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Theorem uncf2 14254
Description: Value of the uncurry functor on a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
Hypotheses
Ref Expression
uncfval.g  |-  F  =  ( <" C D E "> uncurryF  G )
uncfval.c  |-  ( ph  ->  D  e.  Cat )
uncfval.d  |-  ( ph  ->  E  e.  Cat )
uncfval.f  |-  ( ph  ->  G  e.  ( C 
Func  ( D FuncCat  E
) ) )
uncf1.a  |-  A  =  ( Base `  C
)
uncf1.b  |-  B  =  ( Base `  D
)
uncf1.x  |-  ( ph  ->  X  e.  A )
uncf1.y  |-  ( ph  ->  Y  e.  B )
uncf2.h  |-  H  =  (  Hom  `  C
)
uncf2.j  |-  J  =  (  Hom  `  D
)
uncf2.z  |-  ( ph  ->  Z  e.  A )
uncf2.w  |-  ( ph  ->  W  e.  B )
uncf2.r  |-  ( ph  ->  R  e.  ( X H Z ) )
uncf2.s  |-  ( ph  ->  S  e.  ( Y J W ) )
Assertion
Ref Expression
uncf2  |-  ( ph  ->  ( R ( <. X ,  Y >. ( 2nd `  F )
<. Z ,  W >. ) S )  =  ( ( ( ( X ( 2nd `  G
) Z ) `  R ) `  W
) ( <. (
( 1st `  (
( 1st `  G
) `  X )
) `  Y ) ,  ( ( 1st `  ( ( 1st `  G
) `  X )
) `  W ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  Z )
) `  W )
) ( ( Y ( 2nd `  (
( 1st `  G
) `  X )
) W ) `  S ) ) )

Proof of Theorem uncf2
StepHypRef Expression
1 uncfval.g . . . . . . 7  |-  F  =  ( <" C D E "> uncurryF  G )
2 uncfval.c . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
3 uncfval.d . . . . . . 7  |-  ( ph  ->  E  e.  Cat )
4 uncfval.f . . . . . . 7  |-  ( ph  ->  G  e.  ( C 
Func  ( D FuncCat  E
) ) )
51, 2, 3, 4uncfval 14251 . . . . . 6  |-  ( ph  ->  F  =  ( ( D evalF 
E )  o.func  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) )
65fveq2d 5665 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  =  ( 2nd `  ( ( D evalF  E )  o.func  ( ( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) ) )
76oveqd 6030 . . . 4  |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  F
) <. Z ,  W >. )  =  ( <. X ,  Y >. ( 2nd `  ( ( D evalF 
E )  o.func  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) ) <. Z ,  W >. ) )
87oveqd 6030 . . 3  |-  ( ph  ->  ( R ( <. X ,  Y >. ( 2nd `  F )
<. Z ,  W >. ) S )  =  ( R ( <. X ,  Y >. ( 2nd `  (
( D evalF  E )  o.func  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) ) <. Z ,  W >. ) S ) )
9 df-ov 6016 . . . 4  |-  ( R ( <. X ,  Y >. ( 2nd `  (
( D evalF  E )  o.func  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) ) <. Z ,  W >. ) S )  =  ( ( <. X ,  Y >. ( 2nd `  (
( D evalF  E )  o.func  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) ) <. Z ,  W >. ) `  <. R ,  S >. )
10 eqid 2380 . . . . . 6  |-  ( C  X.c  D )  =  ( C  X.c  D )
11 uncf1.a . . . . . 6  |-  A  =  ( Base `  C
)
12 uncf1.b . . . . . 6  |-  B  =  ( Base `  D
)
1310, 11, 12xpcbas 14195 . . . . 5  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
14 eqid 2380 . . . . . 6  |-  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) )  =  ( ( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
)
15 eqid 2380 . . . . . 6  |-  ( ( D FuncCat  E )  X.c  D )  =  ( ( D FuncCat  E )  X.c  D )
16 funcrcl 13980 . . . . . . . . . 10  |-  ( G  e.  ( C  Func  ( D FuncCat  E ) )  -> 
( C  e.  Cat  /\  ( D FuncCat  E )  e.  Cat ) )
174, 16syl 16 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  Cat  /\  ( D FuncCat  E )  e.  Cat ) )
1817simpld 446 . . . . . . . 8  |-  ( ph  ->  C  e.  Cat )
19 eqid 2380 . . . . . . . 8  |-  ( C  1stF  D )  =  ( C  1stF  D )
2010, 18, 2, 191stfcl 14214 . . . . . . 7  |-  ( ph  ->  ( C  1stF  D )  e.  ( ( C  X.c  D
)  Func  C )
)
2120, 4cofucl 14005 . . . . . 6  |-  ( ph  ->  ( G  o.func  ( C  1stF  D ) )  e.  ( ( C  X.c  D ) 
