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Theorem curf2ndf 14021
Description: As shown in diagval 14014, the currying of the first projection is the diagonal functor. On the other hand, the currying of the second projection is  x  e.  C  |->  ( y  e.  D  |->  y ), which is a constant functor of the identity functor at  D. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
curf2ndf.q  |-  Q  =  ( D FuncCat  D )
curf2ndf.c  |-  ( ph  ->  C  e.  Cat )
curf2ndf.d  |-  ( ph  ->  D  e.  Cat )
Assertion
Ref Expression
curf2ndf  |-  ( ph  ->  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  =  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )

Proof of Theorem curf2ndf
Dummy variables  u  f  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 5861 . . . . . . . . . . 11  |-  ( x ( 1st `  ( C  2ndF  D ) ) y )  =  ( ( 1st `  ( C  2ndF  D ) ) `  <. x ,  y >.
)
2 eqid 2283 . . . . . . . . . . . . 13  |-  ( C  X.c  D )  =  ( C  X.c  D )
3 eqid 2283 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2283 . . . . . . . . . . . . . 14  |-  ( Base `  D )  =  (
Base `  D )
52, 3, 4xpcbas 13952 . . . . . . . . . . . . 13  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  ( C  X.c  D ) )
6 eqid 2283 . . . . . . . . . . . . 13  |-  (  Hom  `  ( C  X.c  D ) )  =  (  Hom  `  ( C  X.c  D ) )
7 curf2ndf.c . . . . . . . . . . . . . . 15  |-  ( ph  ->  C  e.  Cat )
87adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  C  e.  Cat )
98adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  C  e.  Cat )
10 curf2ndf.d . . . . . . . . . . . . . . 15  |-  ( ph  ->  D  e.  Cat )
1110adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
1211adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  D  e.  Cat )
13 eqid 2283 . . . . . . . . . . . . 13  |-  ( C  2ndF  D )  =  ( C  2ndF  D )
14 opelxpi 4721 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) )  ->  <. x ,  y >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
1514adantll 694 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  <. x ,  y >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
162, 5, 6, 9, 12, 13, 152ndf1 13969 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  (
( 1st `  ( C  2ndF  D ) ) `  <. x ,  y >.
)  =  ( 2nd `  <. x ,  y
>. ) )
17 vex 2791 . . . . . . . . . . . . 13  |-  x  e. 
_V
18 vex 2791 . . . . . . . . . . . . 13  |-  y  e. 
_V
1917, 18op2nd 6129 . . . . . . . . . . . 12  |-  ( 2nd `  <. x ,  y
>. )  =  y
2016, 19syl6eq 2331 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  (
( 1st `  ( C  2ndF  D ) ) `  <. x ,  y >.
)  =  y )
211, 20syl5eq 2327 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  (
x ( 1st `  ( C  2ndF  D ) ) y )  =  y )
2221mpteq2dva 4106 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( y  e.  ( Base `  D
)  |->  ( x ( 1st `  ( C  2ndF  D ) ) y ) )  =  ( y  e.  ( Base `  D )  |->  y ) )
23 mptresid 5004 . . . . . . . . 9  |-  ( y  e.  ( Base `  D
)  |->  y )  =  (  _I  |`  ( Base `  D ) )
2422, 23syl6eq 2331 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( y  e.  ( Base `  D
)  |->  ( x ( 1st `  ( C  2ndF  D ) ) y ) )  =  (  _I  |`  ( Base `  D ) ) )
25 df-ov 5861 . . . . . . . . . . . . . . 15  |-  ( ( ( Id `  C
) `  x )
( <. x ,  y
>. ( 2nd `  ( C  2ndF  D ) ) <.
x ,  z >.
