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Theorem curf2cl 15033
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 15031 . . 3  |-  ( ph  ->  L  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
14 eqid 2438 . . . . . . . . . 10  |-  ( C  X.c  D )  =  ( C  X.c  D )
1514, 2, 6xpcbas 14980 . . . . . . . . 9  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
16 eqid 2438 . . . . . . . . 9  |-  ( Hom  `  ( C  X.c  D ) )  =  ( Hom  `  ( C  X.c  D ) )
17 eqid 2438 . . . . . . . . 9  |-  ( Hom  `  E )  =  ( Hom  `  E )
18 relfunc 14764 . . . . . . . . . . 11  |-  Rel  (
( C  X.c  D ) 
Func  E )
19 1st2ndbr 6618 . . . . . . . . . . 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 663 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
2120adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  ( 1st `  F ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  F ) )
22 opelxpi 4866 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  z  e.  B )  -> 
<. X ,  z >.  e.  ( A  X.  B
) )
239, 22sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  <. X , 
z >.  e.  ( A  X.  B ) )
24 opelxpi 4866 . . . . . . . . . 10  |-  ( ( Y  e.  A  /\  z  e.  B )  -> 
<. Y ,  z >.  e.  ( A  X.  B
) )
2510, 24sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  <. Y , 
z >.  e.  ( A  X.  B ) )
2615, 16, 17, 21, 23, 25funcf2 14770 . . . . . . . 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 >. ) ) )
27 eqid 2438 . . . . . . . . . 10  |-  ( Hom  `  D )  =  ( Hom  `  D )
289adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  B )  ->  X  e.  A )
29 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  B )  ->  z  e.  B )
3010adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  B )  ->  Y  e.  A )
3114, 2, 6, 7, 27, 28, 29, 30, 29, 16xpchom2 14988 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  ( <. X ,  z >.
( Hom  `  ( C  X.c  D ) ) <. Y ,  z >. )  =  ( ( X H Y )  X.  ( z ( Hom  `  D ) z ) ) )
3231feq2d 5542 . . . . . . . 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 >. )
) ) )
3326, 32mpbid 210 . . . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  K  e.  ( X H Y ) )
354adantr 465 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  D  e.  Cat )
366, 27, 8, 35, 29catidcl 14612 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
I `  z )  e.  ( z ( Hom  `  D ) z ) )
3733, 34, 36fovrnd 6230 . . . . . 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 465 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  C  e.  Cat )
395adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  F  e.  ( ( C  X.c  D
)  Func  E )
)
40 eqid 2438 . . . . . . . . 9  |-  ( ( 1st `  G ) `
 X )  =  ( ( 1st `  G
) `  X )
411, 2, 38, 35, 39, 6, 28, 40, 29curf11 15028 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  z )  =  ( X ( 1st `  F ) z ) )
42 df-ov 6089 . . . . . . . 8  |-  ( X ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. X , 
z >. )
4341, 42syl6eq 2486 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  z )  =  ( ( 1st `  F ) `  <. X ,  z >. )
)
44 eqid 2438 . . . . . . . . 9  |-  ( ( 1st `  G ) `
 Y )  =  ( ( 1st `  G
) `  Y )
451, 2, 38, 35, 39, 6, 30, 44, 29curf11 15028 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  z )  =  ( Y ( 1st `  F ) z ) )
46 df-ov 6089 . . . . . . . 8  |-  ( Y ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. Y , 
z >. )
4745, 46syl6eq 2486 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  z )  =  ( ( 1st `  F ) `  <. Y ,  z >. )
)
4843, 47oveq12d 6104 . . . . . 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 2515 . . . . 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 2794 . . . 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 5696 . . . . . 6  |-  ( Base `  D )  e.  _V
526, 51eqeltri 2508 . . . . 5  |-  B  e. 
