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Theorem uncf2 15360
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 15357 . . . . . 6  |-  ( ph  ->  F  =  ( ( D evalF 
E )  o.func  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) ) ) )
65fveq2d 5868 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  =  ( 2nd `  ( ( D evalF  E )  o.func  ( ( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) ) )
76oveqd 6299 . . . 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 6299 . . 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 6285 . . . 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 2467 . . . . . 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 15301 . . . . 5  |-  ( A  X.  B )  =  ( Base `  ( C  X.c  D ) )
14 eqid 2467 . . . . . 6  |-  ( ( G  o.func  ( C  1stF  D )
) ⟨,⟩F  ( C  2ndF  D ) )  =  ( ( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
)
15 eqid 2467 . . . . . 6  |-  ( ( D FuncCat  E )  X.c  D )  =  ( ( D FuncCat  E )  X.c  D )
16 funcrcl 15086 . . . . . . . . . 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 459 . . . . . . . 8  |-  ( ph  ->  C  e.  Cat )
19 eqid 2467 . . . . . . . 8  |-  ( C  1stF  D )  =  ( C  1stF  D )
2010, 18, 2, 191stfcl 15320 . . . . . . 7  |-  ( ph  ->  ( C  1stF  D )  e.  ( ( C  X.c  D
)  Func  C )
)
2120, 4cofucl 15111 . . . . . 6  |-  ( ph  ->  ( G  o.func  ( C  1stF  D ) )  e.  ( ( C  X.c  D ) 
Func  ( D FuncCat  E
) ) )
22 eqid 2467 . . . . . . 7  |-  ( C  2ndF  D )  =  ( C  2ndF  D )
2310, 18, 2, 222ndfcl 15321 . . . . . 6  |-  ( ph  ->  ( C  2ndF  D )  e.  ( ( C  X.c  D
)  Func  D )
)
2414, 15, 21, 23prfcl 15326 . . . . 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 2467 . . . . . 6  |-  ( D evalF  E
)  =  ( D evalF  E
)
26 eqid 2467 . . . . . 6  |-  ( D FuncCat  E )  =  ( D FuncCat  E )
2725, 26, 2, 3evlfcl 15345 . . . . 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 5030 . . . . . 6  |-  ( ( X  e.  A  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( A  X.  B
) )
3128, 29, 30syl2anc 661 . . . . 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 5030 . . . . . 6  |-  ( ( Z  e.  A  /\  W  e.  B )  -> 
<. Z ,  W >.  e.  ( A  X.  B
) )
3532, 33, 34syl2anc 661 . . . . 5  |-  ( ph  -> 
<. Z ,  W >.  e.  ( A  X.  B
) )
36 eqid 2467 . . . . 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 5030 . . . . . . 7  |-  ( ( R  e.  ( X H Z )  /\  S  e.  ( Y J W ) )  ->  <. R ,  S >.  e.  ( ( X H Z )  X.  ( Y J W ) ) )
4037, 38, 39syl2anc 661 . . . . . 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 15309 . . . . . 6  |-  ( ph  ->  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. Z ,  W >. )  =  ( ( X H Z )  X.  ( Y J W ) ) )
4440, 43eleqtrrd 2558 . . . . 5  |-  ( ph  -> 
<. R ,  S >.  e.  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) )
4513, 24, 27, 31, 35, 36, 44cofu2 15109 . . . 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 2520 . . 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 2508 . 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 15323 . . . . . 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 15107 . . . . . . . 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 15315 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. )  =  ( 1st `  <. X ,  Y >. )
)
51 op1stg 6793 . . . . . . . . . . 11  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
5228, 29, 51syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  <. X ,  Y >. )  =  X )
5350, 52eqtrd 2508 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. )  =  X )
5453fveq2d 5868 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  ( ( 1st `  ( C  1stF  D ) ) `  <. X ,  Y >. ) )  =  ( ( 1st `  G
) `  X )
)
5549, 54eqtrd 2508 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. X ,  Y >. )  =  ( ( 1st `  G ) `  X
) )
5610, 13, 36, 18, 2, 22, 312ndf1 15318 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. X ,  Y >. )  =  ( 2nd `  <. X ,  Y >. )
)
57 op2ndg 6794 . . . . . . . . 9  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
5828, 29, 57syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
5956, 58eqtrd 2508 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. X ,  Y >. )  =  Y )
6055, 59opeq12d 4221 . . . . . 6  |-  ( ph  -> 
<. ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. X ,  Y >. ) ,  ( ( 1st `  ( C  2ndF  D )
) `  <. X ,  Y >. ) >.  =  <. ( ( 1st `  G
) `  X ) ,  Y >. )
6148, 60eqtrd 2508 . . . . 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 15323 . . . . . 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 15107 . . . . . . . 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 15315 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. )  =  ( 1st `  <. Z ,  W >. )
)
65 op1stg 6793 . . . . . . . . . . 11  |-  ( ( Z  e.  A  /\  W  e.  B )  ->  ( 1st `  <. Z ,  W >. )  =  Z )
6632, 33, 65syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  <. Z ,  W >. )  =  Z )
6764, 66eqtrd 2508 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. )  =  Z )
6867fveq2d 5868 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  G
) `  ( ( 1st `  ( C  1stF  D ) ) `  <. Z ,  W >. ) )  =  ( ( 1st `  G
) `  Z )
)
