 7, 57, 58, 59, 60, 61, 62catrid 13586 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( K ( <. X ,  X >. (comp `  C
) Y ) ( ( Id `  C
) `  X )
)  =  K )
642, 7, 57, 58, 59, 60, 61, 62catlid 13585 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( ( Id `  C ) `  Y
) ( <. X ,  Y >. (comp `  C
) Y ) K )  =  K )
6563, 64eqtr4d 2318 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( K ( <. X ,  X >. (comp `  C
) Y ) ( ( Id `  C
) `  X )
)  =  ( ( ( Id `  C
) `  Y )
( <. X ,  Y >. (comp `  C ) Y ) K ) )
664adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  D  e.  Cat )
67 simpr1 961 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  z  e.  B )
68 eqid 2283 . . . . . . . . . 10  |-  (comp `  D )  =  (comp `  D )
69 simpr2 962 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  w  e.  B )
70 simpr3 963 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  f  e.  ( z (  Hom  `  D ) w ) )
716, 30, 8, 66, 67, 68, 69, 70catlid 13585 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( I `  w
) ( <. z ,  w >. (comp `  D
) w ) f )  =  f )
726, 30, 8, 66, 67, 68, 69, 70catrid 13586 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
f ( <. z ,  z >. (comp `  D ) w ) ( I `  z
) )  =  f )
7371, 72eqtr4d 2318 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( I `  w
) ( <. z ,  w >. (comp `  D
) w ) f )  =  ( f ( <. z ,  z
>. (comp `  D )
w ) ( I `
 z ) ) )
7465, 73opeq12d 3804 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. ( K ( <. X ,  X >. (comp `  C
) Y ) ( ( Id `  C
) `  X )
) ,  ( ( I `  w ) ( <. z ,  w >. (comp `  D )
w ) f )
>.  =  <. ( ( ( Id `  C
) `  Y )
( <. X ,  Y >. (comp `  C ) Y ) K ) ,  ( f (
<. z ,  z >.
(comp `  D )
w ) ( I `
 z ) )
>. )
75 eqid 2283 . . . . . . . 8  |-  (comp `  ( C  X.c  D )
)  =  (comp `  ( C  X.c  D )
)
762, 7, 57, 58, 59catidcl 13584 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( Id `  C
) `  X )  e.  ( X H X ) )
776, 30, 8, 66, 69catidcl 13584 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
I `  w )  e.  ( w (  Hom  `  D ) w ) )
7814, 2, 6, 7, 30, 59, 67, 59, 69, 60, 68, 75, 61, 69, 76, 70, 62, 77xpcco2 13961 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( <. K ,  ( I `
 w ) >.
( <. <. X ,  z
>. ,  <. X ,  w >. >. (comp `  ( C  X.c  D ) ) <. Y ,  w >. )
<. ( ( Id `  C ) `  X
) ,  f >.
)  =  <. ( K ( <. X ,  X >. (comp `  C
) Y ) ( ( Id `  C
) `  X )
) ,  ( ( I `  w ) ( <. z ,  w >. (comp `  D )
w ) f )
>. )
79363ad2antr1 1120 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
I `  z )  e.  ( z (  Hom  `  D ) z ) )
802, 7, 57, 58, 61catidcl 13584 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( Id `  C
) `  Y )  e.  ( Y H Y ) )
8114, 2, 6, 7, 30, 59, 67, 61, 67, 60, 68, 75, 61, 69, 62, 79, 80, 70xpcco2 13961 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( <. ( ( Id `  C ) `  Y
) ,  f >.
( <. <. X ,  z
>. ,  <. Y , 
z >. >. (comp `  ( C  X.c  D ) ) <. Y ,  w >. )
<. K ,  ( I `
 z ) >.
)  =  <. (
( ( Id `  C ) `  Y
) ( <. X ,  Y >. (comp `  C
) Y ) K ) ,  ( f ( <. z ,  z
>. (comp `  D )
w ) ( I `
 z ) )
>. )
8274, 78, 813eqtr4d 2325 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( <. K ,  ( I `
 w ) >.