_V
53 mptelixpg 7292 . . . . 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 5 . . . 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 212 . . 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 2512 . 2  |-  ( ph  ->  L  e.  X_ z  e.  B  ( (
( 1st `  (
( 1st `  G
) `  X )
) `  z )
( Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
) )
57 eqid 2438 . . . . . . . . . 10  |-  ( Id
`  C )  =  ( Id `  C
)
583adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  C  e.  Cat )
599adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  X  e.  A )
60 eqid 2438 . . . . . . . . . 10  |-  (comp `  C )  =  (comp `  C )
6110adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  Y  e.  A )
6211adantr 465 . . . . . . . . . 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 14614 . . . . . . . . 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 14613 . . . . . . . . 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 2473 . . . . . . . 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 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  D  e.  Cat )
67 simpr1 994 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  z  e.  B )
68 eqid 2438 . . . . . . . . . 10  |-  (comp `  D )  =  (comp `  D )
69 simpr2 995 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  w  e.  B )
70 simpr3 996 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  f  e.  ( z ( Hom  `  D ) w ) )
716, 27, 8, 66, 67, 68, 69, 70catlid 14613 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  (
( I `  w
) ( <. z ,  w >. (comp `  D
) w ) f )  =  f )
726, 27, 8, 66, 67, 68, 69, 70catrid 14614 . . . . . . . . 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 2473 . . . . . . . 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 4062 . . . . . . 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 2438 . . . . . . . 8  |-  (comp `  ( C  X.c  D )
)  =  (comp `  ( C  X.c  D )
)
762, 7, 57, 58, 59catidcl 14612 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  (
( Id `  C
) `  X )  e.  ( X H X ) )
776, 27, 8, 66, 69catidcl 14612 . . . . . . . 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, 27, 59, 67, 59, 69, 60, 68, 75, 61, 69, 76, 70, 62, 77xpcco2 14989 . . . . . . 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 1153 . . . . . . . 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 14612 . . . . . . . 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, 27, 59, 67, 61, 67, 60, 68, 75, 61, 69, 62, 79, 80, 70xpcco2 14989 . . . . . . 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 2480 . . . . . 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 5690 . . . . 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 2438 . . . . . 6  |-  (comp `  E )  =  (comp `  E )
8520adantr 465 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  ( 1st `  F ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  F ) )
86233ad2antr1 1153 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  <. X , 
z >.  e.  ( A  X.  B ) )
87 opelxpi 4866 . . . . . . 7  |-  ( ( X  e.  A  /\  w  e.  B )  -> 
<. X ,  w >.  e.  ( A  X.  B
) )
8859, 69, 87syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  <. X ,  w >.  e.  ( A  X.  B ) )
89 opelxpi 4866 . . . . . . 7  |-  ( ( Y  e.  A  /\  w  e.  B )  -> 
<. Y ,  w >.  e.  ( A  X.  B
) )
9061, 69, 89syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  <. Y ,  w >.  e.  ( A  X.  B ) )
91 opelxpi 4866 . . . . . . . 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 661 . . . . . . 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, 27, 59, 67, 59, 69, 16xpchom2 14988 . . . . . . 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 2515 . . . . . 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 4866 . . . . . . . 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 661 . . . . . . 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, 27, 59, 69, 61, 69, 16xpchom2 14988 . . . . . . 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 2515 . . . . . 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 14773 . . . . 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 >. )
) )
100253ad2antr1 1153 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  <. Y , 
z >.  e.  ( A  X.  B ) )
101 opelxpi 4866 . . . . . . . 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 661 . . . . . . 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, 27, 59, 67, 61, 67, 16xpchom2 14988 . . . . . . 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 2515 . . . . . 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 4866 . . . . . . . 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 661 . . . . . . 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, 27, 61, 67, 61, 69, 16xpchom2 14988 . . . . . . 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 2515 . . . . . 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 14773 . . . . 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 2478 . . . 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 465 . . . . . . . . 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 15028 . . . . . . . 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 2486 . . . . . . 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 15028 . . . . . . . 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 6089 . . . . . . . 8  |-  ( X ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. X ,  w >. )
116114, 115syl6eq 2486 . . . . . . 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 4062 . . . . . 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 15028 . . . . . . 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 6089 . . . . . . 7  |-  ( Y ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. Y ,  w >. )
120118, 119syl6eq 2486 . . . . . 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 6104 . . . . 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 15032 . . . . . 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 6089 . . . . . 6  |-  ( K ( <. X ,  w >. ( 2nd `  F
) <. Y ,  w >. ) ( I `  w ) )  =  ( ( <. X ,  w >. ( 2nd `  F
) <. Y ,  w >. ) `  <. K , 
( I `  w
) >. )
124122, 123syl6eq 2486 . . . . 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, 27, 57, 69, 70curf12 15029 . . . . . 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 6089 . . . . . 6  |-  ( ( ( Id `  C
) `  X )
( <. X ,  z
>. ( 2nd `  F
) <. X ,  w >. ) f )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  f >. )
127125, 126syl6eq 2486 . . . . 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 6107 . . . 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 15028 . . . . . . . 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 2486 . . . . . . 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 4062 . . . . . 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 6104 . . . . 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, 27, 57, 69, 70curf12 15029 . . . . . 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 6089 . . . . . 6  |-  ( ( ( Id `  C
) `  Y )
( <. Y ,  z
>. ( 2nd `  F
) <. Y ,  w >. ) f )  =  ( ( <. Y , 
z >. ( 2nd `  F
) <. Y ,  w >. ) `  <. (
( Id `  C
) `  Y ) ,  f >. )
135133, 134syl6eq 2486 . . . . 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 15032 . . . . . 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 6089 . . . . . 6  |-  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) `  <. K , 
( I `  z
) >. )
138136, 137syl6eq 2486 . . . . 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 6107 . . . 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 2480 . . 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 2804 . 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 15030 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  X )  e.  ( D  Func  E
) )
1441, 2, 3, 4, 5, 6, 10, 44curf1cl 15030 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  Y )  e.  ( D  Func  E
) )
145142, 6, 27, 17, 84, 143, 144isnat2 14850 . 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 913 1  |-  ( ph  ->  L  e.  ( ( ( 1st `  G
) `  X ) N ( ( 1st `  G ) `  Y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   _Vcvv 2967   <.cop 3878   class class class wbr 4287    e. cmpt 4345    X. cxp 4833   Rel wrel 4840   -->wf 5409   ` cfv 5413  (class class class)co 6086   1stc1st 6570   2ndc2nd 6571   X_cixp 7255   Basecbs 14166   Hom chom 14241  compcco 14242   Catccat 14594   Idccid 14595    Func cfunc 14756   Nat cnat 14843    X.c cxpc 14970   curryF ccurf 15012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-hom 14254  df-cco 14255  df-cat 14598  df-cid 14599  df-func 14760  df-nat 14845  df-xpc 14974  df-curf 15016
This theorem is referenced by:  curfcl  15034
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