) f )  =  ( ( <. x ,  y >. ( 2nd `  ( C  2ndF  D ) ) <. x ,  z
>. ) `  <. (
( Id `  C
) `  x ) ,  f >. )
269ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  C  e.  Cat )
2712ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  D  e.  Cat )
2815ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  <. x ,  y >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
29 simp-4r 743 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  x  e.  ( Base `  C ) )
30 simplr 731 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  -> 
z  e.  ( Base `  D ) )
31 opelxpi 4721 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  ( Base `  C )  /\  z  e.  ( Base `  D
) )  ->  <. x ,  z >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
3229, 30, 31syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  <. x ,  z >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
332, 5, 6, 26, 27, 13, 28, 322ndf2 13970 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  -> 
( <. x ,  y
>. ( 2nd `  ( C  2ndF  D ) ) <.
x ,  z >.
)  =  ( 2nd  |`  ( <. x ,  y
>. (  Hom  `  ( C  X.c  D ) ) <.
x ,  z >.
) ) )
3433fveq1d 5527 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  -> 
( ( <. x ,  y >. ( 2nd `  ( C  2ndF  D ) ) <. x ,  z
>. ) `  <. (
( Id `  C
) `  x ) ,  f >. )  =  ( ( 2nd  |`  ( <. x ,  y
>. (  Hom  `  ( C  X.c  D ) ) <.
x ,  z >.
) ) `  <. ( ( Id `  C
) `  x ) ,  f >. )
)
3525, 34syl5eq 2327 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  -> 
( ( ( Id
`  C ) `  x ) ( <.
x ,  y >.
( 2nd `  ( C  2ndF  D ) ) <.
x ,  z >.
) f )  =  ( ( 2nd  |`  ( <. x ,  y >.
(  Hom  `  ( C  X.c  D ) ) <.
x ,  z >.
) ) `  <. ( ( Id `  C
) `  x ) ,  f >. )
)
36 eqid 2283 . . . . . . . . . . . . . . . . . . . 20  |-  (  Hom  `  C )  =  (  Hom  `  C )
37 eqid 2283 . . . . . . . . . . . . . . . . . . . 20  |-  ( Id
`  C )  =  ( Id `  C
)
38 simpr 447 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
393, 36, 37, 8, 38catidcl 13584 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  C ) `  x )  e.  ( x (  Hom  `  C
) x ) )
4039ad3antrrr 710 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  -> 
( ( Id `  C ) `  x
)  e.  ( x (  Hom  `  C
) x ) )
41 simpr 447 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  -> 
f  e.  ( y (  Hom  `  D
) z ) )
42 opelxpi 4721 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( Id `  C ) `  x
)  e.  ( x (  Hom  `  C
) x )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  <. ( ( Id `  C ) `  x
) ,  f >.  e.  ( ( x (  Hom  `  C )
x )  X.  (
y (  Hom  `  D
) z ) ) )
4340, 41, 42syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  <. ( ( Id `  C ) `  x
) ,  f >.  e.  ( ( x (  Hom  `  C )
x )  X.  (
y (  Hom  `  D
) z ) ) )
44 eqid 2283 . . . . . . . . . . . . . . . . . 18  |-  (  Hom  `  D )  =  (  Hom  `  D )
4538ad3antrrr 710 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  x  e.  ( Base `  C ) )
46 simpllr 735 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  -> 
y  e.  ( Base `  D ) )
472, 3, 4, 36, 44, 45, 46, 45, 30, 6xpchom2 13960 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  -> 
( <. x ,  y
>. (  Hom  `  ( C  X.c  D ) ) <.
x ,  z >.
)  =  ( ( x (  Hom  `  C
) x )  X.  ( y (  Hom  `  D ) z ) ) )
4843, 47eleqtrrd 2360 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  <. ( ( Id `  C ) `  x
) ,  f >.  e.  ( <. x ,  y
>. (  Hom  `  ( C  X.c  D ) ) <.
x ,  z >.
) )
49 fvres 5542 . . . . . . . . . . . . . . . 16  |-  ( <.
( ( Id `  C ) `  x
) ,  f >.  e.  ( <. x ,  y
>. (  Hom  `  ( C  X.c  D ) ) <.
x ,  z >.
)  ->  ( ( 2nd  |`  ( <. x ,  y >. (  Hom  `  ( C  X.c  D
) ) <. x ,  z >. )
) `  <. ( ( Id `  C ) `
 x ) ,  f >. )  =  ( 2nd `  <. (
( Id `  C
) `  x ) ,  f >. )
)
5048, 49syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  -> 
( ( 2nd  |`  ( <. x ,  y >.