Func  ( D FuncCat  E
) ) )
22 eqid 2380 . . . . . . 7  |-  ( C  2ndF  D )  =  ( C  2ndF  D )
2310, 18, 2, 222ndfcl 14215 . . . . . 6  |-  ( ph  ->  ( C  2ndF  D )  e.  ( ( C  X.c  D
)  Func  D )
)
2414, 15, 21, 23prfcl 14220 . . . . 5  |-  ( ph  ->  ( ( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
)  e.  ( ( C  X.c  D )  Func  (
( D FuncCat  E )  X.c  D ) ) )
25 eqid 2380 . . . . . 6  |-  ( D evalF  E
)  =  ( D evalF  E
)
26 eqid 2380 . . . . . 6  |-  ( D FuncCat  E )  =  ( D FuncCat  E )
2725, 26, 2, 3evlfcl 14239 . . . . 5  |-  ( ph  ->  ( D evalF  E )  e.  ( ( ( D FuncCat  E
)  X.c  D )  Func  E
) )
28 uncf1.x . . . . . 6  |-  ( ph  ->  X  e.  A )
29 uncf1.y . . . . . 6  |-  ( ph  ->  Y  e.  B )
30 opelxpi 4843 . . . . . 6  |-  ( ( X  e.  A  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( A  X.  B
) )
3128, 29, 30syl2anc 643 . . . . 5  |-  ( ph  -> 
<. X ,  Y >.  e.  ( A  X.  B
) )
32 uncf2.z . . . . . 6  |-  ( ph  ->  Z  e.  A )
33 uncf2.w . . . . . 6  |-  ( ph  ->  W  e.  B )
34 opelxpi 4843 . . . . . 6  |-  ( ( Z  e.  A  /\  W  e.  B )  -> 
<. Z ,  W >.  e.  ( A  X.  B
) )
3532, 33, 34syl2anc 643 . . . . 5  |-  ( ph  -> 
<. Z ,  W >.  e.  ( A  X.  B
) )
36 eqid 2380 . . . . 5  |-  (  Hom  `  ( C  X.c  D ) )  =  (  Hom  `  ( C  X.c  D ) )
37 uncf2.r . . . . . . 7  |-  ( ph  ->  R  e.  ( X H Z ) )
38 uncf2.s . . . . . . 7  |-  ( ph  ->  S  e.  ( Y J W ) )
39 opelxpi 4843 . . . . . . 7  |-  ( ( R  e.  ( X H Z )  /\  S  e.  ( Y J W ) )  ->  <. R ,  S >.  e.  ( ( X H Z )  X.  ( Y J W ) ) )
4037, 38, 39syl2anc 643 . . . . . 6  |-  ( ph  -> 
<. R ,  S >.  e.  ( ( X H Z )  X.  ( Y J W ) ) )
41 uncf2.h . . . . . . 7  |-  H  =  (  Hom  `  C
)
42 uncf2.j . . . . . . 7  |-  J  =  (  Hom  `  D
)
4310, 11, 12, 41, 42, 28, 29, 32, 33, 36xpchom2 14203 . . . . . 6  |-  ( ph  ->  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. )  =  ( ( X H Z )  X.  ( Y J W ) ) )
4440, 43eleqtrrd 2457 . . . . 5  |-  ( ph  -> 
<. R ,  S >.  e.  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) )
4513, 24, 27, 31, 35, 36, 44cofu2 14003 . . . 4  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  (
( D evalF  E )  o.func  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) ) <. Z ,  W >. ) `  <. R ,  S >. )  =  ( ( ( ( 1st `  ( ( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) `  <. X ,  Y >. )
( 2nd `  ( D evalF  E ) ) ( ( 1st `  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) `
 <. Z ,  W >. ) ) `  (
( <. X ,  Y >. ( 2nd `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) <. Z ,  W >. ) `  <. R ,  S >. )
) )
469, 45syl5eq 2424 . . 3  |-  ( ph  ->  ( R ( <. X ,  Y >. ( 2nd `  ( ( D evalF 
E )  o.func  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) ) <. Z ,  W >. ) S )  =  ( ( ( ( 1st `  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) `
 <. X ,  Y >. ) ( 2nd `  ( D evalF  E ) ) ( ( 1st `  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) `
 <. Z ,  W >. ) ) `  (
( <. X ,  Y >. ( 2nd `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) <. Z ,  W >. ) `  <. R ,  S >. )
) )
478, 46eqtrd 2412 . 2  |-  ( ph  ->  ( R ( <. X ,  Y >. ( 2nd `  F )
<. Z ,  W >. ) S )  =  ( ( ( ( 1st `  ( ( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) `  <. X ,  Y >. )
( 2nd `  ( D evalF  E ) ) ( ( 1st `  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) `
 <. Z ,  W >. ) ) `  (
( <. X ,  Y >. ( 2nd `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) <. Z ,  W >. ) `  <. R ,  S >. )
