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Theorem uncf2 15720
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 15717 . . . . . 6  |-  ( ph  ->  F  =  ( ( D evalF 
E )  o.func  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) )
65fveq2d 5807 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  =  ( 2nd `  ( ( D evalF  E )  o.func  ( ( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) ) )
76oveqd 6249 . . . 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 6249 . . 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 6235 . . . 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 2400 . . . . . 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 15661 . . . . 5  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
14 eqid 2400 . . . . . 6  |-  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) )  =  ( ( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
)
15 eqid 2400 . . . . . 6  |-  ( ( D FuncCat  E )  X.c  D )  =  ( ( D FuncCat  E )  X.c  D )
16 funcrcl 15366 . . . . . . . . . 10  |-  ( G  e.  ( C  Func  ( D FuncCat  E ) )  -> 
( C  e.  Cat  /\  ( D FuncCat  E )  e.  Cat ) )
174, 16syl 17 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  Cat  /\  ( D FuncCat  E )  e.  Cat ) )
1817simpld 457 . . . . . . . 8  |-  ( ph  ->  C  e.  Cat )
19 eqid 2400 . . . . . . . 8  |-  ( C  1stF  D )  =  ( C  1stF  D )
2010, 18, 2, 191stfcl 15680 . . . . . . 7  |-  ( ph  ->  ( C  1stF  D )  e.  ( ( C  X.c  D
)  Func  C )
)
2120, 4cofucl 15391 . . . . . 6  |-  ( ph  ->  ( G  o.func  ( C  1stF  D ) )  e.  ( ( C  X.c  D ) 
Func  ( D FuncCat  E
) ) )
22 eqid 2400 . . . . . . 7  |-  ( C  2ndF  D )  =  ( C  2ndF  D )
2310, 18, 2, 222ndfcl 15681 . . . . . 6  |-  ( ph  ->  ( C  2ndF  D )  e.  ( ( C  X.c  D
)  Func  D )
)
2414, 15, 21, 23prfcl 15686 . . . . 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 2400 . . . . . 6  |-  ( D evalF  E
)  =  ( D evalF  E
)
26 eqid 2400 . . . . . 6  |-  ( D FuncCat  E )  =  ( D FuncCat  E )
2725, 26, 2, 3evlfcl 15705 . . . . 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 4972 . . . . . 6  |-  ( ( X  e.  A  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( A  X.  B
) )
3128, 29, 30syl2anc 659 . . . . 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 4972 . . . . . 6  |-  ( ( Z  e.  A  /\  W  e.  B )  -> 
<. Z ,  W >.  e.  ( A  X.  B
) )
3532, 33, 34syl2anc 659 . . . . 5  |-  ( ph  -> 
<. Z ,  W >.  e.  ( A  X.  B
) )
36 eqid 2400 . . . . 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 4972 . . . . . . 7  |-  ( ( R  e.  ( X H Z )  /\  S  e.  ( Y J W ) )  ->  <. R ,  S >.  e.  ( ( X H Z )  X.  ( Y J W ) ) )
4037, 38, 39syl2anc 659 . . . . . 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 15669 . . . . . 6  |-  ( ph  ->  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. Z ,  W >. )  =  ( ( X H Z )  X.  ( Y J W ) ) )
4440, 43eleqtrrd 2491 . . . . 5  |-  ( ph  -> 
<. R ,  S >.  e.  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) )
4513, 24, 27, 31, 35, 36, 44cofu2 15389 . . . 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 2453 . . 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 2441 . 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 15683 . . . . . 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 15387 . . . . . . . 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 15675 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. )  =  ( 1st `  <. X ,  Y >. )
)
51 op1stg 6748 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
5228, 29, 51syl2anc 659 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  <. X ,  Y >. )  =  X )
5350, 52eqtrd 2441 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. )  =  X )
5453fveq2d 5807 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. ) )  =  ( ( 1st `  G
) `  X )
)
5549, 54eqtrd 2441 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. X ,  Y >. )  =  ( ( 1st `  G ) `  X
) )
5610, 13, 36, 18, 2, 22, 312ndf1 15678 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. X ,  Y >. )  =  ( 2nd `  <. X ,  Y >. )
)
57 op2ndg 6749 . . . . . . . . 9  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
5828, 29, 57syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
5956, 58eqtrd 2441 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. X ,  Y >. )  =  Y )
6055, 59opeq12d 4164 . . . . . 6  |-  ( ph  -> 
<. ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. X ,  Y >. ) ,  ( ( 1st `  ( C  2ndF  D )
) `  <. X ,  Y >. ) >.  =  <. ( ( 1st `  G
) `  X ) ,  Y >. )
6148, 60eqtrd 2441 . . . . 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 15683 . . . . . 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 15387 . . . . . . . 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 15675 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. )  =  ( 1st `  <. Z ,  W >. )
)
65 op1stg 6748 . . . . . . . . . . 11  |-  ( ( Z  e.  A  /\  W  e.  B )  ->  ( 1st `  <. Z ,  W >. )  =  Z )
6632, 33, 65syl2anc 659 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  <. Z ,  W >. )  =  Z )
6764, 66eqtrd 2441 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. )  =  Z )
6867fveq2d 5807 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  ( ( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. ) )  =  ( ( 1st `  G
) `  Z )
)
6963, 68eqtrd 2441 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. Z ,  W >. )  =  ( ( 1st `  G ) `  Z
) )
