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Theorem curf2cl 14283
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 14281 . . 3  |-  ( ph  ->  L  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
14 eqid 2404 . . . . . . . . . 10  |-  ( C  X.c  D )  =  ( C  X.c  D )
1514, 2, 6xpcbas 14230 . . . . . . . . 9  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
16 eqid 2404 . . . . . . . . 9  |-  (  Hom  `  ( C  X.c  D ) )  =  (  Hom  `  ( C  X.c  D ) )
17 eqid 2404 . . . . . . . . 9  |-  (  Hom  `  E )  =  (  Hom  `  E )
18 relfunc 14014 . . . . . . . . . . 11  |-  Rel  (
( C  X.c  D ) 
Func  E )
19 1st2ndbr 6355 . . . . . . . . . . 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 645 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( ( C  X.c  D )  Func  E
) ( 2nd `  F
) )
2120adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  ( 1st `  F ) ( ( C  X.c  D ) 
Func  E ) ( 2nd `  F ) )
22 opelxpi 4869 . . . . . . . . . 10  |-  ( ( X  e.  A  /\  z  e.  B )  -> 
<. X ,  z >.  e.  ( A  X.  B
) )
239, 22sylan 458 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  <. X , 
z >.  e.  ( A  X.  B ) )
24 opelxpi 4869 . . . . . . . . . 10  |-  ( ( Y  e.  A  /\  z  e.  B )  -> 
<. Y ,  z >.  e.  ( A  X.  B
) )
2510, 24sylan 458 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  <. Y , 
z >.  e.  ( A  X.  B ) )
2615, 16, 17, 21, 23, 25funcf2 14020 . . . . . . . 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 2404 . . . . . . . . . 10  |-  (  Hom  `  D )  =  (  Hom  `  D )
289adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  B )  ->  X  e.  A )
29 simpr 448 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  B )  ->  z  e.  B )
3010adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  B )  ->  Y  e.  A )
3114, 2, 6, 7, 27, 28, 29, 30, 29, 16xpchom2 14238 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  ( <. X ,  z >.
(  Hom  `  ( C  X.c  D ) ) <. Y ,  z >. )  =  ( ( X H Y )  X.  ( z (  Hom  `  D ) z ) ) )
3231feq2d 5540 . . . . . . . 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 202 . . . . . . 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 452 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  K  e.  ( X H Y ) )
354adantr 452 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  D  e.  Cat )
366, 27, 8, 35, 29catidcl 13862 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
I `  z )  e.  ( z (  Hom  `  D ) z ) )
3733, 34, 36fovrnd 6177 . . . . . 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 452 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  C  e.  Cat )
395adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  F  e.  ( ( C  X.c  D
)  Func  E )
)
40 eqid 2404 . . . . . . . . 9  |-  ( ( 1st `  G ) `
 X )  =  ( ( 1st `  G
) `  X )
411, 2, 38, 35, 39, 6, 28, 40, 29curf11 14278 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  z )  =  ( X ( 1st `  F ) z ) )
42 df-ov 6043 . . . . . . . 8  |-  ( X ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. X , 
z >. )
4341, 42syl6eq 2452 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  z )  =  ( ( 1st `  F ) `  <. X ,  z >. )
)
44 eqid 2404 . . . . . . . . 9  |-  ( ( 1st `  G ) `
 Y )  =  ( ( 1st `  G
) `  Y )
451, 2, 38, 35, 39, 6, 30, 44, 29curf11 14278 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  z )  =  ( Y ( 1st `  F ) z ) )
46 df-ov 6043 . . . . . . . 8  |-  ( Y ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. Y , 
z >. )
4745, 46syl6eq 2452 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  z )  =  ( ( 1st `  F ) `  <. Y ,  z >. )
)
4843, 47oveq12d 6058 . . . . . 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 2481 . . . . 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 2749 . . . 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 5701 . . . . . 6  |-  ( Base `  D )  e.  _V
526, 51eqeltri 2474 . . . . 5  |-  B  e. 