(  Hom  `  ( C  X.c  D ) ) <.
x ,  z >.
) ) `  <. ( ( Id `  C
) `  x ) ,  f >. )  =  ( 2nd `  <. ( ( Id `  C
) `  x ) ,  f >. )
)
51 fvex 5539 . . . . . . . . . . . . . . . 16  |-  ( ( Id `  C ) `
 x )  e. 
_V
52 vex 2791 . . . . . . . . . . . . . . . 16  |-  f  e. 
_V
5351, 52op2nd 6129 . . . . . . . . . . . . . . 15  |-  ( 2nd `  <. ( ( Id
`  C ) `  x ) ,  f
>. )  =  f
5450, 53syl6eq 2331 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  -> 
( ( 2nd  |`  ( <. x ,  y >.
(  Hom  `  ( C  X.c  D ) ) <.
x ,  z >.
) ) `  <. ( ( Id `  C
) `  x ) ,  f >. )  =  f )
5535, 54eqtrd 2315 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  -> 
( ( ( Id
`  C ) `  x ) ( <.
x ,  y >.
( 2nd `  ( C  2ndF  D ) ) <.
x ,  z >.
) f )  =  f )
5655mpteq2dva 4106 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  -> 
( f  e.  ( y (  Hom  `  D
) z )  |->  ( ( ( Id `  C ) `  x
) ( <. x ,  y >. ( 2nd `  ( C  2ndF  D ) ) <. x ,  z
>. ) f ) )  =  ( f  e.  ( y (  Hom  `  D ) z ) 
|->  f ) )
57 mptresid 5004 . . . . . . . . . . . 12  |-  ( f  e.  ( y (  Hom  `  D )
z )  |->  f )  =  (  _I  |`  (
y (  Hom  `  D
) z ) )
5856, 57syl6eq 2331 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  -> 
( f  e.  ( y (  Hom  `  D
) z )  |->  ( ( ( Id `  C ) `  x
) ( <. x ,  y >. ( 2nd `  ( C  2ndF  D ) ) <. x ,  z
>. ) f ) )  =  (  _I  |`  (
y (  Hom  `  D
) z ) ) )
59583impa 1146 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
)  /\  z  e.  ( Base `  D )
)  ->  ( f  e.  ( y (  Hom  `  D ) z ) 
|->  ( ( ( Id
`  C ) `  x ) ( <.
x ,  y >.
( 2nd `  ( C  2ndF  D ) ) <.
x ,  z >.
) f ) )  =  (  _I  |`  (
y (  Hom  `  D
) z ) ) )
6059mpt2eq3dva 5912 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( y  e.  ( Base `  D
) ,  z  e.  ( Base `  D
)  |->  ( f  e.  ( y (  Hom  `  D ) z ) 
|->  ( ( ( Id
`  C ) `  x ) ( <.
x ,  y >.
( 2nd `  ( C  2ndF  D ) ) <.
x ,  z >.
) f ) ) )  =  ( y  e.  ( Base `  D
) ,  z  e.  ( Base `  D
)  |->  (  _I  |`  (
y (  Hom  `  D
) z ) ) ) )
61 fveq2 5525 . . . . . . . . . . . 12  |-  ( u  =  <. y ,  z
>.  ->  ( (  Hom  `  D ) `  u
)  =  ( (  Hom  `  D ) `  <. y ,  z
>. ) )
62 df-ov 5861 . . . . . . . . . . . 12  |-  ( y (  Hom  `  D
) z )  =  ( (  Hom  `  D
) `  <. y ,  z >. )
6361, 62syl6eqr 2333 . . . . . . . . . . 11  |-  ( u  =  <. y ,  z
>.  ->  ( (  Hom  `  D ) `  u
)  =  ( y (  Hom  `  D
) z ) )
6463reseq2d 4955 . . . . . . . . . 10  |-  ( u  =  <. y ,  z
>.  ->  (  _I  |`  (
(  Hom  `  D ) `
 u ) )  =  (  _I  |`  (
y (  Hom  `  D
) z ) ) )
6564mpt2mpt 5939 . . . . . . . . 9  |-  ( u  e.  ( ( Base `  D )  X.  ( Base `  D ) ) 
|->  (  _I  |`  (
(  Hom  `  D ) `
 u ) ) )  =  ( y  e.  ( Base `  D
) ,  z  e.  ( Base `  D
)  |->  (  _I  |`  (
y (  Hom  `  D
) z ) ) )
6660, 65syl6eqr 2333 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( y  e.  ( Base `  D
) ,  z  e.  ( Base `  D
)  |->  ( f  e.  ( y (  Hom  `  D ) z ) 
|->  ( ( ( Id
`  C ) `  x ) ( <.
