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Theorem curf2ndf 14299
Description: As shown in diagval 14292, 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 6043 . . . . . . . . . . 11  |-  ( x ( 1st `  ( C  2ndF  D ) ) y )  =  ( ( 1st `  ( C  2ndF  D ) ) `  <. x ,  y >.
)
2 eqid 2404 . . . . . . . . . . . . 13  |-  ( C  X.c  D )  =  ( C  X.c  D )
3 eqid 2404 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2404 . . . . . . . . . . . . . 14  |-  ( Base `  D )  =  (
Base `  D )
52, 3, 4xpcbas 14230 . . . . . . . . . . . . 13  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  ( C  X.c  D ) )
6 eqid 2404 . . . . . . . . . . . . 13  |-  (  Hom  `  ( C  X.c  D ) )  =  (  Hom  `  ( C  X.c  D ) )
7 curf2ndf.c . . . . . . . . . . . . . 14  |-  ( ph  ->  C  e.  Cat )
87ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  C  e.  Cat )
9 curf2ndf.d . . . . . . . . . . . . . 14  |-  ( ph  ->  D  e.  Cat )
109ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  D  e.  Cat )
11 eqid 2404 . . . . . . . . . . . . 13  |-  ( C  2ndF  D )  =  ( C  2ndF  D )
12 opelxpi 4869 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  D
) )  ->  <. x ,  y >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
1312adantll 695 . . . . . . . . . . . . 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 14247 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  (
( 1st `  ( C  2ndF  D ) ) `  <. x ,  y >.
)  =  ( 2nd `  <. x ,  y
>. ) )
15 vex 2919 . . . . . . . . . . . . 13  |-  x  e. 
_V
16 vex 2919 . . . . . . . . . . . . 13  |-  y  e. 
_V
1715, 16op2nd 6315 . . . . . . . . . . . 12  |-  ( 2nd `  <. x ,  y
>. )  =  y
1814, 17syl6eq 2452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  (
( 1st `  ( C  2ndF  D ) ) `  <. x ,  y >.
)  =  y )
191, 18syl5eq 2448 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( Base `  C
) )  /\  y  e.  ( Base `  D
) )  ->  (
x ( 1st `  ( C  2ndF  D ) ) y )  =  y )
2019mpteq2dva 4255 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( y  e.  ( Base `  D
)  |->  ( x ( 1st `  ( C  2ndF  D ) ) y ) )  =  ( y  e.  ( Base `  D )  |->  y ) )
21 mptresid 5154 . . . . . . . . 9  |-  ( y  e.  ( Base `  D
)  |->  y )  =  (  _I  |`  ( Base `  D ) )
2220, 21syl6eq 2452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( y  e.  ( Base `  D
)  |->  ( x ( 1st `  ( C  2ndF  D ) ) y ) )  =  (  _I  |`  ( Base `  D ) ) )
23 df-ov 6043 . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  C  e.  Cat )
2510ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  x  e.  ( Base `  C ) )  /\  y  e.  ( Base `  D ) )  /\  z  e.  ( Base `  D ) )  /\  f  e.  ( y
(  Hom  `  D ) z ) )  ->  D  e.  Cat )
2613ad2antrr 707 . . . . . . . . . . . . . . . . 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 744 . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . 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 4869 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  ( Base `  C )  /\  z  e.  ( Base `  D
) )  ->  <. x ,  z >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
3027, 28, 29syl2anc 643 . . . . . . . . . . . . . . . . 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 14248 . . . . . . . . . . . . . . . 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 5689 . . . . . . . . . . . . . . 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 2448 . . . . . . . . . . . . . 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 2404 . . . . . . . . . . . . . . . . . . . 20  |-  (  Hom  `  C )  =  (  Hom  `  C )
35 eqid 2404 . . . . . . . . . . . . . . . . . . . 20  |-  ( Id
`  C )  =  ( Id `  C
)
367adantr 452 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  C  e.  Cat )
37 simpr 448 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
383, 34, 35, 36, 37catidcl 13862 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  C ) `  x )  e.  ( x (  Hom  `  C
) x ) )
3938ad3antrrr 711 . . . . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . . . . 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 4869 . . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . . 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 2404 . . . . . . . . . . . . . . . . . 18  |-  (  Hom  `  D )  =  (  Hom  `  D )
44 simpllr 736 . . . . . . . . . . . . . . . . . 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 14238 . . . . . . . . . . . . . . . . 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 2481 . . . . . . . . . . . . . . . 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 5704 . . . . . . . . . . . . . . . 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 5701 . . . . . . . . . . . . . . . 16  |-  ( ( Id `  C ) `
 x )  e. 
_V
50 vex 2919 . . . . . . . . . . . . . . . 16  |-  f  e. 
