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Theorem curf2ndf 14037
Description: As shown in diagval 14030, 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 5877 . . . . . . . . . . 11  |-  ( x ( 1st `  ( C  2ndF  D ) ) y )  =  ( ( 1st `  ( C  2ndF  D ) ) `  <. x ,  y >.
)
2 eqid 2296 . . . . . . . . . . . . 13  |-  ( C  X.c  D )  =  ( C  X.c  D )
3 eqid 2296 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2296 . . . . . . . . . . . . . 14  |-  ( Base `  D )  =  (
Base `  D )
52, 3, 4xpcbas 13968 . . . . . . . . . . . . 13  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  ( C  X.c  D ) )
6 eqid 2296 . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . 13  |-  ( C  2ndF  D )  =  ( C  2ndF  D )
14 opelxpi 4737 . . . . . . . . . . . . . 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 13985 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  (
( 1st `  ( C  2ndF  D ) ) `  <. x ,  y >.
)  =  ( 2nd `  <. x ,  y
>. ) )
17 vex 2804 . . . . . . . . . . . . 13  |-  x  e. 
_V
18 vex 2804 . . . . . . . . . . . . 13  |-  y  e. 
_V
1917, 18op2nd 6145 . . . . . . . . . . . 12  |-  ( 2nd `  <. x ,  y
>. )  =  y
2016, 19syl6eq 2344 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  (
( 1st `  ( C  2ndF  D ) ) `  <. x ,  y >.
)  =  y )
211, 20syl5eq 2340 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  (
x ( 1st `  ( C  2ndF  D ) ) y )  =  y )
2221mpteq2dva 4122 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( y  e.  ( Base `  D
)  |->  ( x ( 1st `  ( C  2ndF  D ) ) y ) )  =  ( y  e.  ( Base `  D )  |->  y ) )
23 mptresid 5020 . . . . . . . . 9  |-  ( y  e.  ( Base `  D
)  |->  y )  =  (  _I  |`  ( Base `  D ) )
2422, 23syl6eq 2344 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( y  e.  ( Base `  D
)  |->  ( x ( 1st `  ( C  2ndF  D ) ) y ) )  =  (  _I  |`  ( Base `  D ) ) )
25 df-ov 5877 . . . . . . . . . . . . . . 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 4737 . . . . . . . . . . . . . . . . . 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 13986 . . . . . . . . . . . . . . . 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 5543 . . . . . . . . . . . . . . 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 2340 . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . . . . . 20  |-  (  Hom  `  C )  =  (  Hom  `  C )
37 eqid 2296 . . . . . . . . . . . . . . . . . . . 20  |-  ( Id
`  C )  =  ( Id `  C
)
38 simpr 447 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
393, 36, 37, 8, 38catidcl 13600 . . . . . . . . . . . . . . . . . . 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 4737 . . . . . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . . . 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 13976 . . . . . . . . . . . . . . . . 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 2373 . . . . . . . . . . . . . . . 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 5558 . . . . . . . . . . . . . . . 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 5555 . . . . . . . . . . . . . . . 16  |-  ( ( Id `  C ) `
 x )  e. 
_V
52 vex 2804 . . . . . . . . . . . . . . . 16  |-  f  e. 
_V
5351, 52op2nd 6145 . . . . . . . . . . . . . . 15  |-  ( 2nd `  <. ( ( Id
`  C ) `  x ) ,  f
>. )  =  f
5450, 53syl6eq 2344 . . . . . . . . . . . . . 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 2328 . . . . . . . . . . . . 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 4122 . . . . . . . . . . . 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 5020 . . . . . . . . . . . 12  |-  ( f  e.  ( y (  Hom  `  D )
z )  |->  f )  =  (  _I  |`  (
y (  Hom  `  D
) z ) )
5856, 57syl6eq 2344 . . . . . . . . . . 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 5928 . . . . . . . . 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 5541 . . . . . . . . . . . 12  |-  ( u  =  <. y ,  z
>.  ->  ( (  Hom  `  D ) `  u
)  =  ( (  Hom  `  D ) `  <. y ,  z
>. ) )
62 df-ov 5877 . . . . . . . . . . . 12  |-  ( y (  Hom  `  D
) z )  =  ( (  Hom  `  D
) `  <. y ,  z >. )
6361, 62syl6eqr 2346 . . . . . . . . . . 11  |-  ( u  =  <. y ,  z
>.  ->  ( (  Hom  `  D ) `  u
)  =  ( y (  Hom  `  D
) z ) )
6463reseq2d 4971 . . . . . . . . . 10  |-  ( u  =  <. y ,  z
>.  ->  (  _I  |`  (
(  Hom  `  D ) `
 u ) )  =  (  _I  |`  (
y (  Hom  `  D
) z ) ) )
6564mpt2mpt 5955 . . . . . . . . 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 2346 . . . . . . . 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 3820 . . . . . . 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 2296 . . . . . . . 8  |-  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  =  ( <. C ,  D >. curryF  ( C  2ndF  D ) )
692, 7, 10, 132ndfcl 13988 . . . . . . . . 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 2296 . . . . . . . 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 14015 . . . . . . 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 2296 . . . . . . . 8  |-  (idfunc `  D
)  =  (idfunc `  D
)
7473, 4, 11, 44idfuval 13766 . . . . . . 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 2338 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x )  =  (idfunc `  D ) )
76 eqid 2296 . . . . . . 7  |-  ( QΔfunc C )  =  ( QΔfunc C )
77 curf2ndf.q . . . . . . . . 9  |-  Q  =  ( D FuncCat  D )
7877, 10, 10fuccat 13860 . . . . . . . 8  |-  ( ph  ->  Q  e.  Cat )
7978adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  Q  e.  Cat )
8077fucbas 13850 . . . . . . 7  |-  ( D 
Func  D )  =  (
Base `  Q )
8173idfucl 13771 . . . . . . . . 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 2296 . . . . . . 7  |-  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D
) )  =  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) )
8576, 79, 8, 80, 83, 84, 3, 38diag11 14033 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x )  =  (idfunc `  D ) )
