MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  curf2cl Structured version   Unicode version

Theorem curf2cl 15354
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 15352 . . 3  |-  ( ph  ->  L  =  ( z  e.  B  |->  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) ) ) )
14 eqid 2467 . . . . . . . . . 10  |-  ( C  X.c  D )  =  ( C  X.c  D )
1514, 2, 6xpcbas 15301 . . . . . . . . 9  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
16 eqid 2467 . . . . . . . . 9  |-  ( Hom  `  ( C  X.c  D ) )  =  ( Hom  `  ( C  X.c  D ) )
17 eqid 2467 . . . . . . . . 9  |-  ( Hom  `  E )  =  ( Hom  `  E )
18 relfunc 15085 . . . . . . . . . . 11  |-  Rel  (
( C  X.c  D ) 
Func  E )
19 1st2ndbr 6830 . . . . . . . . . . 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 5030 . . . . . . . . . 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 5030 . . . . . . . . . 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 15091 . . . . . . . 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 2467 . . . . . . . . . 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 15309 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  ( <. X ,  z >.
( Hom  `  ( C  X.c  D ) ) <. Y ,  z >. )  =  ( ( X H Y )  X.  ( z ( Hom  `  D ) z ) ) )
3231feq2d 5716 . . . . . . . 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 14933 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
I `  z )  e.  ( z ( Hom  `  D ) z ) )
3733, 34, 36fovrnd 6429 . . . . . 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 2467 . . . . . . . . 9  |-  ( ( 1st `  G ) `
 X )  =  ( ( 1st `  G
) `  X )
411, 2, 38, 35, 39, 6, 28, 40, 29curf11 15349 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  z )  =  ( X ( 1st `  F ) z ) )
42 df-ov 6285 . . . . . . . 8  |-  ( X ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. X , 
z >. )
4341, 42syl6eq 2524 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  X )
) `  z )  =  ( ( 1st `  F ) `  <. X ,  z >. )
)
44 eqid 2467 . . . . . . . . 9  |-  ( ( 1st `  G ) `
 Y )  =  ( ( 1st `  G
) `  Y )
451, 2, 38, 35, 39, 6, 30, 44, 29curf11 15349 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  z )  =  ( Y ( 1st `  F ) z ) )
46 df-ov 6285 . . . . . . . 8  |-  ( Y ( 1st `  F
) z )  =  ( ( 1st `  F
) `  <. Y , 
z >. )
4745, 46syl6eq 2524 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  G
) `  Y )
) `  z )  =  ( ( 1st `  F ) `  <. Y ,  z >. )
)
4843, 47oveq12d 6300 . . . . . 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 2558 . . . . 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 2878 . . . 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 5874 . . . . . 6  |-  ( Base `  D )  e.  _V
526, 51eqeltri 2551 . . . . 5  |-  B  e. 
