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Theorem curf2ndf 14346
Description: As shown in diagval 14339, 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 6086 . . . . . . . . . . 11  |-  ( x ( 1st `  ( C  2ndF  D ) ) y )  =  ( ( 1st `  ( C  2ndF  D ) ) `  <. x ,  y >.
)
2 eqid 2438 . . . . . . . . . . . . 13  |-  ( C  X.c  D )  =  ( C  X.c  D )
3 eqid 2438 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2438 . . . . . . . . . . . . . 14  |-  ( Base `  D )  =  (
Base `  D )
52, 3, 4xpcbas 14277 . . . . . . . . . . . . 13  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  ( C  X.c  D ) )
6 eqid 2438 . . . . . . . . . . . . 13  |-  (  Hom  `  ( C  X.c  D ) )  =  (  Hom  `  ( C  X.c  D ) )
7 curf2ndf.c . . . . . . . . . . . . . 14  |-  ( ph  ->  C  e.  Cat )
87ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  C  e.  Cat )
9 curf2ndf.d . . . . . . . . . . . . . 14  |-  ( ph  ->  D  e.  Cat )
109ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  D  e.  Cat )
11 eqid 2438 . . . . . . . . . . . . 13  |-  ( C  2ndF  D )  =  ( C  2ndF  D )
12 opelxpi 4912 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) )  ->  <. x ,  y >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
1312adantll 696 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  <. x ,  y >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
142, 5, 6, 8, 10, 11, 132ndf1 14294 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  (
( 1st `  ( C  2ndF  D ) ) `  <. x ,  y >.
)  =  ( 2nd `  <. x ,  y
>. ) )
15 vex 2961 . . . . . . . . . . . . 13  |-  x  e. 
_V
16 vex 2961 . . . . . . . . . . . . 13  |-  y  e. 
_V
1715, 16op2nd 6358 . . . . . . . . . . . 12  |-  ( 2nd `  <. x ,  y
>. )  =  y
1814, 17syl6eq 2486 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  (
( 1st `  ( C  2ndF  D ) ) `  <. x ,  y >.
)  =  y )
191, 18syl5eq 2482 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  (
x ( 1st `  ( C  2ndF  D ) ) y )  =  y )
2019mpteq2dva 4297 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( y  e.  ( Base `  D
)  |->  ( x ( 1st `  ( C  2ndF  D ) ) y ) )  =  ( y  e.  ( Base `  D )  |->  y ) )
21 mptresid 5197 . . . . . . . . 9  |-  ( y  e.  ( Base `  D
)  |->  y )  =  (  _I  |`  ( Base `  D ) )
2220, 21syl6eq 2486 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( y  e.  ( Base `  D
)  |->  ( x ( 1st `  ( C  2ndF  D ) ) y ) )  =  (  _I  |`  ( Base `  D ) ) )
23 df-ov 6086 . . . . . . . . . . . . . . 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 >. )
248ad2antrr 708 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  C  e.  Cat )
2510ad2antrr 708 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  D  e.  Cat )
2613ad2antrr 708 . . . . . . . . . . . . . . . . 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 ) ) )
27 simp-4r 745 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  x  e.  ( Base `  C ) )
28 simplr 733 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  -> 
z  e.  ( Base `  D ) )
29 opelxpi 4912 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  ( Base `  C )  /\  z  e.  ( Base `  D
) )  ->  <. x ,  z >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
3027, 28, 29syl2anc 644 . . . . . . . . . . . . . . . . 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 ) ) )
312, 5, 6, 24, 25, 11, 26, 302ndf2 14295 . . . . . . . . . . . . . . . 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 >.
