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Theorem curf2cl 14054
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 14052 . . 3  |-  ( ph  ->  L  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
14 eqid 2316 . . . . . . . . . 10  |-  ( C  X.c  D )  =  ( C  X.c  D )
1514, 2, 6xpcbas 14001 . . . . . . . . 9  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
16 eqid 2316 . . . . . . . . 9  |-  (  Hom  `  ( C  X.c  D ) )  =  (  Hom  `  ( C  X.c  D ) )
17 eqid 2316 . . . . . . . . 9  |-  (  Hom  `  E )  =  (  Hom  `  E )
18 relfunc 13785 . . . . . . . . . . 11  |-  Rel  (
( C  X.c  D ) 
Func  E )
19 1st2ndbr 6211 . . . . . . . . . . 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 4758 . . . . . . . . . 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 4758 . . . . . . . . . 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 13791 . . . . . . . 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 2316 . . . . . . . . . 10  |-  (  Hom  `  D )  =  (  Hom  `  D )
3114, 2, 6, 7, 30, 22, 23, 26, 23, 16xpchom2 14009 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  ( <. X ,  z >.
(  Hom  `  ( C  X.c  D ) ) <. Y ,  z >. )  =  ( ( X H Y )  X.  ( z (  Hom  `  D ) z ) ) )
3231feq2d 5417 . . . . . . . 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 13633 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
I `  z )  e.  ( z (  Hom  `  D ) z ) )
3733, 34, 36fovrnd 6034 . . . . . 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 2316 . . . . . . . . 9  |-  ( ( 1st `  G ) `
 X )  =  ( ( 1st `  G
) `  X )
411, 2, 38, 35, 39, 6, 22, 40, 23curf11 14049 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  z )  =  ( X ( 1st `  F ) z ) )
42 df-ov 5903 . . . . . . . 8  |-  ( X ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. X , 
z >. )
4341, 42syl6eq 2364 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  z )  =  ( ( 1st `  F ) `  <. X ,  z >. )
)
44 eqid 2316 . . . . . . . . 9  |-  ( ( 1st `  G ) `
 Y )  =  ( ( 1st `  G
) `  Y )
451, 2, 38, 35, 39, 6, 26, 44, 23curf11 14049 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  z )  =  ( Y ( 1st `  F ) z ) )
46 df-ov 5903 . . . . . . . 8  |-  ( Y ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. Y , 
z >. )
4745, 46syl6eq 2364 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  z )  =  ( ( 1st `  F ) `  <. Y ,  z >. )
)
4843, 47oveq12d 5918 . . . . . 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 2393 . . . . 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 2660 . . . 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 5577 . . . . . 6  |-  ( Base `  D )  e.  _V
526, 51eqeltri 2386 . . . . 5  |-  B  e. 
_V
53 mptelixpg 6896 . . . . 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 2390 . 2  |-  ( ph  ->  L  e.  X_ z  e.  B  ( (
( 1st `  (
( 1st `  G
) `  X )
) `  z )
(  Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
) )
57 eqid 2316 . . . . . . . . . 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 2316 . . . . . . . . . 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 13635 . . . . . . . . 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 13634 . . . . . . . . 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 2351 . . . . . . . 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 2316 . . . . . . . . . 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 13634 . . . . . . . . 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 13635 . . . . . . . . 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 2351 . . . . . . . 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 3841 . . . . . . 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 2316 . . . . . . . 8  |-  (comp `  ( C  X.c  D )
)  =  (comp `  ( C  X.c  D )
)
762, 7, 57, 58, 59catidcl 13633 . . . . . . . 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 13633 . . . . . . . 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 14010 . . . . . . 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 13633 . . . . . . . 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 14010 . . . . . . 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 2358 . . . . . 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 5567 . . . . 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 2316 . . . . . 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 4758 . . . . . . 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 4758 . . . . . . 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 4758 . . . . . . . 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 14009 . . . . . . 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 2393 . . . . . 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 4758 . . . . . . . 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 14009 . . . . . . 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 2393 . . . . . 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 13794 . . . . 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 4758 . . . . . . . 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 14009 . . . . . . 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 2393 . . . . . 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 4758 . . . . . . . 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 14009 . . . . . . 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 2393 . . . . . 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 13794 . . . . 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 2356 . . . 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 14049 . . . . . . . 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 2364 . . . . . . 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 14049 . . . . . . . 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 5903 . . . . . . . 8  |-  ( X ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. X ,  w >. )
116114, 115syl6eq 2364 . . . . . . 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 3841 . . . . . 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 14049 . . . . . . 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 5903 . . . . . . 7  |-  ( Y ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. Y ,  w >. )
120118, 119syl6eq 2364 . . . . . 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 5918 . . . . 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 14053 . . . . . 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 5903 . . . . . 6  |-  ( K ( <. X ,  w >. ( 2nd `  F
) <. Y ,  w >. ) ( I `  w ) )  =  ( ( <. X ,  w >. ( 2nd `  F
) <. Y ,  w >. ) `  <. K , 
( I `  w
) >. )
124122, 123syl6eq 2364 . . . . 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 14050 . . . . . 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 5903 . . . . . 6  |-  ( ( ( Id `  C
) `  X )
( <. X ,  z
>. ( 2nd `  F
) <. X ,  w >. ) f )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  f >. )
127125, 126syl6eq 2364 . . . . 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 5921 . . . 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 14049 . . . . . . . 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 2364 . . . . . . 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 3841 . . . . . 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 5918 . . . . 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 14050 . . . . . 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 5903 . . . . . 6  |-  ( ( ( Id `  C
) `  Y )
( <. Y ,  z
>. ( 2nd `  F
) <. Y ,  w >. ) f )  =  ( ( <. Y , 
z >. ( 2nd `  F
) <. Y ,  w >. ) `  <. (
( Id `  C
) `  Y ) ,  f >. )
135133, 134syl6eq 2364 . . . . 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 14053 . . . . . 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 5903 . . . . . 6  |-  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) `  <. K , 
( I `  z
) >. )
138136, 137syl6eq 2364 . . . . 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 5921 . . . 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 2358 . . 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 2670 . 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 14051 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  X )  e.  ( D  Func  E
) )
1441, 2, 3, 4, 5, 6, 10, 44curf1cl 14051 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  Y )  e.  ( D  Func  E
) )
145142, 6, 30, 17, 84, 143, 144isnat2 13871 . 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 1633    e. wcel 1701   A.wral 2577   _Vcvv 2822   <.cop 3677   class class class wbr 4060    e. cmpt 4114    X. cxp 4724   Rel wrel 4731   -->wf 5288   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163   X_cixp 6860   Basecbs 13195    Hom chom 13266  compcco 13267   Catccat 13615   Idccid 13616    Func cfunc 13777   Nat cnat 13864    X.c cxpc 13991   curryF ccurf 14033
This theorem is referenced by:  curfcl  14055
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-hom 13279  df-cco 13280  df-cat 13619  df-cid 13620  df-func 13781  df-nat 13866  df-xpc 13995  df-curf 14037
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