) )
4814, 13, 36, 21, 23, 31prf1 14217 . . . . . 6  |-  ( ph  ->  ( ( 1st `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) `  <. X ,  Y >. )  =  <. ( ( 1st `  ( G  o.func  ( C  1stF  D ) ) ) `  <. X ,  Y >. ) ,  ( ( 1st `  ( C  2ndF  D )
) `  <. X ,  Y >. ) >. )
4913, 20, 4, 31cofu1 14001 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. X ,  Y >. )  =  ( ( 1st `  G ) `  (
( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. ) ) )
5010, 13, 36, 18, 2, 19, 311stf1 14209 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. )  =  ( 1st `  <. X ,  Y >. )
)
51 op1stg 6291 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
5228, 29, 51syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  <. X ,  Y >. )  =  X )
5350, 52eqtrd 2412 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. )  =  X )
5453fveq2d 5665 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. ) )  =  ( ( 1st `  G
) `  X )
)
5549, 54eqtrd 2412 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. X ,  Y >. )  =  ( ( 1st `  G ) `  X
) )
5610, 13, 36, 18, 2, 22, 312ndf1 14212 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. X ,  Y >. )  =  ( 2nd `  <. X ,  Y >. )
)
57 op2ndg 6292 . . . . . . . . 9  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
5828, 29, 57syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
5956, 58eqtrd 2412 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. X ,  Y >. )  =  Y )
6055, 59opeq12d 3927 . . . . . 6  |-  ( ph  -> 
<. ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. X ,  Y >. ) ,  ( ( 1st `  ( C  2ndF  D )
) `  <. X ,  Y >. ) >.  =  <. ( ( 1st `  G
) `  X ) ,  Y >. )
6148, 60eqtrd 2412 . . . . 5  |-  ( ph  ->  ( ( 1st `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) `  <. X ,  Y >. )  =  <. ( ( 1st `  G ) `  X
) ,  Y >. )
6214, 13, 36, 21, 23, 35prf1 14217 . . . . . 6  |-  ( ph  ->  ( ( 1st `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) `  <. Z ,  W >. )  =  <. ( ( 1st `  ( G  o.func  ( C  1stF  D ) ) ) `  <. Z ,  W >. ) ,  ( ( 1st `  ( C  2ndF  D )
) `  <. Z ,  W >. ) >. )
6313, 20, 4, 35cofu1 14001 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. Z ,  W >. )  =  ( ( 1st `  G ) `  (
( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. ) ) )
6410, 13, 36, 18, 2, 19, 351stf1 14209 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. )  =  ( 1st `  <. Z ,  W >. )
)
65 op1stg 6291 . . . . . . . . . . 11  |-  ( ( Z  e.  A  /\  W  e.  B )  ->  ( 1st `  <. Z ,  W >. )  =  Z )
6632, 33, 65syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  <. Z ,  W >. )  =  Z )
6764, 66eqtrd 2412 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. )  =  Z )
6867fveq2d 5665 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  ( ( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. ) )  =  ( ( 1st `  G
) `  Z )
)
6963, 68eqtrd 2412 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. Z ,  W >. )  =  ( ( 1st `  G ) `  Z
) )
7010, 13, 36, 18, 2, 22, 352ndf1 14212 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. Z ,  W >. )  =  ( 2nd `  <. Z ,  W >. )
)
71 op2ndg 6292 . . . . . . . . 9  |-  ( ( Z  e.  A  /\  W  e.  B )  ->  ( 2nd `  <. Z ,  W >. )  =  W )
7232, 33, 71syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. Z ,  W >. )  =  W )
7370, 72eqtrd 2412 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. Z ,  W >. )  =  W )
7469, 73opeq12d 3927 . . . . . 6  |-  ( ph  -> 
<. ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. Z ,  W >. ) ,  ( ( 1st `  ( C  2ndF  D )