( <. <. X ,  z
>. ,  <. X ,  w >. >. (comp `  ( C  X.c  D ) ) <. Y ,  w >. )
<. ( ( Id `  C ) `  X
) ,  f >.
)  =  ( <.
( ( Id `  C ) `  Y
) ,  f >.
( <. <. X ,  z
>. ,  <. Y , 
z >. >. (comp `  ( C  X.c  D ) ) <. Y ,  w >. )
<. K ,  ( I `
 z ) >.
) )
8382fveq2d 5529 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( <. X ,  z
>. ( 2nd `  F
) <. Y ,  w >. ) `  ( <. K ,  ( I `  w ) >. ( <. <. X ,  z
>. ,  <. X ,  w >. >. (comp `  ( C  X.c  D ) ) <. Y ,  w >. )
<. ( ( Id `  C ) `  X
) ,  f >.
) )  =  ( ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  w >. ) `  ( <.
( ( Id `  C ) `  Y
) ,  f >.
( <. <. X ,  z
>. ,  <. Y , 
z >. >. (comp `  ( C  X.c  D ) ) <. Y ,  w >. )
<. K ,  ( I `
 z ) >.
) ) )
84 eqid 2283 . . . . . 6  |-  (comp `  E )  =  (comp `  E )
8520adantr 451 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( 1st `  F ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  F ) )
86253ad2antr1 1120 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. X , 
z >.  e.  ( A  X.  B ) )
87 opelxpi 4721 . . . . . . 7  |-  ( ( X  e.  A  /\  w  e.  B )  -> 
<. X ,  w >.  e.  ( A  X.  B
) )
8859, 69, 87syl2anc 642 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. X ,  w >.  e.  ( A  X.  B ) )
89 opelxpi 4721 . . . . . . 7  |-  ( ( Y  e.  A  /\  w  e.  B )  -> 
<. Y ,  w >.  e.  ( A  X.  B
) )
9061, 69, 89syl2anc 642 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. Y ,  w >.  e.  ( A  X.  B ) )
91 opelxpi 4721 . . . . . . . 8  |-  ( ( ( ( Id `  C ) `  X
)  e.  ( X H X )  /\  f  e.  ( z
(  Hom  `  D ) w ) )  ->  <. ( ( Id `  C ) `  X
) ,  f >.  e.  ( ( X H X )  X.  (
z (  Hom  `  D
) w ) ) )
9276, 70, 91syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. (
( Id `  C
) `  X ) ,  f >.  e.  ( ( X H X )  X.  ( z (  Hom  `  D
) w ) ) )
9314, 2, 6, 7, 30, 59, 67, 59, 69, 16xpchom2 13960 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( <. X ,  z >.
(  Hom  `  ( C  X.c  D ) ) <. X ,  w >. )  =  ( ( X H X )  X.  ( z (  Hom  `  D ) w ) ) )
9492, 93eleqtrrd 2360 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. (
( Id `  C
) `  X ) ,  f >.  e.  (
<. X ,  z >.
(  Hom  `  ( C  X.c  D ) ) <. X ,  w >. ) )
95 opelxpi 4721 . . . . . . . 8  |-  ( ( K  e.  ( X H Y )  /\  ( I `  w
)  e.  ( w (  Hom  `  D
) w ) )  ->  <. K ,  ( I `  w )
>.  e.  ( ( X H Y )  X.  ( w (  Hom  `  D ) w ) ) )
9662, 77, 95syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. K , 
( I `  w
) >.  e.  ( ( X H Y )  X.  ( w (  Hom  `  D )
w ) ) )
9714, 2, 6, 7, 30, 59, 69, 61, 69, 16xpchom2 13960 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( <. X ,  w >. (  Hom  `  ( C  X.c  D ) ) <. Y ,  w >. )  =  ( ( X H Y )  X.  ( w (  Hom  `  D ) w ) ) )
9896, 97eleqtrrd 2360 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. K , 
( I `  w
) >.  e.  ( <. X ,  w >. (  Hom  `  ( C  X.c  D ) ) <. Y ,  w >. ) )
9915, 16, 75, 84, 85, 86, 88, 90, 94, 98funcco 13745 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( <. X ,  z
>. ( 2nd `  F
) <. Y ,  w >. ) `  ( <. K ,  ( I `  w ) >. ( <. <. X ,  z
>. ,  <. X ,  w >. >. (comp `  ( C  X.c  D ) ) <. Y ,  w >. )
<. ( ( Id `  C ) `  X
) ,  f >.
