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Theorem uncf2 14339
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 14336 . . . . . 6  |-  ( ph  ->  F  =  ( ( D evalF 
E )  o.func  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) )
65fveq2d 5735 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  =  ( 2nd `  ( ( D evalF  E )  o.func  ( ( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) ) )
76oveqd 6101 . . . 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 6101 . . 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 6087 . . . 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 2438 . . . . . 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 14280 . . . . 5  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
14 eqid 2438 . . . . . 6  |-  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) )  =  ( ( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
)
15 eqid 2438 . . . . . 6  |-  ( ( D FuncCat  E )  X.c  D )  =  ( ( D FuncCat  E )  X.c  D )
16 funcrcl 14065 . . . . . . . . . 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 447 . . . . . . . 8  |-  ( ph  ->  C  e.  Cat )
19 eqid 2438 . . . . . . . 8  |-  ( C  1stF  D )  =  ( C  1stF  D )
2010, 18, 2, 191stfcl 14299 . . . . . . 7  |-  ( ph  ->  ( C  1stF  D )  e.  ( ( C  X.c  D
)  Func  C )
)
2120, 4cofucl 14090 . . . . . 6  |-  ( ph  ->  ( G  o.func  ( C  1stF  D ) )  e.  ( ( C  X.c  D ) 
Func  ( D FuncCat  E
) ) )
22 eqid 2438 . . . . . . 7  |-  ( C  2ndF  D )  =  ( C  2ndF  D )
2310, 18, 2, 222ndfcl 14300 . . . . . 6  |-  ( ph  ->  ( C  2ndF  D )  e.  ( ( C  X.c  D
)  Func  D )
)
2414, 15, 21, 23prfcl 14305 . . . . 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 2438 . . . . . 6  |-  ( D evalF  E
)  =  ( D evalF  E
)
26 eqid 2438 . . . . . 6  |-  ( D FuncCat  E )  =  ( D FuncCat  E )
2725, 26, 2, 3evlfcl 14324 . . . . 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 4913 . . . . . 6  |-  ( ( X  e.  A  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( A  X.  B
) )
3128, 29, 30syl2anc 644 . . . . 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 4913 . . . . . 6  |-  ( ( Z  e.  A  /\  W  e.  B )  -> 
<. Z ,  W >.  e.  ( A  X.  B
) )
3532, 33, 34syl2anc 644 . . . . 5  |-  ( ph  -> 
<. Z ,  W >.  e.  ( A  X.  B
) )
36 eqid 2438 . . . . 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 4913 . . . . . . 7  |-  ( ( R  e.  ( X H Z )  /\  S  e.  ( Y J W ) )  ->  <. R ,  S >.  e.  ( ( X H Z )  X.  ( Y J W ) ) )
4037, 38, 39syl2anc 644 . . . . . 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 14288 . . . . . 6  |-  ( ph  ->  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. )  =  ( ( X H Z )  X.  ( Y J W ) ) )
4440, 43eleqtrrd 2515 . . . . 5  |-  ( ph  -> 
<. R ,  S >.  e.  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) )
4513, 24, 27, 31, 35, 36, 44cofu2 14088 . . . 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 2482 . . 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 2470 . 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 14302 . . . . . 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 14086 . . . . . . . 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 14294 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. )  =  ( 1st `  <. X ,  Y >. )
)
51 op1stg 6362 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
5228, 29, 51syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  <. X ,  Y >. )  =  X )
5350, 52eqtrd 2470 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. )  =  X )
5453fveq2d 5735 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. ) )  =  ( ( 1st `  G
) `  X )
)
5549, 54eqtrd 2470 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. X ,  Y >. )  =  ( ( 1st `  G ) `  X
) )
5610, 13, 36, 18, 2, 22, 312ndf1 14297 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. X ,  Y >. )  =  ( 2nd `  <. X ,  Y >. )
)
57 op2ndg 6363 . . . . . . . . 9  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
5828, 29, 57syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
5956, 58eqtrd 2470 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. X ,  Y >. )  =  Y )
6055, 59opeq12d 3994 . . . . . 6  |-  ( ph  -> 
<. ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. X ,  Y >. ) ,  ( ( 1st `  ( C  2ndF  D )
) `  <. X ,  Y >. ) >.  =  <. ( ( 1st `  G
) `  X ) ,  Y >. )
6148, 60eqtrd 2470 . . . . 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 14302 . . . . . 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 14086 . . . . . . . 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 14294 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. )  =  ( 1st `  <. Z ,  W >. )
)
65 op1stg 6362 . . . . . . . . . . 11  |-  ( ( Z  e.  A  /\  W  e.  B )  ->  ( 1st `  <. Z ,  W >. )  =  Z )
6632, 33, 65syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  <. Z ,  W >. )  =  Z )
6764, 66eqtrd 2470 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. )  =  Z )
6867fveq2d 5735 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  ( ( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. ) )  =  ( ( 1st `  G
) `  Z )
)
6963, 68eqtrd 2470 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. Z ,  W >. )  =  ( ( 1st `  G ) `  Z