7010, 13, 36, 18, 2, 22, 352ndf1 15678 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. Z ,  W >. )  =  ( 2nd `  <. Z ,  W >. )
)
71 op2ndg 6749 . . . . . . . . 9  |-  ( ( Z  e.  A  /\  W  e.  B )  ->  ( 2nd `  <. Z ,  W >. )  =  W )
7232, 33, 71syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. Z ,  W >. )  =  W )
7370, 72eqtrd 2441 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. Z ,  W >. )  =  W )
7469, 73opeq12d 4164 . . . . . 6  |-  ( ph  -> 
<. ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. Z ,  W >. ) ,  ( ( 1st `  ( C  2ndF  D )
) `  <. Z ,  W >. ) >.  =  <. ( ( 1st `  G
) `  Z ) ,  W >. )
7562, 74eqtrd 2441 . . . . 5  |-  ( ph  ->  ( ( 1st `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) `  <. Z ,  W >. )  =  <. ( ( 1st `  G ) `  Z
) ,  W >. )
7661, 75oveq12d 6250 . . . 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 15685 . . . . 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 15389 . . . . . . 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 6250 . . . . . . . 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 15676 . . . . . . . . . 10  |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. Z ,  W >. )  =  ( 1st  |`  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) )
8180fveq1d 5805 . . . . . . . . 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 5817 . . . . . . . . . 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 17 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st  |`  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) `  <. R ,  S >. )  =  ( 1st `  <. R ,  S >. ) )
84 op1stg 6748 . . . . . . . . . 10  |-  ( ( R  e.  ( X H Z )  /\  S  e.  ( Y J W ) )  -> 
( 1st `  <. R ,  S >. )  =  R )
8537, 38, 84syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  <. R ,  S >. )  =  R )
8681, 83, 853eqtrd 2445 . . . . . . . 8  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. )  =  R )
8779, 86fveq12d 5809 . . . . . . 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 2441 . . . . . 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 15679 . . . . . . . 8  |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  ( C  2ndF  D ) ) <. Z ,  W >. )  =  ( 2nd  |`  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) )
9089fveq1d 5805 . . . . . . 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 5817 . . . . . . . 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 17 . . . . . . 7  |-  ( ph  ->  ( ( 2nd  |`  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) `  <. R ,  S >. )  =  ( 2nd `  <. R ,  S >. ) )
93 op2ndg 6749 . . . . . . . 8  |-  ( ( R  e.  ( X H Z )  /\  S  e.  ( Y J W ) )  -> 
( 2nd `  <. R ,  S >. )  =  S )
9437, 38, 93syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. R ,  S >. )  =  S )
9590, 92, 943eqtrd 2445 . . . . . 6  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  ( C  2ndF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. )  =  S )
9688, 95opeq12d 4164 . . . . 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 2441 . . . 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 5809 . . 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 6235 . . 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 2459 . 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 2400 . . 3  |-  (comp `  E )  =  (comp `  E )
102 eqid 2400 . . 3  |-  ( D Nat 
E )  =  ( D Nat  E )
10326fucbas 15463 . . . . 5  |-  ( D 
Func  E )  =  (
Base `  ( D FuncCat  E ) )
104 relfunc 15365 . . . . . 6  |-  Rel  ( C  Func  ( D FuncCat  E
) )
105 1st2ndbr 6785 . . . . . 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 661 . . . . 5  |-  ( ph  ->  ( 1st `  G
) ( C  Func  ( D FuncCat  E ) ) ( 2nd `  G ) )
10711, 103, 106funcf1 15369 . . . 4  |-  ( ph  ->  ( 1st `  G
) : A --> ( D 
Func  E ) )
108107, 28ffvelrnd 5964 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  X )  e.  ( D  Func  E
) )
109107, 32ffvelrnd 5964 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  Z )  e.  ( D  Func  E
) )
110 eqid 2400 . . 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 15464 . . . . 5  |-  ( D Nat 
E )  =  ( Hom  `  ( D FuncCat  E ) )
11211, 41, 111, 106, 28, 32funcf2 15371 . . . 4  |-  ( ph  ->  ( X ( 2nd `  G ) Z ) : ( X H Z ) --> ( ( ( 1st `  G
) `  X )
( D Nat  E ) ( ( 1st `  G
) `  Z )
) )
113112, 37ffvelrnd 5964 . . 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 15702 . 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 2445 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840   <.cop 3975   class class class wbr 4392    X. cxp 4938    |` cres 4942   Rel wrel 4945   ` cfv 5523  (class class class)co 6232   1stc1st 6734   2ndc2nd 6735   <"cs3 12768   Basecbs 14731   Hom chom 14810  compcco 14811   Catccat 15168    Func cfunc 15357    o.func ccofu 15359   Nat cnat 15444   FuncCat cfuc 15445    X.c cxpc 15651    1stF c1stf 15652    2ndF c2ndf 15653   ⟨,⟩F cprf 15654   evalF cevlf 15692   uncurryF cuncf 15694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-ixp 7426  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-fz 11642  df-fzo 11766  df-hash 12358  df-word 12496  df-concat 12498  df-s1 12499  df-s2 12774  df-s3 12775  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-hom 14823  df-cco 14824  df-cat 15172  df-cid 15173  df-func 15361  df-cofu 15363  df-nat 15446  df-fuc 15447  df-xpc 15655  df-1stf 15656  df-2ndf 15657  df-prf 15658  df-evlf 15696  df-uncf 15698
This theorem is referenced by:  curfuncf  15721  uncfcurf  15722
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