x ,  y >.
( 2nd `  ( C  2ndF  D ) ) <.
x ,  z >.
) f ) ) )  =  ( u  e.  ( ( Base `  D )  X.  ( Base `  D ) ) 
|->  (  _I  |`  (
(  Hom  `  D ) `
 u ) ) ) )
6724, 66opeq12d 3804 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  <. ( y  e.  ( Base `  D
)  |->  ( x ( 1st `  ( C  2ndF  D ) ) y ) ) ,  ( y  e.  ( Base `  D ) ,  z  e.  ( Base `  D
)  |->  ( f  e.  ( y (  Hom  `  D ) z ) 
|->  ( ( ( Id
`  C ) `  x ) ( <.
x ,  y >.
( 2nd `  ( C  2ndF  D ) ) <.
x ,  z >.
) f ) ) ) >.  =  <. (  _I  |`  ( Base `  D ) ) ,  ( u  e.  ( ( Base `  D
)  X.  ( Base `  D ) )  |->  (  _I  |`  ( (  Hom  `  D ) `  u ) ) )
>. )
68 eqid 2283 . . . . . . . 8  |-  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  =  ( <. C ,  D >. curryF  ( C  2ndF  D ) )
692, 7, 10, 132ndfcl 13972 . . . . . . . . 9  |-  ( ph  ->  ( C  2ndF  D )  e.  ( ( C  X.c  D
)  Func  D )
)
7069adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( C  2ndF  D )  e.  ( ( C  X.c  D )  Func  D
) )
71 eqid 2283 . . . . . . . 8  |-  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x )  =  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x )
7268, 3, 8, 11, 70, 4, 38, 71, 44, 37curf1 13999 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x )  = 
<. ( y  e.  (
Base `  D )  |->  ( x ( 1st `  ( C  2ndF  D )
) y ) ) ,  ( y  e.  ( Base `  D
) ,  z  e.  ( Base `  D
)  |->  ( f  e.  ( y (  Hom  `  D ) z ) 
|->  ( ( ( Id
`  C ) `  x ) ( <.
x ,  y >.
( 2nd `  ( C  2ndF  D ) ) <.
x ,  z >.