) )
7010, 13, 36, 18, 2, 22, 352ndf1 14297 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. Z ,  W >. )  =  ( 2nd `  <. Z ,  W >. )
)
71 op2ndg 6363 . . . . . . . . 9  |-  ( ( Z  e.  A  /\  W  e.  B )  ->  ( 2nd `  <. Z ,  W >. )  =  W )
7232, 33, 71syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. Z ,  W >. )  =  W )
7370, 72eqtrd 2470 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. Z ,  W >. )  =  W )
7469, 73opeq12d 3994 . . . . . 6  |-  ( ph  -> 
<. ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. Z ,  W >. ) ,  ( ( 1st `  ( C  2ndF  D )
) `  <. Z ,  W >. ) >.  =  <. ( ( 1st `  G
) `  Z ) ,  W >. )
7562, 74eqtrd 2470 . . . . 5  |-  ( ph  ->  ( ( 1st `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) `  <. Z ,  W >. )  =  <. ( ( 1st `  G ) `  Z
) ,  W >. )
7661, 75oveq12d 6102 . . . 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 14304 . . . . 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 14088 . . . . . . 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 6102 . . . . . . . 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 14295 . . . . . . . . . 10  |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. Z ,  W >. )  =  ( 1st  |`  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) )
8180fveq1d 5733 . . . . . . . . 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 5748 . . . . . . . . . 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 6362 . . . . . . . . . 10  |-  ( ( R  e.  ( X H Z )  /\  S  e.  ( Y J W ) )  -> 
( 1st `  <. R ,  S >. )  =  R )
8537, 38, 84syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  <. R ,  S >. )  =  R )
8681, 83, 853eqtrd 2474 . . . . . . . 8  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. )  =  R )
8779, 86fveq12d 5737 . . . . . . 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 2470 . . . . . 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 14298 . . . . . . . 8  |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  ( C  2ndF  D ) ) <. Z ,  W >. )  =  ( 2nd  |`  ( <. X ,  Y >. (  Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) )
9089fveq1d 5733 . . . . . . 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 5748 . . . . . . . 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 6363 . . . . . . . 8  |-  ( ( R  e.  ( X H Z )  /\  S  e.  ( Y J W ) )  -> 
( 2nd `  <. R ,  S >. )  =  S )
9437, 38, 93syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. R ,  S >. )  =  S )
9590, 92, 943eqtrd 2474 . . . . . 6  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  ( C  2ndF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. )  =  S )
9688, 95opeq12d 3994 . . . . 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 2470 . . . 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 5737 . . 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 6087 . . 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 2488 . 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 2438 . . 3  |-  (comp `  E )  =  (comp `  E )
102 eqid 2438 . . 3  |-  ( D Nat 
E )  =  ( D Nat  E )
10326fucbas 14162 . . . . 5  |-  ( D 
Func  E )  =  (
Base `  ( D FuncCat  E ) )
104 relfunc 14064 . . . . . 6  |-  Rel  ( C  Func  ( D FuncCat  E
) )
105 1st2ndbr 6399 . . . . . 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 646 . . . . 5  |-  ( ph  ->  ( 1st `  G
) ( C  Func  ( D FuncCat  E ) ) ( 2nd `  G ) )
10711, 103, 106funcf1 14068 . . . 4  |-  ( ph  ->  ( 1st `  G
) : A --> ( D 
Func  E ) )
108107, 28ffvelrnd 5874 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  X )  e.  ( D  Func  E
) )
109107, 32ffvelrnd 5874 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  Z )  e.  ( D  Func  E
) )
110 eqid 2438 . . 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 14163 . . . . 5  |-  ( D Nat 
E )  =  (  Hom  `  ( D FuncCat  E ) )
11211, 41, 111, 106, 28, 32funcf2 14070 . . . 4  |-  ( ph  ->  ( X ( 2nd `  G ) Z ) : ( X H Z ) --> ( ( ( 1st `  G
) `  X )
( D Nat  E ) ( ( 1st `  G
) `  Z )
) )
113112, 37ffvelrnd 5874 . . 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 14321 . 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 2474 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 360    = wceq 1653    e. wcel 1726   <.cop 3819   class class class wbr 4215    X. cxp 4879    |` cres 4883   Rel wrel 4886   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   <"cs3 11811   Basecbs 13474    Hom chom 13545  compcco 13546   Catccat 13894    Func cfunc 14056    o.func ccofu 14058   Nat cnat 14143   FuncCat cfuc 14144    X.c cxpc 14270    1stF c1stf 14271    2ndF c2ndf 14272   ⟨,⟩F cprf 14273   evalF cevlf 14311   uncurryF cuncf 14313
This theorem is referenced by:  curfuncf  14340  uncfcurf  14341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-fzo 11141  df-hash 11624  df-word 11728  df-concat 11729  df-s1 11730  df-s2 11817  df-s3 11818  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-hom 13558  df-cco 13559  df-cat 13898  df-cid 13899  df-func 14060  df-cofu 14062  df-nat 14145  df-fuc 14146  df-xpc 14274  df-1stf 14275  df-2ndf 14276  df-prf 14277  df-evlf 14315  df-uncf 14317
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