_V
53 mptelixpg 7058 . . . . 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 204 . . 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 2478 . 2  |-  ( ph  ->  L  e.  X_ z  e.  B  ( (
( 1st `  (
( 1st `  G
) `  X )
) `  z )
(  Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
) )
57 eqid 2404 . . . . . . . . . 10  |-  ( Id
`  C )  =  ( Id `  C
)
583adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  C  e.  Cat )
599adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  X  e.  A )
60 eqid 2404 . . . . . . . . . 10  |-  (comp `  C )  =  (comp `  C )
6110adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  Y  e.  A )
6211adantr 452 . . . . . . . . . 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 13864 . . . . . . . . 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 13863 . . . . . . . . 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 2439 . . . . . . . 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 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  D  e.  Cat )
67 simpr1 963 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  z  e.  B )
68 eqid 2404 . . . . . . . . . 10  |-  (comp `  D )  =  (comp `  D )
69 simpr2 964 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  w  e.  B )
70 simpr3 965 . . . . . . . . . 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 13863 . . . . . . . . 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 13864 . . . . . . . . 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 2439 . . . . . . . 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 3952 . . . . . . 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 2404 . . . . . . . 8  |-  (comp `  ( C  X.c  D )
)  =  (comp `  ( C  X.c  D )
)
762, 7, 57, 58, 59catidcl 13862 . . . . . . . 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 13862 . . . . . . . 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 14239 . . . . . . 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 1122 . . . . . . . 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 13862 . . . . . . . 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 14239 . . . . . . 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 2446 . . . . . 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 5691 . . . . 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 2404 . . . . . 6  |-  (comp `  E )  =  (comp `  E )
8520adantr 452 . . . . . 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 1122 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. X , 
z >.  e.  ( A  X.  B ) )
87 opelxpi 4869 . . . . . . 7  |-  ( ( X  e.  A  /\  w  e.  B )  -> 
<. X ,  w >.  e.  ( A  X.  B
) )
8859, 69, 87syl2anc 643 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. X ,  w >.  e.  ( A  X.  B ) )
89 opelxpi 4869 . . . . . . 7  |-  ( ( Y  e.  A  /\  w  e.  B )  -> 
<. Y ,  w >.  e.  ( A  X.  B
) )
9061, 69, 89syl2anc 643 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. Y ,  w >.  e.  ( A  X.  B ) )
91 opelxpi 4869 . . . . . . . 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 643 . . . . . . 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 14238 . . . . . . 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 2481 . . . . . 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 4869 . . . . . . . 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 643 . . . . . . 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 14238 . . . . . . 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 2481 . . . . . 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 14023 . . . . 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 1122 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z (  Hom  `  D ) w ) ) )  ->  <. Y , 
z >.  e.  ( A  X.  B ) )
101 opelxpi 4869 . . . . . . . 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 643 . . . . . . 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 14238 . . . . . . 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 2481 . . . . . 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 4869 . . . . . . . 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 643 . . . . . . 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 14238 . . . . . . 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 2481 . . . . . 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 14023 . . . . 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 2444 . . . 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 452 . . . . . . . . 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 14278 . . . . . . . 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 2452 . . . . . . 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 14278 . . . . . . . 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 6043 . . . . . . . 8  |-  ( X ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. X ,  w >. )
116114, 115syl6eq 2452 . . . . . . 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 3952 . . . . . 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 14278 . . . . . . 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 6043 . . . . . . 7  |-  ( Y ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. Y ,  w >. )
120118, 119syl6eq 2452 . . . . . 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 6058 . . . . 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 14282 . . . . . 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 6043 . . . . . 6  |-  ( K ( <. X ,  w >. ( 2nd `  F
) <. Y ,  w >. ) ( I `  w ) )  =  ( ( <. X ,  w >. ( 2nd `  F
) <. Y ,  w >. ) `  <. K , 
( I `  w
) >. )
124122, 123syl6eq 2452 . . . . 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 14279 . . . . . 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 6043 . . . . . 6  |-  ( ( ( Id `  C
) `  X )
( <. X ,  z
>. ( 2nd `  F
) <. X ,  w >. ) f )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  f >. )
127125, 126syl6eq 2452 . . . . 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 6061 . . . 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 14278 . . . . . . . 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 2452 . . . . . . 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 3952 . . . . . 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 6058 . . . . 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 14279 . . . . . 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 6043 . . . . . 6  |-  ( ( ( Id `  C
) `  Y )
( <. Y ,  z
>. ( 2nd `  F
) <. Y ,  w >. ) f )  =  ( ( <. Y , 
z >. ( 2nd `  F
) <. Y ,  w >. ) `  <. (
( Id `  C
) `  Y ) ,  f >. )
135133, 134syl6eq 2452 . . . . 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 14282 . . . . . 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 6043 . . . . . 6  |-  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) `  <. K , 
( I `  z
) >. )
138136, 137syl6eq 2452 . . . . 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 6061 . . . 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 2446 . . 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 2759 . 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 14280 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  X )  e.  ( D  Func  E
) )
1441, 2, 3, 4, 5, 6, 10, 44curf1cl 14280 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  Y )  e.  ( D  Func  E
) )
145142, 6, 27, 17, 84, 143, 144isnat2 14100 . 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 889 1  |-  ( ph  ->  L  e.  ( ( ( 1st `  G
) `  X ) N ( ( 1st `  G ) `  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   <.cop 3777   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   Rel wrel 4842   -->wf 5409   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   X_cixp 7022   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844   Idccid 13845    Func cfunc 14006   Nat cnat 14093    X.c cxpc 14220   curryF ccurf 14262
This theorem is referenced by:  curfcl  14284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-hom 13508  df-cco 13509  df-cat 13848  df-cid 13849  df-func 14010  df-nat 14095  df-xpc 14224  df-curf 14266
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