) ) )
3231fveq1d 5732 . . . . . . . . . . . . . . 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 >. )
)
3323, 32syl5eq 2482 . . . . . . . . . . . . . 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 >. )
)
34 eqid 2438 . . . . . . . . . . . . . . . . . . . 20  |-  (  Hom  `  C )  =  (  Hom  `  C )
35 eqid 2438 . . . . . . . . . . . . . . . . . . . 20  |-  ( Id
`  C )  =  ( Id `  C
)
367adantr 453 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  C  e.  Cat )
37 simpr 449 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
383, 34, 35, 36, 37catidcl 13909 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  C ) `  x )  e.  ( x (  Hom  `  C
) x ) )
3938ad3antrrr 712 . . . . . . . . . . . . . . . . . 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 ) )
40 simpr 449 . . . . . . . . . . . . . . . . . 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 ) )
41 opelxpi 4912 . . . . . . . . . . . . . . . . . 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 ) ) )
4239, 40, 41syl2anc 644 . . . . . . . . . . . . . . . . 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 ) ) )
43 eqid 2438 . . . . . . . . . . . . . . . . . 18  |-  (  Hom  `  D )  =  (  Hom  `  D )
44 simpllr 737 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  -> 
y  e.  ( Base `  D ) )
452, 3, 4, 34, 43, 27, 44, 27, 28, 6xpchom2 14285 . . . . . . . . . . . . . . . . 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 ) ) )
4642, 45eleqtrrd 2515 . . . . . . . . . . . . . . . 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 >.
) )
47 fvres 5747 . . . . . . . . . . . . . . . 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 >. )
)
4846, 47syl 16 . . . . . . . . . . . . . . 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 >. )
)
49 fvex 5744 . . . . . . . . . . . . . . . 16  |-  ( ( Id `  C ) `
 x )  e. 
_V
50 vex 2961 . . . . . . . . . . . . . . . 16  |-  f  e. 
_V
5149, 50op2nd 6358 . . . . . . . . . . . . . . 15  |-  ( 2nd `  <. ( ( Id
`  C ) `  x ) ,  f
>. )  =  f
5248, 51syl6eq 2486 . . . . . . . . . . . . . 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 )
5333, 52eqtrd 2470 . . . . . . . . . . . . 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 )
5453mpteq2dva 4297 . . . . . . . . . . . 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 ) )
55 mptresid 5197 . . . . . . . . . . . 12  |-  ( f  e.  ( y (  Hom  `  D )
z )  |->  f )  =  (  _I  |`  (
y (  Hom  `  D
) z ) )
5654, 55syl6eq 2486 . . . . . . . . . . 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 ) ) )
57563impa 1149 . . . . . . . . . 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 ) ) )
5857mpt2eq3dva 6140 . . . . . . . . 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 ) ) ) )
59 fveq2 5730 . . . . . . . . . . . 12  |-  ( u  =  <. y ,  z
>.  ->  ( (  Hom  `  D ) `  u
)  =  ( (  Hom  `  D ) `  <. y ,  z
>. ) )
60 df-ov 6086 . . . . . . . . . . . 12  |-  ( y (  Hom  `  D
) z )  =  ( (  Hom  `  D
) `  <. y ,  z >. )
6159, 60syl6eqr 2488 . . . . . . . . . . 11  |-  ( u  =  <. y ,  z
>.  ->  ( (  Hom  `  D ) `  u
)  =  ( y (  Hom  `  D
) z ) )
6261reseq2d 5148 . . . . . . . . . 10  |-  ( u  =  <. y ,  z
>.  ->  (  _I  |`  (
(  Hom  `  D ) `
 u ) )  =  (  _I  |`  (
y (  Hom  `  D
) z ) ) )
6362mpt2mpt 6167 . . . . . . . . 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 ) ) )
6458, 63syl6eqr 2488 . . . . . . . 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 ) ) ) )
6522, 64opeq12d 3994 . . . . . . 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 ) ) )
>. )
66 eqid 2438 . . . . . . . 8  |-  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  =  ( <. C ,  D >. curryF  ( C  2ndF  D ) )
679adantr 453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