) f ) ) ) >. )
73 eqid 2283 . . . . . . . 8  |-  (idfunc `  D
)  =  (idfunc `  D
)
7473, 4, 11, 44idfuval 13750 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  (idfunc `  D )  =  <. (  _I  |`  ( Base `  D ) ) ,  ( u  e.  ( ( Base `  D
)  X.  ( Base `  D ) )  |->  (  _I  |`  ( (  Hom  `  D ) `  u ) ) )
>. )
7567, 72, 743eqtr4d 2325 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x )  =  (idfunc `  D ) )
76 eqid 2283 . . . . . . 7  |-  ( QΔfunc C )  =  ( QΔfunc C )
77 curf2ndf.q . . . . . . . . 9  |-  Q  =  ( D FuncCat  D )
7877, 10, 10fuccat 13844 . . . . . . . 8  |-  ( ph  ->  Q  e.  Cat )
7978adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  Q  e.  Cat )
8077fucbas 13834 . . . . . . 7  |-  ( D 
Func  D )  =  (
Base `  Q )
8173idfucl 13755 . . . . . . . . 9  |-  ( D  e.  Cat  ->  (idfunc `  D
)  e.  ( D 
Func  D ) )
8210, 81syl 15 . . . . . . . 8  |-  ( ph  ->  (idfunc `  D )  e.  ( D  Func  D )
)
8382adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  (idfunc `  D )  e.  ( D  Func  D )
)
84 eqid 2283 . . . . . . 7  |-  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D
) )  =  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) )
8576, 79, 8, 80, 83, 84, 3, 38diag11 14017 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x )  =  (idfunc `  D ) )
8675, 85eqtr4d 2318 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x )  =  ( ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x ) )
8786mpteq2dva 4106 . . . 4  |-  ( ph  ->  ( x  e.  (
Base `  C )  |->  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x ) )  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x ) ) )
88 relfunc 13736 . . . . . . 7  |-  Rel  ( C  Func  Q )
8968, 77, 7, 10, 69curfcl 14006 . . . . . . 7  |-  ( ph  ->  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  e.  ( C  Func  Q
) )
90 1st2ndbr 6169 . . . . . . 7  |-  ( ( Rel  ( C  Func  Q )  /\  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  e.  ( C  Func  Q
) )  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) ( C  Func  Q
) ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) )
9188, 89, 90sylancr 644 . . . . . 6  |-  ( ph  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) ( C  Func  Q
) ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) )
923, 80, 91funcf1 13740 . . . . 5  |-  ( ph  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) : ( Base `  C
) --> ( D  Func  D ) )
9392feqmptd 5575 . . . 4  |-  ( ph  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x ) ) )
9476, 78, 7, 80, 82, 84diag1cl 14016 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) )  e.  ( C  Func  Q
) )
95 1st2ndbr 6169 . . . . . . 7  |-  ( ( Rel  ( C  Func  Q )  /\  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D
) )  e.  ( C  Func  Q )
)  ->  ( 1st `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) ( C  Func  Q
) ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) )
9688, 94, 95sylancr 644 . . . . . 6  |-  ( ph  ->  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) ( C  Func  Q
) ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) )
973, 80, 96funcf1 13740 . . . . 5  |-  ( ph  ->  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) : ( Base `  C
) --> ( D  Func  D ) )
9897feqmptd 5575 . . . 4  |-  ( ph  ->  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x ) ) )
9987, 93, 983eqtr4d 2325 . . 3  |-  ( ph  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  =  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) )
10010ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  D  e.  Cat )
10173, 4, 100idfu1st 13753 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( 1st `  (idfunc `  D
) )  =  (  _I  |`  ( Base `  D ) ) )
102101coeq2d 4846 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( Id `  D )  o.  ( 1st `  (idfunc `  D ) ) )  =  ( ( Id
`  D )  o.  (  _I  |`  ( Base `  D ) ) ) )
103 eqid 2283 . . . . . . . . . . 11  |-  ( Id
`  Q )  =  ( Id `  Q
)
104 eqid 2283 . . . . . . . . . . 11  |-  ( Id
`  D )  =  ( Id `  D
)
10582ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
(idfunc `  D )  e.  ( D  Func  D )
)