6963, 68eqtrd 2508 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. Z ,  W >. )  =  ( ( 1st `  G ) `  Z
) )
7010, 13, 36, 18, 2, 22, 352ndf1 15318 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. Z ,  W >. )  =  ( 2nd `  <. Z ,  W >. )
)
71 op2ndg 6794 . . . . . . . . 9  |-  ( ( Z  e.  A  /\  W  e.  B )  ->  ( 2nd `  <. Z ,  W >. )  =  W )
7232, 33, 71syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. Z ,  W >. )  =  W )
7370, 72eqtrd 2508 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( C  2ndF  D ) ) `  <. Z ,  W >. )  =  W )
7469, 73opeq12d 4221 . . . . . 6  |-  ( ph  -> 
<. ( ( 1st `  ( G  o.func  ( C  1stF  D )
) ) `  <. Z ,  W >. ) ,  ( ( 1st `  ( C  2ndF  D )
) `  <. Z ,  W >. ) >.  =  <. ( ( 1st `  G
) `  Z ) ,  W >. )
7562, 74eqtrd 2508 . . . . 5  |-  ( ph  ->  ( ( 1st `  (
( G  o.func  ( C  1stF  D ) ) ⟨,⟩F  ( C  2ndF  D )
) ) `  <. Z ,  W >. )  =  <. ( ( 1st `  G ) `  Z
) ,  W >. )
7661, 75oveq12d 6300 . . . 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 15325 . . . . 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 15109 . . . . . . 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 6300 . . . . . . . 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 15316 . . . . . . . . . 10  |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. Z ,  W >. )  =  ( 1st  |`  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) )
8180fveq1d 5866 . . . . . . . . 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 5878 . . . . . . . . . 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 6793 . . . . . . . . . 10  |-  ( ( R  e.  ( X H Z )  /\  S  e.  ( Y J W ) )  -> 
( 1st `  <. R ,  S >. )  =  R )
8537, 38, 84syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  <. R ,  S >. )  =  R )
8681, 83, 853eqtrd 2512 . . . . . . . 8  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  ( C  1stF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. )  =  R )
8779, 86fveq12d 5870 . . . . . . 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 2508 . . . . . 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 15319 . . . . . . . 8  |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  ( C  2ndF  D ) ) <. Z ,  W >. )  =  ( 2nd  |`  ( <. X ,  Y >. ( Hom  `  ( C  X.c  D ) ) <. Z ,  W >. ) ) )
9089fveq1d 5866 . . . . . . 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 5878 . . . . . . . 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 6794 . . . . . . . 8  |-  ( ( R  e.  ( X H Z )  /\  S  e.  ( Y J W ) )  -> 
( 2nd `  <. R ,  S >. )  =  S )
9437, 38, 93syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. R ,  S >. )  =  S )
9590, 92, 943eqtrd 2512 . . . . . 6  |-  ( ph  ->  ( ( <. X ,  Y >. ( 2nd `  ( C  2ndF  D ) ) <. Z ,  W >. ) `
 <. R ,  S >. )  =  S )
9688, 95opeq12d 4221 . . . . 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 2508 . . . 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 5870 . . 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 6285 . . 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 2526 . 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 2467 . . 3  |-  (comp `  E )  =  (comp `  E )
102 eqid 2467 . . 3  |-  ( D Nat 
E )  =  ( D Nat  E )
10326fucbas 15183 . . . . 5  |-  ( D 
Func  E )  =  (
Base `  ( D FuncCat  E ) )
104 relfunc 15085 . . . . . 6  |-  Rel  ( C  Func  ( D FuncCat  E
) )
105 1st2ndbr 6830 . . . . . 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 663 . . . . 5  |-  ( ph  ->  ( 1st `  G
) ( C  Func  ( D FuncCat  E ) ) ( 2nd `  G ) )
10711, 103, 106funcf1 15089 . . . 4  |-  ( ph  ->  ( 1st `  G
) : A --> ( D 
Func  E ) )
108107, 28ffvelrnd 6020 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  X )  e.  ( D  Func  E
) )
109107, 32ffvelrnd 6020 . . 3  |-  ( ph  ->  ( ( 1st `  G
) `  Z )  e.  ( D  Func  E
) )
110 eqid 2467 . . 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 15184 . . . . 5  |-  ( D Nat 
E )  =  ( Hom  `  ( D FuncCat  E ) )
11211, 41, 111, 106, 28, 32funcf2 15091 . . . 4  |-  ( ph  ->  ( X ( 2nd `  G ) Z ) : ( X H Z ) --> ( ( ( 1st `  G
) `  X )
( D Nat  E ) ( ( 1st `  G
) `  Z )
) )
113112, 37ffvelrnd 6020 . . 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 15342 . 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 2512 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 369    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447    X. cxp 4997    |` cres 5001   Rel wrel 5004   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   <"cs3 12766   Basecbs 14486   Hom chom 14562  compcco 14563   Catccat 14915    Func cfunc 15077    o.func ccofu 15079   Nat cnat 15164   FuncCat cfuc 15165    X.c cxpc 15291    1stF c1stf 15292    2ndF c2ndf 15293   ⟨,⟩F cprf 15294   evalF cevlf 15332   uncurryF cuncf 15334
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-card 8316  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-fzo 11789  df-hash 12370  df-word 12504  df-concat 12506  df-s1 12507  df-s2 12772  df-s3 12773  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-cofu 15083  df-nat 15166  df-fuc 15167  df-xpc 15295  df-1stf 15296  df-2ndf 15297  df-prf 15298  df-evlf 15336  df-uncf 15338
This theorem is referenced by:  curfuncf  15361  uncfcurf  15362
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