) )  =  ( ( ( <. X ,  w >. ( 2nd `  F
) <. Y ,  w >. ) `  <. K , 
( I `  w
) >. ) ( <.
( ( 1st `  F
) `  <. X , 
z >. ) ,  ( ( 1st `  F
) `  <. X ,  w >. ) >. (comp `  E ) ( ( 1st `  F ) `
 <. Y ,  w >. ) ) ( (
<. X ,  z >.
( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  f >. )
) )
100283ad2antr1 1120 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. Y , 
z >.  e.  ( A  X.  B ) )
101 opelxpi 4721 . . . . . . . 8  |-  ( ( K  e.  ( X H Y )  /\  ( I `  z
)  e.  ( z (  Hom  `  D
) z ) )  ->  <. K ,  ( I `  z )
>.  e.  ( ( X H Y )  X.  ( z (  Hom  `  D ) z ) ) )
10262, 79, 101syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. K , 
( I `  z
) >.  e.  ( ( X H Y )  X.  ( z (  Hom  `  D )
z ) ) )
10314, 2, 6, 7, 30, 59, 67, 61, 67, 16xpchom2 13960 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( <. X ,  z >.
(  Hom  `  ( C  X.c  D ) ) <. Y ,  z >. )  =  ( ( X H Y )  X.  ( z (  Hom  `  D ) z ) ) )
104102, 103eleqtrrd 2360 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. K , 
( I `  z
) >.  e.  ( <. X ,  z >. (  Hom  `  ( C  X.c  D ) ) <. Y ,  z >. ) )
105 opelxpi 4721 . . . . . . . 8  |-  ( ( ( ( Id `  C ) `  Y
)  e.  ( Y H Y )  /\  f  e.  ( z
(  Hom  `  D ) w ) )  ->  <. ( ( Id `  C ) `  Y
) ,  f >.  e.  ( ( Y H Y )  X.  (
z (  Hom  `  D
) w ) ) )
10680, 70, 105syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. (
( Id `  C
) `  Y ) ,  f >.  e.  ( ( Y H Y )  X.  ( z (  Hom  `  D
) w ) ) )
10714, 2, 6, 7, 30, 61, 67, 61, 69, 16xpchom2 13960 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( <. Y ,  z >.
(  Hom  `  ( C  X.c  D ) ) <. Y ,  w >. )  =  ( ( Y H Y )  X.  ( z (  Hom  `  D ) w ) ) )
108106, 107eleqtrrd 2360 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. (
( Id `  C
) `  Y ) ,  f >.  e.  (
<. Y ,  z >.
(  Hom  `  ( C  X.c  D ) ) <. Y ,  w >. ) )
10915, 16, 75, 84, 85, 86, 100, 90, 104, 108funcco 13745 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( <. X ,  z
>. ( 2nd `  F
) <. Y ,  w >. ) `  ( <.
( ( Id `  C ) `  Y
) ,  f >.
( <. <. X ,  z
>. ,  <. Y , 
z >. >. (comp `  ( C  X.c  D ) ) <. Y ,  w >. )
<. K ,  ( I `
 z ) >.