10677, 103, 104, 105fucid 13845 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( Id `  Q ) `  (idfunc `  D
) )  =  ( ( Id `  D
)  o.  ( 1st `  (idfunc `  D ) ) ) )
1074, 104cidfn 13581 . . . . . . . . . . . . . 14  |-  ( D  e.  Cat  ->  ( Id `  D )  Fn  ( Base `  D
) )
108100, 107syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( Id `  D
)  Fn  ( Base `  D ) )
109 dffn2 5390 . . . . . . . . . . . . 13  |-  ( ( Id `  D )  Fn  ( Base `  D
)  <->  ( Id `  D ) : (
Base `  D ) --> _V )
110108, 109sylib 188 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( Id `  D
) : ( Base `  D ) --> _V )
111110feqmptd 5575 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( Id `  D
)  =  ( z  e.  ( Base `  D
)  |->  ( ( Id
`  D ) `  z ) ) )
112 fcoi1 5415 . . . . . . . . . . . 12  |-  ( ( Id `  D ) : ( Base `  D
) --> _V  ->  ( ( Id `  D )  o.  (  _I  |`  ( Base `  D ) ) )  =  ( Id
`  D ) )
113110, 112syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( Id `  D )  o.  (  _I  |`  ( Base `  D
) ) )  =  ( Id `  D
) )
1147ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  C  e.  Cat )
115114adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  C  e.  Cat )
116100adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  D  e.  Cat )
117 simplrl 736 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  x  e.  ( Base `  C ) )
118117, 31sylan 457 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  <. x ,  z >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
119 simplrr 737 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
y  e.  ( Base `  C ) )
120119adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  y  e.  ( Base `  C )
)
121 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  z  e.  ( Base `  D )
)
122 opelxpi 4721 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  ( Base `  C )  /\  z  e.  ( Base `  D
) )  ->  <. y ,  z >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
123120, 121, 122syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  <. y ,  z >.  e.  (
( Base `  C )  X.  ( Base `  D
) ) )
1242, 5, 6, 115, 116, 13, 118, 1232ndf2 13970 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  ( <. x ,  z >. ( 2nd `  ( C  2ndF  D ) ) <. y ,  z
>. )  =  ( 2nd  |`  ( <. x ,  z >. (  Hom  `  ( C  X.c  D
) ) <. y ,  z >. )
) )
125124oveqd 5875 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  ( f
( <. x ,  z
>. ( 2nd `  ( C  2ndF  D ) ) <.
y ,  z >.
) ( ( Id
`  D ) `  z ) )  =  ( f ( 2nd  |`  ( <. x ,  z
>. (  Hom  `  ( C  X.c  D ) ) <.
y ,  z >.
) ) ( ( Id `  D ) `
 z ) ) )
126 df-ov 5861 . . . . . . . . . . . . . . 15  |-  ( f ( 2nd  |`  ( <. x ,  z >.
(  Hom  `  ( C  X.c  D ) ) <.
y ,  z >.
) ) ( ( Id `  D ) `
 z ) )  =  ( ( 2nd  |`  ( <. x ,  z
>. (  Hom  `  ( C  X.c  D ) ) <.
y ,  z >.
) ) `  <. f ,  ( ( Id
`  D ) `  z ) >. )
127 simplr 731 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  f  e.  ( x (  Hom  `  C ) y ) )
1284, 44, 104, 116, 121catidcl 13584 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  ( ( Id `  D ) `  z )  e.  ( z (  Hom  `  D
) z ) )
129 opelxpi 4721 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  ( x (  Hom  `  C
) y )  /\  ( ( Id `  D ) `  z
)  e.  ( z (  Hom  `  D
) z ) )  ->  <. f ,  ( ( Id `  D
) `  z ) >.  e.  ( ( x (  Hom  `  C
) y )  X.  ( z (  Hom  `  D ) z ) ) )
130127, 128, 129syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  <. f ,  ( ( Id `  D ) `  z
) >.  e.  ( ( x (  Hom  `  C
) y )  X.  ( z (  Hom  `  D ) z ) ) )
131117adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  x  e.  ( Base `  C )
)
1322, 3, 4, 36, 44, 131, 121, 120, 121, 6xpchom2 13960 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  ( <. x ,  z >. (  Hom  `  ( C  X.c  D
) ) <. y ,  z >. )  =  ( ( x (  Hom  `  C
) y )  X.  ( z (  Hom  `  D ) z ) ) )
133130, 132eleqtrrd 2360 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  <. f ,  ( ( Id `  D ) `  z
) >.  e.  ( <.
x ,  z >.
(  Hom  `  ( C  X.c  D ) ) <.
y ,  z >.
) )
134 fvres 5542 . . . . . . . . . . . . . . . 16  |-  ( <.
f ,  ( ( Id `  D ) `
 z ) >.  e.  ( <. x ,  z
>. (  Hom  `  ( C  X.c  D ) ) <.
y ,  z >.