_V
53 mptelixpg 7503 . . . . 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 2555 . 2  |-  ( ph  ->  L  e.  X_ z  e.  B  ( (
( 1st `  (
( 1st `  G
) `  X )
) `  z )
( Hom  `  E ) ( ( 1st `  (
( 1st `  G
) `  Y )
) `  z )
) )
57 eqid 2467 . . . . . . . . . 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 2467 . . . . . . . . . 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 14935 . . . . . . . . 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 14934 . . . . . . . . 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 2511 . . . . . . . 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 1002 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  z  e.  B )
68 eqid 2467 . . . . . . . . . 10  |-  (comp `  D )  =  (comp `  D )
69 simpr2 1003 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  w  e.  B )
70 simpr3 1004 . . . . . . . . . 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 14934 . . . . . . . . 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 14935 . . . . . . . . 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 2511 . . . . . . . 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 4221 . . . . . . 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 2467 . . . . . . . 8  |-  (comp `  ( C  X.c  D )
)  =  (comp `  ( C  X.c  D )
)
762, 7, 57, 58, 59catidcl 14933 . . . . . . . 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 14933 . . . . . . . 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 15310 . . . . . . 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 1161 . . . . . . . 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 14933 . . . . . . . 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 15310 . . . . . . 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 2518 . . . . . 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 5868 . . . . 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 2467 . . . . . 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 1161 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  <. X , 
z >.  e.  ( A  X.  B ) )
87 opelxpi 5030 . . . . . . 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 5030 . . . . . . 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 5030 . . . . . . . 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 15309 . . . . . . 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 2558 . . . . . 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 5030 . . . . . . . 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 15309 . . . . . . 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 2558 . . . . . 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 15094 . . . . 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 1161 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  f  e.  ( z ( Hom  `  D ) w ) ) )  ->  <. Y , 
z >.  e.  ( A  X.  B ) )
101 opelxpi 5030 . . . . . . . 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 15309 . . . . . . 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 2558 . . . . . 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 5030 . . . . . . . 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 15309 . . . . . . 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 2558 . . . . . 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 15094 . . . . 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 2516 . . . 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 15349 . . . . . . . 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 2524 . . . . . . 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 15349 . . . . . . . 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 6285 . . . . . . . 8  |-  ( X ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. X ,  w >. )
116114, 115syl6eq 2524 . . . . . . 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 4221 . . . . . 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 15349 . . . . . . 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 6285 . . . . . . 7  |-  ( Y ( 1st `  F
) w )  =  ( ( 1st `  F
) `  <. Y ,  w >. )
120118, 119syl6eq 2524 . . . . . 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 6300 . . . . 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 15353 . . . . . 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 6285 . . . . . 6  |-  ( K ( <. X ,  w >. ( 2nd `  F
) <. Y ,  w >. ) ( I `  w ) )  =  ( ( <. X ,  w >. ( 2nd `  F
) <. Y ,  w >. ) `  <. K , 
( I `  w
) >. )
124122, 123syl6eq 2524 . . . . 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 15350 . . . . . 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 6285 . . . . . 6  |-  ( ( ( Id `  C
) `  X )
( <. X ,  z
>. ( 2nd `  F
) <. X ,  w >. ) f )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. X ,  w >. ) `  <. (
( Id `  C
) `  X ) ,  f >. )
127125, 126syl6eq 2524 . . . . 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 6303 . . . 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 15349 . . . . . . . 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 2524 . . . . . . 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 4221 . . . . . 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 6300 . . . . 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 15350 . . . . . 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 6285 . . . . . 6  |-  ( ( ( Id `  C
) `  Y )
( <. Y ,  z
>. ( 2nd `  F
) <. Y ,  w >. ) f )  =  ( ( <. Y , 
z >. ( 2nd `  F
) <. Y ,  w >. ) `  <. (
( Id `  C
) `  Y ) ,  f >. )
135133, 134syl6eq 2524 . . . . 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 15353 . . . . . 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 6285 . . . . . 6  |-  ( K ( <. X ,  z
>. ( 2nd `  F
) <. Y ,  z
>. ) ( I `  z ) )  =  ( ( <. X , 
z >. ( 2nd `  F
) <. Y ,  z
>. ) `  <. K , 
( I `  z
) >. )
138136, 137syl6eq 2524 . . . . 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 6303 . . . 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 2518 . . 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 2886 . 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 15351 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  X )  e.  ( D  Func  E
) )
1441, 2, 3, 4, 5, 6, 10, 44curf1cl 15351 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  Y )  e.  ( D  Func  E
) )
145142, 6, 27, 17, 84, 143, 144isnat2 15171 . 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 920 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   Rel wrel 5004   -->wf 5582   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   X_cixp 7466   Basecbs 14486   Hom chom 14562  compcco 14563   Catccat 14915   Idccid 14916    Func cfunc 15077   Nat cnat 15164    X.c cxpc 15291   curryF ccurf 15333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-hom 14575  df-cco 14576  df-cat 14919  df-cid 14920  df-func 15081  df-nat 15166  df-xpc 15295  df-curf 15337
This theorem is referenced by:  curfcl  15355
  Copyright terms: Public domain W3C validator