) )  =  ( ( ( <. Y , 
z >. ( 2nd `  F
) <. Y ,  w >. ) `  <. (
( Id `  C
) `  Y ) ,  f >. )
( <. ( ( 1st `  F ) `  <. X ,  z >. ) ,  ( ( 1st `  F ) `  <. Y ,  z >. ) >. (comp `  E )
( ( 1st `  F
) `  <. Y ,  w >. ) ) ( ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) `  <. K , 
( I `  z
) >. ) ) )
11083, 99, 1093eqtr3d 2323 . . . 4  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( ( <. X ,  w >. ( 2nd `  F
) <. Y ,  w >. ) `  <. K , 
( I `  w
) >. ) ( <.
( ( 1st `  F
) `  <. X , 
z >. ) ,  ( ( 1st `  F
) `  <. X ,  w >. ) >. (comp `  E ) ( ( 1st `  F ) `
 <. Y ,  w >. ) ) ( (
<. X ,  z >.
( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  f >. )
)  =  ( ( ( <. Y ,  z
>. ( 2nd `  F
) <. Y ,  w >. ) `  <. (
( Id `  C
) `  Y ) ,  f >. )
( <. ( ( 1st `  F ) `  <. X ,  z >. ) ,  ( ( 1st `  F ) `  <. Y ,  z >. ) >. (comp `  E )
( ( 1st `  F
) `  <. Y ,  w >. ) ) ( ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) `  <. K , 
( I `  z
) >. ) ) )
1115adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  F  e.  ( ( C  X.c  D
)  Func  E )
)
1121, 2, 58, 66, 111, 6, 59, 40, 67curf11 14000 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  z )  =  ( X ( 1st `  F ) z ) )
113112, 42syl6eq 2331 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  z )  =  ( ( 1st `  F ) `  <. X ,  z >. )
)
1141, 2, 58, 66, 111, 6, 59, 40, 69curf11 14000 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  w )  =  ( X ( 1st `  F ) w ) )
115 df-ov 5861 . . . . . . . 8  |-  ( X ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. X ,  w >. )
116114, 115syl6eq 2331 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  w )  =  ( ( 1st `  F ) `  <. X ,  w >. )
)
117113, 116opeq12d 3804 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. (
( 1st `  (
( 1st `  G
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  G
) `  X )
) `  w ) >.  =  <. ( ( 1st `  F ) `  <. X ,  z >. ) ,  ( ( 1st `  F ) `  <. X ,  w >. ) >. )
1181, 2, 58, 66, 111, 6, 61, 44, 69curf11 14000 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  w )  =  ( Y ( 1st `  F ) w ) )
119 df-ov 5861 . . . . . . 7  |-  ( Y ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. Y ,  w >. )
120118, 119syl6eq 2331 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  w )  =  ( ( 1st `  F ) `  <. Y ,  w >. )
)
121117, 120oveq12d 5876 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( <. ( ( 1st `  (
( 1st `  G
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  G
) `  X )
) `  w ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  Y )
) `  w )
)  =  ( <.