)  ->  ( ( 2nd  |`  ( <. x ,  z >. (  Hom  `  ( C  X.c  D
) ) <. y ,  z >. )
) `  <. f ,  ( ( Id `  D ) `  z
) >. )  =  ( 2nd `  <. f ,  ( ( Id
`  D ) `  z ) >. )
)
135133, 134syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  ( ( 2nd  |`  ( <. x ,  z >. (  Hom  `  ( C  X.c  D
) ) <. y ,  z >. )
) `  <. f ,  ( ( Id `  D ) `  z
) >. )  =  ( 2nd `  <. f ,  ( ( Id
`  D ) `  z ) >. )
)
136126, 135syl5eq 2327 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  ( f
( 2nd  |`  ( <.
x ,  z >.
(  Hom  `  ( C  X.c  D ) ) <.
y ,  z >.
) ) ( ( Id `  D ) `
 z ) )  =  ( 2nd `  <. f ,  ( ( Id
`  D ) `  z ) >. )
)
137 fvex 5539 . . . . . . . . . . . . . . 15  |-  ( ( Id `  D ) `
 z )  e. 
_V
13852, 137op2nd 6129 . . . . . . . . . . . . . 14  |-  ( 2nd `  <. f ,  ( ( Id `  D
) `  z ) >. )  =  ( ( Id `  D ) `
 z )
139136, 138syl6eq 2331 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  ( f
( 2nd  |`  ( <.
x ,  z >.
(  Hom  `  ( C  X.c  D ) ) <.
y ,  z >.
) ) ( ( Id `  D ) `
 z ) )  =  ( ( Id
`  D ) `  z ) )
140125, 139eqtrd 2315 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  ( f
( <. x ,  z
>. ( 2nd `  ( C  2ndF  D ) ) <.
y ,  z >.
) ( ( Id
`  D ) `  z ) )  =  ( ( Id `  D ) `  z
) )
141140mpteq2dva 4106 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( z  e.  (
Base `  D )  |->  ( f ( <.
x ,  z >.
( 2nd `  ( C  2ndF  D ) ) <.
y ,  z >.
) ( ( Id
`  D ) `  z ) ) )  =  ( z  e.  ( Base `  D
)  |->  ( ( Id
`  D ) `  z ) ) )
142111, 113, 1413eqtr4rd 2326 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( z  e.  (
Base `  D )  |->  ( f ( <.
x ,  z >.
( 2nd `  ( C  2ndF  D ) ) <.
y ,  z >.
) ( ( Id
`  D ) `  z ) ) )  =  ( ( Id
`  D )  o.  (  _I  |`  ( Base `  D ) ) ) )
143102, 106, 1423eqtr4rd 2326 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( z  e.  (
Base `  D )  |->  ( f ( <.
x ,  z >.
( 2nd `  ( C  2ndF  D ) ) <.
y ,  z >.
) ( ( Id
`  D ) `  z ) ) )  =  ( ( Id
`  Q ) `  (idfunc `  D ) ) )
14469ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( C  2ndF  D )  e.  ( ( C  X.c  D
)  Func  D )
)
145 simpr 447 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
f  e.  ( x (  Hom  `  C
) y ) )
146 eqid 2283 . . . . . . . . . 10  |-  ( ( x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y ) `  f
)  =  ( ( x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y ) `  f
)
14768, 3, 114, 100, 144, 4, 36, 104, 117, 119, 145, 146curf2 14003 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y ) `  f
)  =  ( z  e.  ( Base `  D
)  |->  ( f (
<. x ,  z >.
( 2nd `  ( C  2ndF  D ) ) <.
y ,  z >.