682, 7, 9, 112ndfcl 14297 . . . . . . . . 9  |-  ( ph  ->  ( C  2ndF  D )  e.  ( ( C  X.c  D
)  Func  D )
)
6968adantr 453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( C  2ndF  D )  e.  ( ( C  X.c  D )  Func  D
) )
70 eqid 2438 . . . . . . . 8  |-  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x )  =  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x )
7166, 3, 36, 67, 69, 4, 37, 70, 43, 35curf1 14324 . . . . . . 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 ) ) ) >. )
72 eqid 2438 . . . . . . . 8  |-  (idfunc `  D
)  =  (idfunc `  D
)
7372, 4, 67, 43idfuval 14075 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  (idfunc `  D )  =  <. (  _I  |`  ( Base `  D ) ) ,  ( u  e.  ( ( Base `  D
)  X.  ( Base `  D ) )  |->  (  _I  |`  ( (  Hom  `  D ) `  u ) ) )
>. )
7465, 71, 733eqtr4d 2480 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x )  =  (idfunc `  D ) )
75 eqid 2438 . . . . . . 7  |-  ( QΔfunc C )  =  ( QΔfunc C )
76 curf2ndf.q . . . . . . . . 9  |-  Q  =  ( D FuncCat  D )
7776, 9, 9fuccat 14169 . . . . . . . 8  |-  ( ph  ->  Q  e.  Cat )
7877adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  Q  e.  Cat )
7976fucbas 14159 . . . . . . 7  |-  ( D 
Func  D )  =  (
Base `  Q )
8072idfucl 14080 . . . . . . . . 9  |-  ( D  e.  Cat  ->  (idfunc `  D
)  e.  ( D 
Func  D ) )
819, 80syl 16 . . . . . . . 8  |-  ( ph  ->  (idfunc `  D )  e.  ( D  Func  D )
)
8281adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  (idfunc `  D )  e.  ( D  Func  D )
)
83 eqid 2438 . . . . . . 7  |-  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D
) )  =  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) )
8475, 78, 36, 79, 82, 83, 3, 37diag11 14342 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x )  =  (idfunc `  D ) )
8574, 84eqtr4d 2473 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x )  =  ( ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x ) )
8685mpteq2dva 4297 . . . 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 ) ) )
87 relfunc 14061 . . . . . . 7  |-  Rel  ( C  Func  Q )
8866, 76, 7, 9, 68curfcl 14331 . . . . . . 7  |-  ( ph  ->  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  e.  ( C  Func  Q
) )
89 1st2ndbr 6398 . . . . . . 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 ) ) ) )
9087, 88, 89sylancr 646 . . . . . 6  |-  ( ph  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) ( C  Func  Q
) ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) )
913, 79, 90funcf1 14065 . . . . 5  |-  ( ph  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) : ( Base `  C
) --> ( D  Func  D ) )
9291feqmptd 5781 . . . 4  |-  ( ph  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x ) ) )
9375, 77, 7, 79, 81, 83diag1cl 14341 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) )  e.  ( C  Func  Q
) )
94 1st2ndbr 6398 . . . . . . 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 ) ) ) )
9587, 93, 94sylancr 646 . . . . . 6  |-  ( ph  ->  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) ( C  Func  Q
) ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) )
963, 79, 95funcf1 14065 . . . . 5  |-  ( ph  ->  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) : ( Base `  C
) --> ( D  Func  D ) )
9796feqmptd 5781 . . . 4  |-  ( ph  ->  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x ) ) )
9886, 92, 973eqtr4d 2480 . . 3  |-  ( ph  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  =  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) )
999ad2antrr 708 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  D  e.  Cat )
10072, 4, 99idfu1st 14078 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( 1st `  (idfunc `  D
) )  =  (  _I  |`  ( Base `  D ) ) )
101100coeq2d 5037 . . . . . . . . . 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 ) ) ) )
