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Theorem fucidcl 14164
Description: The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fucidcl.q  |-  Q  =  ( C FuncCat  D )
fucidcl.n  |-  N  =  ( C Nat  D )
fucidcl.x  |-  .1.  =  ( Id `  D )
fucidcl.f  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
Assertion
Ref Expression
fucidcl  |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  e.  ( F N F ) )

Proof of Theorem fucidcl
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucidcl.f . . . . . . . 8  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
2 funcrcl 14062 . . . . . . . 8  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
43simprd 451 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
5 eqid 2438 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
6 fucidcl.x . . . . . . 7  |-  .1.  =  ( Id `  D )
75, 6cidfn 13906 . . . . . 6  |-  ( D  e.  Cat  ->  .1.  Fn  ( Base `  D
) )
84, 7syl 16 . . . . 5  |-  ( ph  ->  .1.  Fn  ( Base `  D ) )
9 dffn2 5594 . . . . 5  |-  (  .1. 
Fn  ( Base `  D
)  <->  .1.  : ( Base `  D ) --> _V )
108, 9sylib 190 . . . 4  |-  ( ph  ->  .1.  : ( Base `  D ) --> _V )
11 eqid 2438 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
12 relfunc 14061 . . . . . 6  |-  Rel  ( C  Func  D )
13 1st2ndbr 6398 . . . . . 6  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1412, 1, 13sylancr 646 . . . . 5  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
1511, 5, 14funcf1 14065 . . . 4  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
16 fcompt 5906 . . . 4  |-  ( (  .1.  : ( Base `  D ) --> _V  /\  ( 1st `  F ) : ( Base `  C
) --> ( Base `  D
) )  ->  (  .1.  o.  ( 1st `  F
) )  =  ( x  e.  ( Base `  C )  |->  (  .1.  `  ( ( 1st `  F
) `  x )
) ) )
1710, 15, 16syl2anc 644 . . 3  |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  =  ( x  e.  ( Base `  C
)  |->  (  .1.  `  ( ( 1st `  F
) `  x )
) ) )
18 eqid 2438 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
194adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
2015ffvelrnda 5872 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
215, 18, 6, 19, 20catidcl 13909 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  (  .1.  `  ( ( 1st `  F
) `  x )
)  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) )
2221ralrimiva 2791 . . . 4  |-  ( ph  ->  A. x  e.  (
Base `  C )
(  .1.  `  (
( 1st `  F
) `  x )
)  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) )
23 fvex 5744 . . . . 5  |-  ( Base `  C )  e.  _V
24 mptelixpg 7101 . . . . 5  |-  ( (
Base `  C )  e.  _V  ->  ( (
x  e.  ( Base `  C )  |->  (  .1.  `  ( ( 1st `  F
) `  x )
) )  e.  X_ x  e.  ( Base `  C ) ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
)  <->  A. x  e.  (
Base `  C )
(  .1.  `  (
( 1st `  F
) `  x )
)  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) ) )
2523, 24ax-mp 8 . . . 4  |-  ( ( x  e.  ( Base `  C )  |->  (  .1.  `  ( ( 1st `  F
) `  x )
) )  e.  X_ x  e.  ( Base `  C ) ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
)  <->  A. x  e.  (
Base `  C )
(  .1.  `  (
( 1st `  F
) `  x )
)  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) )
2622, 25sylibr 205 . . 3  |-  ( ph  ->  ( x  e.  (
Base `  C )  |->  (  .1.  `  (
( 1st `  F
) `  x )
) )  e.  X_ x  e.  ( Base `  C ) ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) )
2717, 26eqeltrd 2512 . 2  |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  e.  X_ x  e.  (
Base `  C )
( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  x
) ) )
284adantr 453 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  D  e.  Cat )
29 simpr1 964 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  x  e.  ( Base `  C )
)
3029, 20syldan 458 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
31 eqid 2438 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
3215adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( 1st `  F ) : (
Base `  C ) --> ( Base `  D )
)
33 simpr2 965 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  y  e.  ( Base `  C )
)
3432, 33ffvelrnd 5873 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
35 eqid 2438 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
3614adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( 1st `  F ) ( C 
Func  D ) ( 2nd `  F ) )
3711, 35, 18, 36, 29, 33funcf2 14067 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( x
( 2nd `  F