) ( ( Id
`  D ) `  z ) ) ) )
14878ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  Q  e.  Cat )
14976, 148, 114, 80, 105, 84, 3, 117, 36, 103, 119, 145diag12 14018 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D
) ) ) y ) `  f )  =  ( ( Id
`  Q ) `  (idfunc `  D ) ) )
150143, 147, 1493eqtr4d 2325 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y ) `  f
)  =  ( ( x ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) y ) `  f
) )
151150mpteq2dva 4106 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
f  e.  ( x (  Hom  `  C
) y )  |->  ( ( x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y ) `  f
) )  =  ( f  e.  ( x (  Hom  `  C
) y )  |->  ( ( x ( 2nd `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) y ) `  f
) ) )
152 eqid 2283 . . . . . . . . . 10  |-  ( D Nat 
D )  =  ( D Nat  D )
15377, 152fuchom 13835 . . . . . . . . 9  |-  ( D Nat 
D )  =  (  Hom  `  Q )
15491adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) ( C  Func  Q
) ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) )
155 simprl 732 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
156 simprr 733 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
1573, 36, 153, 154, 155, 156funcf2 13742 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y ) : ( x (  Hom  `  C
) y ) --> ( ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x ) ( D Nat  D ) ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 y ) ) )
158157feqmptd 5575 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y )  =  ( f  e.  ( x (  Hom  `  C
) y )  |->  ( ( x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y ) `  f
) ) )
15996adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) ( C  Func  Q
) ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) )
16038adantrr 697 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
1613, 36, 153, 159, 160, 156funcf2 13742 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) y ) : ( x (  Hom  `  C
) y ) --> ( ( ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x ) ( D Nat  D ) ( ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 y ) ) )
162161feqmptd 5575 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) y )  =  ( f  e.  ( x (  Hom  `  C
) y )  |->  ( ( x ( 2nd `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) y ) `  f
) ) )
163151, 158, 1623eqtr4d 2325 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y )  =  ( x ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) y ) )
1641633impb 1147 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y )  =  ( x ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) y ) )
165164mpt2eq3dva 5912 . . . 4  |-  ( ph  ->  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y ) )  =  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( x ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) y ) ) )
1663, 91funcfn2 13743 . . . . 5  |-  ( ph  ->  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  Fn  ( ( Base `  C )  X.  ( Base `  C ) ) )
167 fnov 5952 . . . . 5  |-  ( ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  Fn  ( ( Base `  C )  X.  ( Base `  C ) )  <-> 
( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y ) ) )
168166, 167sylib 188 . . . 4  |-  ( ph  ->  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y ) ) )
1693, 96funcfn2 13743 . . . . 5  |-  ( ph  ->  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )  Fn  ( ( Base `  C )  X.  ( Base `  C ) ) )
170 fnov 5952 . . . . 5  |-  ( ( 2nd `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D
) ) )  Fn  ( ( Base `  C
)  X.  ( Base `  C ) )  <->  ( 2nd `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D
) ) ) y ) ) )
171169, 170sylib 188 . . . 4  |-  ( ph  ->  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D
) ) ) y ) ) )
172165, 168, 1713eqtr4d 2325 . . 3  |-  ( ph  ->  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  =  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) )
17399, 172opeq12d 3804 . 2  |-  ( ph  -> 
<. ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) ,  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )
>.  =  <. ( 1st `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) ,  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )
>. )
174 1st2nd 6166 . . 3  |-  ( ( Rel  ( C  Func  Q )  /\  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  e.  ( C  Func  Q
) )  ->  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  = 
<. ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) ,  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )
>. )
17588, 89, 174sylancr 644 . 2  |-  ( ph  ->  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  = 
<. ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) ,  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )
>. )
176 1st2nd 6166 . . 3  |-  ( ( Rel  ( C  Func  Q )  /\  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D
) )  e.  ( C  Func  Q )
)  ->  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D
) )  =  <. ( 1st `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D
) ) ) ,  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )
>. )
17788, 94, 176sylancr 644 . 2  |-  ( ph  ->  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) )  = 
<. ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) ,  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )
>. )
178173, 175, 1773eqtr4d 2325 1  |-  ( ph  ->  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  =  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687    |` cres 4691    o. ccom 4693   Rel wrel 4694    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   Basecbs 13148    Hom chom 13219   Catccat 13566   Idccid 13567    Func cfunc 13728  idfunccidfu 13729   Nat cnat 13815   FuncCat cfuc 13816    X.c cxpc 13942    2ndF c2ndf 13944   curryF ccurf 13984  Δfunccdiag 13986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-hom 13232  df-cco 13233  df-cat 13570  df-cid 13571  df-func 13732  df-idfu 13733  df-nat 13817  df-fuc 13818  df-xpc 13946  df-1stf 13947  df-2ndf 13948  df-curf 13988  df-diag 13990
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