102 eqid 2438 . . . . . . . . . . 11  |-  ( Id
`  Q )  =  ( Id `  Q
)
103 eqid 2438 . . . . . . . . . . 11  |-  ( Id
`  D )  =  ( Id `  D
)
10481ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
(idfunc `  D )  e.  ( D  Func  D )
)
10576, 102, 103, 104fucid 14170 . . . . . . . . . 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 ) ) ) )
1064, 103cidfn 13906 . . . . . . . . . . . . . 14  |-  ( D  e.  Cat  ->  ( Id `  D )  Fn  ( Base `  D
) )
10799, 106syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( Id `  D
)  Fn  ( Base `  D ) )
108 dffn2 5594 . . . . . . . . . . . . 13  |-  ( ( Id `  D )  Fn  ( Base `  D
)  <->  ( Id `  D ) : (
Base `  D ) --> _V )
109107, 108sylib 190 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( Id `  D
) : ( Base `  D ) --> _V )
110109feqmptd 5781 . . . . . . . . . . 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 ) ) )
111 fcoi1 5619 . . . . . . . . . . . 12  |-  ( ( Id `  D ) : ( Base `  D
) --> _V  ->  ( ( Id `  D )  o.  (  _I  |`  ( Base `  D ) ) )  =  ( Id
`  D ) )
112109, 111syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( Id `  D )  o.  (  _I  |`  ( Base `  D
) ) )  =  ( Id `  D
) )
1137ad2antrr 708 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  C  e.  Cat )
114113adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  C  e.  Cat )
11599adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  D  e.  Cat )
116 simplrl 738 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  x  e.  ( Base `  C ) )
117116, 29sylan 459 . . . . . . . . . . . . . . 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
) ) )
118 simplrr 739 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
y  e.  ( Base `  C ) )
119 opelxpi 4912 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  ( Base `  C )  /\  z  e.  ( Base `  D
) )  ->  <. y ,  z >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
120118, 119sylan 459 . . . . . . . . . . . . . . 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
) ) )
1212, 5, 6, 114, 115, 11, 117, 1202ndf2 14295 . . . . . . . . . . . . . 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 >. )
) )
122121oveqd 6100 . . . . . . . . . . . . 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 ) ) )
123 df-ov 6086 . . . . . . . . . . . . . . 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 ) >. )
124 simplr 733 . . . . . . . . . . . . . . . . . 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 ) )
125 simpr 449 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  z  e.  ( Base `  D )
)
1264, 43, 103, 115, 125catidcl 13909 . . . . . . . . . . . . . . . . . 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 ) )
127 opelxpi 4912 . . . . . . . . . . . . . . . . . 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 ) ) )
128124, 126, 127syl2anc 644 . . . . . . . . . . . . . . . . 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 ) ) )
129116adantr 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  x  e.  ( Base `  C )
)
130118adantr 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  y  e.  ( Base `  C )
)
1312, 3, 4, 34, 43, 129, 125, 130, 125, 6xpchom2 14285 . . . . . . . . . . . . . . . . 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 ) ) )
132128, 131eleqtrrd 2515 . . . . . . . . . . . . . . . 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 >.
) )
133 fvres 5747 . . . . . . . . . . . . . . . 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 ) >. )
)
134132, 133syl 16 . . . . . . . . . . . . . . 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 ) >. )
)
135123, 134syl5eq 2482 . . . . . . . . . . . . . 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 ) >. )
)
136 fvex 5744 . . . . . . . . . . . . . . 15  |-  ( ( Id `  D ) `
 z )  e. 