) `  <. Z ,  W >. ) >.  =  <. ( ( 1st `  G
) `  Z ) ,  W >. )
7562, 74eqtrd 2412 . . . . 5  |-  ( ph  ->  ( ( 1st `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) `  <. Z ,  W >. )  =  <. ( ( 1st `  G ) `  Z
) ,  W >. )
7661, 75oveq12d 6031 . . . 4  |-  ( ph  ->  ( ( ( 1st `  ( ( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) `  <. X ,  Y >. )
( 2nd `  ( D evalF  E ) ) ( ( 1st `  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) `
 <. Z ,  W >. ) )  =  (
<. ( ( 1st `  G
) `  X ) ,  Y >. ( 2nd `  ( D evalF  E ) ) <. (
( 1st `  G
) `  Z ) ,  W >. ) )
7714, 13, 36, 21, 23, 31, 35, 44prf2 14219 . . . . 5  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) <. Z ,  W >. ) `  <. R ,  S >. )  =  <. ( ( <. X ,  Y >. ( 2nd `  ( G  o.func  ( C  1stF  D )
) ) <. Z ,  W >. ) `  <. R ,  S >. ) ,  ( ( <. X ,  Y >. ( 2nd `  ( C  2ndF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. ) >. )
7813, 20, 4, 31, 35, 36, 44cofu2 14003 . . . . . . 7  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  ( G  o.func  ( C  1stF  D )
) ) <. Z ,  W >. ) `  <. R ,  S >. )  =  ( ( ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. ) ( 2nd `  G
) ( ( 1st `  ( C  1stF  D )
) `  <. Z ,  W >. ) ) `  ( ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. ) ) )
7953, 67oveq12d 6031 . . . . . . . 8  |-  ( ph  ->  ( ( ( 1st `  ( C  1stF  D )
) `  <. X ,  Y >. ) ( 2nd `  G ) ( ( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. ) )  =  ( X ( 2nd `  G
) Z ) )
8010, 13, 36, 18, 2, 19, 31, 351stf2 14210 . . . . . . . . . 10  |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. Z ,  W >. )  =  ( 1st  |`  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) )
8180fveq1d 5663 . . . . . . . . 9  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. )  =  ( ( 1st  |`  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D
) ) <. Z ,  W >. ) ) `  <. R ,  S >. ) )
82 fvres 5678 . . . . . . . . . 10  |-  ( <. R ,  S >.  e.  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. )  ->  ( ( 1st  |`  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) `  <. R ,  S >. )  =  ( 1st `  <. R ,  S >. ) )
8344, 82syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st  |`  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) `  <. R ,  S >. )  =  ( 1st `  <. R ,  S >. ) )
84 op1stg 6291 . . . . . . . . . 10  |-  ( ( R  e.  ( X H Z )  /\  S  e.  ( Y J W ) )  -> 
( 1st `  <. R ,  S >. )  =  R )
8537, 38, 84syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  <. R ,  S >. )  =  R )
8681, 83, 853eqtrd 2416 . . . . . . . 8  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. )  =  R )
8779, 86fveq12d 5667 . . . . . . 7  |-  ( ph  ->  ( ( ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. ) ( 2nd `  G
) ( ( 1st `  ( C  1stF  D )
) `  <. Z ,  W >. ) ) `  ( ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. ) )  =  ( ( X ( 2nd `  G ) Z ) `
 R ) )
8878, 87eqtrd 2412 . . . . . 6  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  ( G  o.func  ( C  1stF  D )
) ) <. Z ,  W >. ) `  <. R ,  S >. )  =  ( ( X ( 2nd `  G
) Z ) `  R ) )
8910, 13, 36, 18, 2, 22, 31, 352ndf2 14213 . . . . . . . 8  |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  ( C  2ndF  D ) ) <. Z ,  W >. )  =  ( 2nd  |`  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) )
9089fveq1d 5663 . . . . . . 7  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  ( C  2ndF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. )  =  ( ( 2nd  |`  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D