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) ) )
38 simpr3 966 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  f  e.  ( x (  Hom  `  C ) y ) )
3937, 38ffvelrnd 5873 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
x ( 2nd `  F
) y ) `  f )  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) )
405, 18, 6, 28, 30, 31, 34, 39catlid 13910 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (  .1.  `  ( ( 1st `  F ) `  y
) ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( x ( 2nd `  F ) y ) `  f
) )
415, 18, 6, 28, 30, 31, 34, 39catrid 13911 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
( x ( 2nd `  F ) y ) `
 f ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) (  .1.  `  ( ( 1st `  F
) `  x )
) )  =  ( ( x ( 2nd `  F ) y ) `
 f ) )
4240, 41eqtr4d 2473 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (  .1.  `  ( ( 1st `  F ) `  y
) ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  F
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) (  .1.  `  ( ( 1st `  F
) `  x )
) ) )
43 fvco3 5802 . . . . . 6  |-  ( ( ( 1st `  F
) : ( Base `  C ) --> ( Base `  D )  /\  y  e.  ( Base `  C
) )  ->  (
(  .1.  o.  ( 1st `  F ) ) `
 y )  =  (  .1.  `  (
( 1st `  F
) `  y )
) )
4432, 33, 43syl2anc 644 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (  .1.  o.  ( 1st `  F
) ) `  y
)  =  (  .1.  `  ( ( 1st `  F
) `  y )
) )
4544oveq1d 6098 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
(  .1.  o.  ( 1st `  F ) ) `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( (  .1.  `  ( ( 1st `  F
) `  y )
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) )
46 fvco3 5802 . . . . . 6  |-  ( ( ( 1st `  F
) : ( Base `  C ) --> ( Base `  D )  /\  x  e.  ( Base `  C
) )  ->  (
(  .1.  o.  ( 1st `  F ) ) `
 x )  =  (  .1.  `  (
( 1st `  F
) `  x )
) )
4732, 29, 46syl2anc 644 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (  .1.  o.  ( 1st `  F
) ) `  x
)  =  (  .1.  `  ( ( 1st `  F
) `  x )
) )
4847oveq2d 6099 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
( x ( 2nd `  F ) y ) `
 f ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) ( (  .1.  o.  ( 1st `  F ) ) `  x ) )  =  ( ( ( x ( 2nd `  F
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) (  .1.  `  ( ( 1st `  F
) `  x )
) ) )
4942, 45, 483eqtr4d 2480 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
(  .1.  o.  ( 1st `  F ) ) `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  F
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) ( (  .1.  o.  ( 1st `  F ) ) `  x ) ) )
5049ralrimivvva 2801 . 2  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) A. f  e.  ( x
(  Hom  `  C ) y ) ( ( (  .1.  o.  ( 1st `  F ) ) `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  F
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) ( (  .1.  o.  ( 1st `  F ) ) `  x ) ) )
51 fucidcl.n . . 3  |-  N  =  ( C Nat  D )
5251, 11, 35, 18, 31, 1, 1isnat2 14147 . 2  |-  ( ph  ->  ( (  .1.  o.  ( 1st `  F ) )  e.  ( F N F )  <->  ( (  .1.  o.  ( 1st `  F
) )  e.  X_ x  e.  ( Base `  C ) ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
)  /\  A. x  e.  ( Base `  C
) A. y  e.  ( Base `  C
) A. f  e.  ( x (  Hom  `  C ) y ) ( ( (  .1. 
o.  ( 1st `  F
) ) `  y
) ( <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  F
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  x
) >. (comp `  D
) ( ( 1st `  F ) `  y
) ) ( (  .1.  o.  ( 1st `  F ) ) `  x ) ) ) ) )
5327, 50, 52mpbir2and 890 1  |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  e.  ( F N F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958   <.cop 3819   class class class wbr 4214    e. cmpt 4268    o. ccom 4884   Rel wrel 4885    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350   X_cixp 7065   Basecbs 13471    Hom chom 13542  compcco 13543   Catccat 13891   Idccid 13892    Func cfunc 14053   Nat cnat 14140   FuncCat cfuc 14141
This theorem is referenced by:  fuclid  14165  fucrid  14166  fuccatid  14168
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
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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-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-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  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-map 7022  df-ixp 7066  df-cat 13895  df-cid 13896  df-func 14057  df-nat 14142
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