( ( 1st `  F
) `  <. X , 
z >. ) ,  ( ( 1st `  F
) `  <. X ,  w >. ) >. (comp `  E ) ( ( 1st `  F ) `
 <. Y ,  w >. ) ) )
1221, 2, 58, 66, 111, 6, 7, 8, 59, 61, 62, 12, 69curf2val 14004 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( L `  w )  =  ( K (
<. X ,  w >. ( 2nd `  F )
<. Y ,  w >. ) ( I `  w
) ) )
123 df-ov 5861 . . . . . 6  |-  ( K ( <. X ,  w >. ( 2nd `  F
) <. Y ,  w >. ) ( I `  w ) )  =  ( ( <. X ,  w >. ( 2nd `  F
) <. Y ,  w >. ) `  <. K , 
( I `  w
) >. )
124122, 123syl6eq 2331 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( L `  w )  =  ( ( <. X ,  w >. ( 2nd `  F )
<. Y ,  w >. ) `
 <. K ,  ( I `  w )
>. ) )
1251, 2, 58, 66, 111, 6, 59, 40, 67, 30, 57, 69, 70curf12 14001 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( z ( 2nd `  ( ( 1st `  G
) `  X )
) w ) `  f )  =  ( ( ( Id `  C ) `  X
) ( <. X , 
z >. ( 2nd `  F
) <. X ,  w >. ) f ) )
126 df-ov 5861 . . . . . 6  |-  ( ( ( Id `  C
) `  X )
( <. X ,  z
>. ( 2nd `  F
) <. X ,  w >. ) f )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  f >. )
127125, 126syl6eq 2331 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( z ( 2nd `  ( ( 1st `  G
) `  X )
) w ) `  f )  =  ( ( <. X ,  z
>. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  f >. )
)
128121, 124, 127oveq123d 5879 . . . 4  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( L `  w
) ( <. (
( 1st `  (
( 1st `  G
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  G
) `  X )
) `  w ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  Y )
) `  w )
) ( ( z ( 2nd `  (
( 1st `  G
) `  X )
) w ) `  f ) )  =  ( ( ( <. X ,  w >. ( 2nd `  F )
<. Y ,  w >. ) `
 <. K ,  ( I `  w )
>. ) ( <. (
( 1st `  F
) `  <. X , 
z >. ) ,  ( ( 1st `  F
) `  <. X ,  w >. ) >. (comp `  E ) ( ( 1st `  F ) `
 <. Y ,  w >. ) ) ( (
<. X ,  z >.
( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  f >. )
) )
1291, 2, 58, 66, 111, 6, 61, 44, 67curf11 14000 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  z )  =  ( Y ( 1st `  F ) z ) )
130129, 46syl6eq 2331 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  z )  =  ( ( 1st `  F ) `  <. Y ,  z >. )
)
131113, 130opeq12d 3804 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. (
( 1st `  (
( 1st `  G
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  G
) `  Y )
) `  z ) >.  =  <. ( ( 1st `  F ) `  <. X ,  z >. ) ,  ( ( 1st `  F ) `  <. Y ,  z >. ) >. )
132131, 120oveq12d 5876 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( <. ( ( 1st `  (
( 1st `  G
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  G
) `  Y )
) `  z ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  Y )
) `  w )
)  =  ( <.
( ( 1st `  F
) `  <. X , 
z >. ) ,  ( ( 1st `  F
) `  <. Y , 
z >. ) >. (comp `  E ) ( ( 1st `  F ) `
 <. Y ,  w >. ) ) )
1331, 2, 58, 66, 111, 6, 61, 44, 67, 30, 57, 69, 70curf12 14001 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( z ( 2nd `  ( ( 1st `  G
) `  Y )
) w ) `  f )  =  ( ( ( Id `  C ) `  Y
) ( <. Y , 
z >. ( 2nd `  F
) <. Y ,  w >. ) f ) )
134 df-ov 5861 . . . . . 6  |-  ( ( ( Id `  C
) `  Y )
( <. Y ,  z
>. ( 2nd `  F
) <. Y ,  w >. ) f )  =  ( ( <. Y , 
z >. ( 2nd `  F
) <. Y ,  w >. ) `  <. (
( Id `  C
) `  Y ) ,  f >. )
135133, 134syl6eq 2331 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( z ( 2nd `  ( ( 1st `  G
) `  Y )
) w ) `  f )  =  ( ( <. Y ,  z
>. ( 2nd `  F
) <. Y ,  w >. ) `  <. (
( Id `  C
) `  Y ) ,  f >. )
)
1361, 2, 58, 66, 111, 6, 7, 8, 59, 61, 62, 12, 67curf2val 14004 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( L `  z )  =  ( K (
<. X ,  z >.
( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) )
137 df-ov 5861 . . . . . 6  |-  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) `  <. K , 
( I `  z
) >. )
138136, 137syl6eq 2331 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  ( L `  z )  =  ( ( <. X ,  z >. ( 2nd `  F )
<. Y ,  z >.