) ) <. Z ,  W >. ) ) `  <. R ,  S >. ) )
91 fvres 5678 . . . . . . . 8  |-  ( <. R ,  S >.  e.  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. )  ->  ( ( 2nd  |`  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) `  <. R ,  S >. )  =  ( 2nd `  <. R ,  S >. ) )
9244, 91syl 16 . . . . . . 7  |-  ( ph  ->  ( ( 2nd  |`  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) `  <. R ,  S >. )  =  ( 2nd `  <. R ,  S >. ) )
93 op2ndg 6292 . . . . . . . 8  |-  ( ( R  e.  ( X H Z )  /\  S  e.  ( Y J W ) )  -> 
( 2nd `  <. R ,  S >. )  =  S )
9437, 38, 93syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. R ,  S >. )  =  S )
9590, 92, 943eqtrd 2416 . . . . . 6  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  ( C  2ndF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. )  =  S )
9688, 95opeq12d 3927 . . . . 5  |-  ( ph  -> 
<. ( ( <. X ,  Y >. ( 2nd `  ( G  o.func  ( C  1stF  D )
) ) <. Z ,  W >. ) `  <. R ,  S >. ) ,  ( ( <. X ,  Y >. ( 2nd `  ( C  2ndF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. ) >.  =  <. ( ( X ( 2nd `  G ) Z ) `
 R ) ,  S >. )
9777, 96eqtrd 2412 . . . 4  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) <. Z ,  W >. ) `  <. R ,  S >. )  =  <. ( ( X ( 2nd `  G
) Z ) `  R ) ,  S >. )
9876, 97fveq12d 5667 . . 3  |-  ( ph  ->  ( ( ( ( 1st `  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) `
 <. X ,  Y >. ) ( 2nd `  ( D evalF  E ) ) ( ( 1st `  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) `
 <. Z ,  W >. ) ) `  (
( <. X ,  Y >. ( 2nd `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) <. Z ,  W >. ) `  <. R ,  S >. )
)  =  ( (
<. ( ( 1st `  G
) `  X ) ,  Y >. ( 2nd `  ( D evalF  E ) ) <. (
( 1st `  G
) `  Z ) ,  W >. ) `  <. ( ( X ( 2nd `  G ) Z ) `
 R ) ,  S >. ) )
99 df-ov 6016 . . 3  |-  ( ( ( X ( 2nd `  G ) Z ) `
 R ) (
<. ( ( 1st `  G
) `  X ) ,  Y >. ( 2nd `  ( D evalF  E ) ) <. (
( 1st `  G
) `  Z ) ,  W >. ) S )  =  ( ( <.
( ( 1st `  G
) `  X ) ,  Y >. ( 2nd `  ( D evalF  E ) ) <. (
( 1st `  G
) `  Z ) ,  W >. ) `  <. ( ( X ( 2nd `  G ) Z ) `
 R ) ,  S >. )
10098, 99syl6eqr 2430 . 2  |-  ( ph  ->  ( ( ( ( 1st `  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) `
 <. X ,  Y >. ) ( 2nd `  ( D evalF  E ) ) ( ( 1st `  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) `
 <. Z ,  W >. ) ) `  (
( <. X ,  Y >. ( 2nd `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) <. Z ,  W >. ) `  <. R ,  S >. )
)  =  ( ( ( X ( 2nd `  G ) Z ) `
 R ) (
<. ( ( 1st `  G
) `  X ) ,  Y >. ( 2nd `  ( D evalF  E ) ) <. (
( 1st `  G
) `  Z ) ,  W >. ) S ) )
101 eqid 2380 . . 3  |-  (comp `  E )  =  (comp `  E )
102 eqid 2380 . . 3  |-  ( D Nat 
E )  =  ( D Nat  E )
10326fucbas 14077 . . . . 5  |-  ( D 
Func  E )  =  (
Base `  ( D FuncCat  E ) )
104 relfunc 13979 . . . . . 6  |-  Rel  ( C  Func  ( D FuncCat  E
) )
105 1st2ndbr 6328 . . . . . 6  |-  ( ( Rel  ( C  Func  ( D FuncCat  E ) )  /\  G  e.  ( C  Func  ( D FuncCat  E )
) )  ->  ( 1st `  G ) ( C  Func  ( D FuncCat  E ) ) ( 2nd `  G ) )
106104, 4, 105sylancr 645 . . . . 5  |-  ( ph  ->  ( 1st `  G
) ( C  Func  ( D FuncCat  E ) ) ( 2nd `  G ) )
10711, 103, 106funcf1 13983 . . . 4  |-  ( ph  ->  ( 1st `  G
) : A --> ( D 
Func  E ) )
108107, 28ffvelrnd 5803 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  X )  e.  ( D  Func  E
) )
109107, 32ffvelrnd 5803 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  Z )  e.  ( D  Func  E
) )
110 eqid 2380 . . 3  |-  ( <.