_V
5149, 50op2nd 6315 . . . . . . . . . . . . . . 15  |-  ( 2nd `  <. ( ( Id
`  C ) `  x ) ,  f
>. )  =  f
5248, 51syl6eq 2452 . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . 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 4255 . . . . . . . . . . . 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 5154 . . . . . . . . . . . 12  |-  ( f  e.  ( y (  Hom  `  D )
z )  |->  f )  =  (  _I  |`  (
y (  Hom  `  D
) z ) )
5654, 55syl6eq 2452 . . . . . . . . . . 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 1148 . . . . . . . . . 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 6097 . . . . . . . . 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 5687 . . . . . . . . . . . 12  |-  ( u  =  <. y ,  z
>.  ->  ( (  Hom  `  D ) `  u
)  =  ( (  Hom  `  D ) `  <. y ,  z
>. ) )
60 df-ov 6043 . . . . . . . . . . . 12  |-  ( y (  Hom  `  D
) z )  =  ( (  Hom  `  D
) `  <. y ,  z >. )
6159, 60syl6eqr 2454 . . . . . . . . . . 11  |-  ( u  =  <. y ,  z
>.  ->  ( (  Hom  `  D ) `  u
)  =  ( y (  Hom  `  D
) z ) )
6261reseq2d 5105 . . . . . . . . . 10  |-  ( u  =  <. y ,  z
>.  ->  (  _I  |`  (
(  Hom  `  D ) `
 u ) )  =  (  _I  |`  (
y (  Hom  `  D
) z ) ) )
6362mpt2mpt 6124 . . . . . . . . 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 2454 . . . . . . . 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 3952 . . . . . . 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 2404 . . . . . . . 8  |-  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  =  ( <. C ,  D >. curryF  ( C  2ndF  D ) )
679adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
682, 7, 9, 112ndfcl 14250 . . . . . . . . 9  |-  ( ph  ->  ( C  2ndF  D )  e.  ( ( C  X.c  D
)  Func  D )
)
6968adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( C  2ndF  D )  e.  ( ( C  X.c  D )  Func  D
) )
70 eqid 2404 . . . . . . . 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 14277 . . . . . . 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 2404 . . . . . . . 8  |-  (idfunc `  D
)  =  (idfunc `  D
)
7372, 4, 67, 43idfuval 14028 . . . . . . 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 2446 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x )  =  (idfunc `  D ) )
75 eqid 2404 . . . . . . 7  |-  ( QΔfunc C )  =  ( QΔfunc C )
76 curf2ndf.q . . . . . . . . 9  |-  Q  =  ( D FuncCat  D )
7776, 9, 9fuccat 14122 . . . . . . . 8  |-  ( ph  ->  Q  e.  Cat )
7877adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  Q  e.  Cat )
7976fucbas 14112 . . . . . . 7  |-  ( D 
Func  D )  =  (
Base `  Q )
8072idfucl 14033 . . . . . . . . 9  |-  ( D  e.  Cat  ->  (idfunc `  D
)  e.  ( D 
Func  D ) )
819, 80syl 16 . . . . . . . 8  |-  ( ph  ->  (idfunc `  D )  e.  ( D  Func  D )
)
8281adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  (idfunc `  D )  e.  ( D  Func  D )
)
83 eqid 2404 . . . . . . 7  |-  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D
) )  =  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) )
8475, 78, 36, 79, 82, 83, 3, 37diag11 14295 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x )  =  (idfunc `  D ) )
8574, 84eqtr4d 2439 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) `
 x )  =  ( ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x ) )
8685mpteq2dva 4255 . . . 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 14014 . . . . . . 7  |-  Rel  ( C  Func  Q )
8866, 76, 7, 9, 68curfcl 14284 . . . . . . 7  |-  ( ph  ->  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  e.  ( C  Func  Q
) )
89 1st2ndbr 6355 . . . . . . 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 645 . . . . . 6  |-  ( ph  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) ( C  Func  Q
) ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) )
913, 79, 90funcf1 14018 . . . . 5  |-  ( ph  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) : ( Base `  C
) --> ( D  Func  D ) )
9291feqmptd 5738 . . . 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 14294 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) )  e.  ( C  Func  Q
) )
94 1st2ndbr 6355 . . . . . . 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 645 . . . . . 6  |-  ( ph  ->  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) ( C  Func  Q
) ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) )
963, 79, 95funcf1 14018 . . . . 5  |-  ( ph  ->  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) : ( Base `  C
) --> ( D  Func  D ) )
9796feqmptd 5738 . . . 4  |-  ( ph  ->  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) `
 x ) ) )
9886, 92, 973eqtr4d 2446 . . 3  |-  ( ph  ->  ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  =  ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) )
999ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  D  e.  Cat )
10072, 4, 99idfu1st 14031 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( 1st `  (idfunc `  D
) )  =  (  _I  |`  ( Base `  D ) ) )
101100coeq2d 4994 . . . . . . . . . 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 2404 . . . . . . . . . . 11  |-  ( Id
`  Q )  =  ( Id `  Q
)
103 eqid 2404 . . . . . . . . . . 11  |-  ( Id
`  D )  =  ( Id `  D
)
10481ad2antrr 707 . . . . . . . . . . 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 14123 . . . . . . . . . 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 13859 . . . . . . . . . . . . . 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 5551 . . . . . . . . . . . . 13  |-  ( ( Id `  D )  Fn  ( Base `  D
)  <->  ( Id `  D ) : (
Base `  D ) --> _V )
109107, 108sylib 189 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( Id `  D
) : ( Base `  D ) --> _V )
110109feqmptd 5738 . . . . . . . . . . 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 5576 . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  C  e.  Cat )
114113adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  /\  f  e.  ( x (  Hom  `  C