) `  <. K , 
( I `  z
) >. ) )
139132, 135, 138oveq123d 5879 . . . 4  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( ( z ( 2nd `  ( ( 1st `  G ) `
 Y ) ) w ) `  f
) ( <. (
( 1st `  (
( 1st `  G
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  G
) `  Y )
) `  z ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  Y )
) `  w )
) ( L `  z ) )  =  ( ( ( <. Y ,  z >. ( 2nd `  F )
<. Y ,  w >. ) `
 <. ( ( Id
`  C ) `  Y ) ,  f
>. ) ( <. (
( 1st `  F
) `  <. X , 
z >. ) ,  ( ( 1st `  F
) `  <. Y , 
z >. ) >. (comp `  E ) ( ( 1st `  F ) `
 <. Y ,  w >. ) ) ( (
<. X ,  z >.
( 2nd `  F
) <. Y ,  z
>. ) `  <. K , 
( I `  z
) >. ) ) )
140110, 128, 1393eqtr4d 2325 . . 3  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  (
( L `  w
) ( <. (
( 1st `  (
( 1st `  G
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  G
) `  X )
) `  w ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  Y )
) `  w )
) ( ( z ( 2nd `  (
( 1st `  G
) `  X )
) w ) `  f ) )  =  ( ( ( z ( 2nd `  (
( 1st `  G
) `  Y )
) w ) `  f ) ( <.
( ( 1st `  (
( 1st `  G
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  G
) `  Y )
) `  z ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  Y )
) `  w )
) ( L `  z ) ) )
141140ralrimivvva 2636 . 2  |-  ( ph  ->  A. z  e.  B  A. w  e.  B  A. f  e.  (
z (  Hom  `  D
) w ) ( ( L `  w
) ( <. (
( 1st `  (
( 1st `  G
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  G
) `  X )
) `  w ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  Y )
) `  w )
) ( ( z ( 2nd `  (
( 1st `  G
) `  X )
) w ) `  f ) )  =  ( ( ( z ( 2nd `  (
( 1st `  G
) `  Y )
) w ) `  f ) ( <.
( ( 1st `  (
( 1st `  G
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  G
) `  Y )
) `  z ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  Y )
) `  w )
) ( L `  z ) ) )
142 curf2.n . . 3  |-  N  =  ( D Nat  E )
1431, 2, 3, 4, 5, 6, 9, 40curf1cl 14002 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  X )  e.  ( D  Func  E
) )
1441, 2, 3, 4, 5, 6, 10, 44curf1cl 14002 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  Y )  e.  ( D  Func  E
) )
145142, 6, 30, 17, 84, 143, 144isnat2 13822 . 2  |-  ( ph  ->  ( L  e.  ( ( ( 1st `  G
) `  X ) N ( ( 1st `  G ) `  Y
) )  <->  ( L  e.  X_ z  e.  B  ( ( ( 1st `  ( ( 1st `  G
) `  X )
) `  z )
(  Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
)  /\  A. z  e.  B  A. w  e.  B  A. f  e.  ( z (  Hom  `  D ) w ) ( ( L `  w ) ( <.
( ( 1st `  (
( 1st `  G
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  G
) `  X )
) `  w ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  Y )
) `  w )
) ( ( z ( 2nd `  (
( 1st `  G
) `  X )
) w ) `  f ) )  =  ( ( ( z ( 2nd `  (
( 1st `  G
) `  Y )
) w ) `  f ) ( <.
( ( 1st `  (
( 1st `  G
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  G
) `  Y )
) `  z ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  Y )
) `  w )
) ( L `  z ) ) ) ) )
14656, 141, 145mpbir2and 888 1  |-  ( ph  ->  L  e.  ( ( ( 1st `  G
) `  X ) N ( ( 1st `  G ) `  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   Rel wrel 4694   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   X_cixp 6817   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567    Func cfunc 13728   Nat cnat 13815    X.c cxpc 13942   curryF ccurf 13984
This theorem is referenced by:  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-nat 13817  df-xpc 13946  df-curf 13988
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