( ( 1st `  G
) `  X ) ,  Y >. ( 2nd `  ( D evalF  E ) ) <. (
( 1st `  G
) `  Z ) ,  W >. )  =  (
<. ( ( 1st `  G
) `  X ) ,  Y >. ( 2nd `  ( D evalF  E ) ) <. (
( 1st `  G
) `  Z ) ,  W >. )
11126, 102fuchom 14078 . . . . 5  |-  ( D Nat 
E )  =  (  Hom  `  ( D FuncCat  E ) )
11211, 41, 111, 106, 28, 32funcf2 13985 . . . 4  |-  ( ph  ->  ( X ( 2nd `  G ) Z ) : ( X H Z ) --> ( ( ( 1st `  G
) `  X )
( D Nat  E ) ( ( 1st `  G
) `  Z )
) )
113112, 37ffvelrnd 5803 . . 3  |-  ( ph  ->  ( ( X ( 2nd `  G ) Z ) `  R
)  e.  ( ( ( 1st `  G
) `  X )
( D Nat  E ) ( ( 1st `  G
) `  Z )
) )
11425, 2, 3, 12, 42, 101, 102, 108, 109, 29, 33, 110, 113, 38evlf2val 14236 . 2  |-  ( ph  ->  ( ( ( X ( 2nd `  G
) Z ) `  R ) ( <.
( ( 1st `  G
) `  X ) ,  Y >. ( 2nd `  ( D evalF  E ) ) <. (
( 1st `  G
) `  Z ) ,  W >. ) S )  =  ( ( ( ( X ( 2nd `  G ) Z ) `
 R ) `  W ) ( <.
( ( 1st `  (
( 1st `  G
) `  X )
) `  Y ) ,  ( ( 1st `  ( ( 1st `  G
) `  X )
) `  W ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  Z )
) `  W )
) ( ( Y ( 2nd `  (
( 1st `  G
) `  X )
) W ) `  S ) ) )
11547, 100, 1143eqtrd 2416 1  |-  ( ph  ->  ( R ( <. X ,  Y >. ( 2nd `  F )
<. Z ,  W >. ) S )  =  ( ( ( ( X ( 2nd `  G
) Z ) `  R ) `  W
) ( <. (
( 1st `  (
( 1st `  G
) `  X )
) `  Y ) ,  ( ( 1st `  ( ( 1st `  G
) `  X )
) `  W ) >. (comp `  E )
( ( 1st `  (
( 1st `  G
) `  Z )
) `  W )
) ( ( Y ( 2nd `  (
( 1st `  G
) `  X )
) W ) `  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3753   class class class wbr 4146    X. cxp 4809    |` cres 4813   Rel wrel 4816   ` cfv 5387  (class class class)co 6013   1stc1st 6279   2ndc2nd 6280   <"cs3 11726   Basecbs 13389    Hom chom 13460  compcco 13461   Catccat 13809    Func cfunc 13971    o.func ccofu 13973   Nat cnat 14058   FuncCat cfuc 14059    X.c cxpc 14185    1stF c1stf 14186    2ndF c2ndf 14187   ⟨,⟩F cprf 14188   evalF cevlf 14226   uncurryF cuncf 14228
This theorem is referenced by:  curfuncf  14255  uncfcurf  14256
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-fz 10969  df-fzo 11059  df-hash 11539  df-word 11643  df-concat 11644  df-s1 11645  df-s2 11732  df-s3 11733  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-hom 13473  df-cco 13474  df-cat 13813  df-cid 13814  df-func 13975  df-cofu 13977  df-nat 14060  df-fuc 14061  df-xpc 14189  df-1stf 14190  df-2ndf 14191  df-prf 14192  df-evlf 14230  df-uncf 14232
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