) y ) )  /\  z  e.  (
Base `  D )
)  ->  C  e.  Cat )
11599adantr 452 . . . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  x  e.  ( Base `  C ) )
117116, 29sylan 458 . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
y  e.  ( Base `  C ) )
119 opelxpi 4869 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  ( Base `  C )  /\  z  e.  ( Base `  D
) )  ->  <. y ,  z >.  e.  ( ( Base `  C
)  X.  ( Base `  D ) ) )
120118, 119sylan 458 . . . . . . . . . . . . . . 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 14248 . . . . . . . . . . . . . 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 6057 . . . . . . . . . . . . 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 6043 . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . . . . . 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 13862 . . . . . . . . . . . . . . . . . 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 4869 . . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . 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 14238 . . . . . . . . . . . . . . . . 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 2481 . . . . . . . . . . . . . . . 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 5704 . . . . . . . . . . . . . . . 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 2448 . . . . . . . . . . . . . 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 5701 . . . . . . . . . . . . . . 15  |-  ( ( Id `  D ) `
 z )  e. 
_V
13750, 136op2nd 6315 . . . . . . . . . . . . . 14  |-  ( 2nd `  <. f ,  ( ( Id `  D
) `  z ) >. )  =  ( ( Id `  D ) `
 z )
138135, 137syl6eq 2452 . . . . . . . . . . . . 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 2436 . . . . . . . . . . . 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 4255 . . . . . . . . . . 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 2447 . . . . . . . . . 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 2447 . . . . . . . . 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 707 . . . . . . . . . 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 448 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
f  e.  ( x (  Hom  `  C
) y ) )
145 eqid 2404 . . . . . . . . . 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 14281 . . . . . . . . 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 707 . . . . . . . . . 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 14296 . . . . . . . . 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 2446 . . . . . . . 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 4255 . . . . . . 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 2404 . . . . . . . . . 10  |-  ( D Nat 
D )  =  ( D Nat  D )
15276, 151fuchom 14113 . . . . . . . . 9  |-  ( D Nat 
D )  =  (  Hom  `  Q )
15390adantr 452 . . . . . . . . 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 733 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
155 simprr 734 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
1563, 34, 152, 153, 154, 155funcf2 14020 . . . . . . . 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 5738 . . . . . . 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 452 . . . . . . . . 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 14020 . . . . . . . 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 5738 . . . . . . 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 2446 . . . . . 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 1149 . . . . 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 6097 . . . 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 14021 . . . . 5  |-  ( ph  ->  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  Fn  ( ( Base `  C )  X.  ( Base `  C ) ) )
165 fnov 6137 . . . . 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 189 . . . 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 14021 . . . . 5  |-  ( ph  ->  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )  Fn  ( ( Base `  C )  X.  ( Base `  C ) ) )
168 fnov 6137 . . . . 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 189 . . . 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 2446 . . 3  |-  ( ph  ->  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )  =  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) )
17198, 170opeq12d 3952 . 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 6352 . . 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 645 . 2  |-  ( ph  ->  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  = 
<. ( 1st `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) ) ,  ( 2nd `  ( <. C ,  D >. curryF  ( C  2ndF  D ) ) )
>. )
174 1st2nd 6352 . . 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 645 . 2  |-  ( ph  ->  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) )  = 
<. ( 1st `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) ) ,  ( 2nd `  (
( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )
>. )
176171, 173, 1753eqtr4d 2446 1  |-  ( ph  ->  ( <. C ,  D >. curryF  ( C  2ndF  D ) )  =  ( ( 1st `  ( QΔfunc C ) ) `  (idfunc `  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777   class class class wbr 4172    e. cmpt 4226    _I cid 4453    X. cxp 4835    |` cres 4839    o. ccom 4841   Rel wrel 4842    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   Basecbs 13424    Hom chom 13495   Catccat 13844   Idccid 13845    Func cfunc 14006  idfunccidfu 14007   Nat cnat 14093   FuncCat cfuc 14094    X.c cxpc 14220    2ndF c2ndf 14222   curryF ccurf 14262  Δfunccdiag 14264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-hom 13508  df-cco 13509  df-cat 13848  df-cid 13849  df-func 14010  df-idfu 14011  df-nat 14095  df-fuc 14096  df-xpc 14224  df-1stf 14225  df-2ndf 14226  df-curf 14266  df-diag 14268
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