8675, 85eqtr4d 2331 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x )  =  ( ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x ) )
8786mpteq2dva 4122 . . . 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 13752 . . . . . . 7  |-  Rel  ( C  Func  Q )
8968, 77, 7, 10, 69curfcl 14022 . . . . . . 7  |-  ( ph  ->  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  e.  ( C  Func  Q
) )
90 1st2ndbr 6185 . . . . . . 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 13756 . . . . 5  |-  ( ph  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) : ( Base `  C
) --> ( D  Func  D ) )
9392feqmptd 5591 . . . 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 14032 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) )  e.  ( C  Func  Q
) )
95 1st2ndbr 6185 . . . . . . 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 13756 . . . . 5  |-  ( ph  ->  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) : ( Base `  C
) --> ( D  Func  D ) )
9897feqmptd 5591 . . . 4  |-  ( ph  ->  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x ) ) )
9987, 93, 983eqtr4d 2338 . . 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 13769 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( 1st `  (idfunc `  D
) )  =  (  _I  |`  ( Base `  D ) ) )
102101coeq2d 4862 . . . . . . . . . 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 2296 . . . . . . . . . . 11  |-  ( Id
`  Q )  =  ( Id `  Q
)
104 eqid 2296 . . . . . . . . . . 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 13861 . . . . . . . . . 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 13597 . . . . . . . . . . . . . 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 5406 . . . . . . . . . . . . 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 5591 . . . . . . . . . . 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 5431 . . . . . . . . . . . 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 4737 . . . . . . . . . . . . . . . 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 13986 . . . . . . . . . . . . . 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 5891 . . . . . . . . . . . . 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 5877 . . . . . . . . . . . . . . 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 13600 . . . . . . . . . . . . . . . . . 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 4737 . . . . . . . . . . . . . . . . . 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 13976 . . . . . . . . . . . . . . . . 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 2373 . . . . . . . . . . . . . . . 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 5558 . . . . . . . . . . . . . . . 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 2340 . . . . . . . . . . . . . 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 5555 . . . . . . . . . . . . . . 15  |-  ( ( Id `  D ) `
 z )  e. 
_V
13852, 137op2nd 6145 . . . . . . . . . . . . . 14  |-  ( 2nd `  <. f ,  ( ( Id `  D
) `  z ) >. )  =  ( ( Id `  D ) `
 z )
139136, 138syl6eq 2344 . . . . . . . . . . . . 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 2328 . . . . . . . . . . . 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 4122 . . . . . . . . . . 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 2339 . . . . . . . . . 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 2339 . . . . . . . . 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 2296 . . . . . . . . . 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 14019 . . . . . . . . 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 14034 . . . . . . . . 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 2338 . . . . . . . 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 4122 . . . . . . 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 2296 . . . . . . . . . 10  |-  ( D Nat 
D )  =  ( D Nat  D )
15377, 152fuchom 13851 . . . . . . . . 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 13758 . . . . . . . 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 5591 . . . . . . 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 13758 . . . . . . . 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 5591 . . . . . . 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 2338 . . . . . 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 5928 . . . 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 13759 . . . . 5  |-  ( ph  ->  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  Fn  ( ( Base `  C )  X.  ( Base `  C ) ) )
167 fnov 5968 . . . . 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 13759 . . . . 5  |-  ( ph  ->  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )  Fn  ( ( Base `  C )  X.  ( Base `  C ) ) )
170 fnov 5968 . . . . 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 2338 . . 3  |-  ( ph  ->  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  =  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) )
17399, 172opeq12d 3820 . 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 6182 . . 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 6182 . . 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 2338 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703    |` cres 4707    o. ccom 4709   Rel wrel 4710    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   Basecbs 13164    Hom chom 13235   Catccat 13582   Idccid 13583    Func cfunc 13744  idfunccidfu 13745   Nat cnat 13831   FuncCat cfuc 13832    X.c cxpc 13958    2ndF c2ndf 13960   curryF ccurf 14000  Δfunccdiag 14002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-hom 13248  df-cco 13249  df-cat 13586  df-cid 13587  df-func 13748  df-idfu 13749  df-nat 13833  df-fuc 13834  df-xpc 13962  df-1stf 13963  df-2ndf 13964  df-curf 14004  df-diag 14006
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