_V
13750, 136op2nd 6358 . . . . . . . . . . . . . 14  |-  ( 2nd `  <. f ,  ( ( Id `  D
) `  z ) >. )  =  ( ( Id `  D ) `
 z )
138135, 137syl6eq 2486 . . . . . . . . . . . . 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 ) )
139122, 138eqtrd 2470 . . . . . . . . . . . 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
) )
140139mpteq2dva 4297 . . . . . . . . . . 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 ) ) )
141110, 112, 1403eqtr4rd 2481 . . . . . . . . . 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 ) ) ) )
142101, 105, 1413eqtr4rd 2481 . . . . . . . . 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 ) ) )
14368ad2antrr 708 . . . . . . . . . 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 )
)
144 simpr 449 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
f  e.  ( x (  Hom  `  C
) y ) )
145 eqid 2438 . . . . . . . . . 10  |-  ( ( x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y ) `  f
)  =  ( ( x ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) y ) `  f
)
14666, 3, 113, 99, 143, 4, 34, 103, 116, 118, 144, 145curf2 14328 . . . . . . . . 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 ) ) ) )
14777ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  Q  e.  Cat )
14875, 147, 113, 79, 104, 83, 3, 116, 34, 102, 118, 144diag12 14343 . . . . . . . . 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 ) ) )
149142, 146, 1483eqtr4d 2480 . . . . . . . 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
) )
150149mpteq2dva 4297 . . . . . . 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
) ) )
151 eqid 2438 . . . . . . . . . 10  |-  ( D Nat 
D )  =  ( D Nat  D )
15276, 151fuchom 14160 . . . . . . . . 9  |-  ( D Nat 
D )  =  (  Hom  `  Q )
15390adantr 453 . . . . . . . . 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 ) ) ) )
154 simprl 734 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
155 simprr 735 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
1563, 34, 152, 153, 154, 155funcf2 14067 . . . . . . . 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 ) ) )
157156feqmptd 5781 . . . . . . 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
) ) )
15895adantr 453 . . . . . . . . 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 ) ) ) )
1593, 34, 152, 158, 154, 155funcf2 14067 . . . . . . . 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 ) ) )
160159feqmptd 5781 . . . . . . 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
) ) )
161150, 157, 1603eqtr4d 2480 . . . . . 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 ) )
1621613impb 1150 . . . . 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 ) )
163162mpt2eq3dva 6140 . . . 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 ) ) )
1643, 90funcfn2 14068 . . . . 5  |-  ( ph  ->  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  Fn  ( ( Base `  C )  X.  ( Base `  C ) ) )
165 fnov 6180 . . . . 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 ) ) )
166164, 165sylib 190 . . . 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 ) ) )
1673, 95funcfn2 14068 . . . . 5  |-  ( ph  ->  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )  Fn  ( ( Base `  C )  X.  ( Base `  C ) ) )
168 fnov 6180 . . . . 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 ) ) )
169167, 168sylib 190 . . . 4  |-  ( ph  ->  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D
) ) ) y ) ) )
170163, 166, 1693eqtr4d 2480 . . 3  |-  ( ph  ->  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  =  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) )
17198, 170opeq12d 3994 . 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 ) ) )
>. )
172 1st2nd 6395 . . 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 ) ) )
>. )
17387, 88, 172sylancr 646 . 2  |-  ( ph  ->  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  = 
<. ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) ,  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )
>. )
174 1st2nd 6395 . . 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 ) ) )
>. )
17587, 93, 174sylancr 646 . 2  |-  ( ph  ->  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) )  = 
<. ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) ,  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )
>. )
176171, 173, 1753eqtr4d 2480 1  |-  ( ph  ->  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  =  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   class class class wbr 4214    e. cmpt 4268    _I cid 4495    X. cxp 4878    |` cres 4882    o. ccom 4884   Rel wrel 4885    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1stc1st 6349   2ndc2nd 6350   Basecbs 13471    Hom chom 13542   Catccat 13891   Idccid 13892    Func cfunc 14053  idfunccidfu 14054   Nat cnat 14140   FuncCat cfuc 14141    X.c cxpc 14267    2ndF c2ndf 14269   curryF ccurf 14309  Δfunccdiag 14311
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-hom 13555  df-cco 13556  df-cat 13895  df-cid 13896  df-func 14057  df-idfu 14058  df-nat 14142  df-fuc 14143  df-xpc 14271  df-1stf 14272  df-2ndf 14273  df-curf 14313  df-diag 14315
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