MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fucidcl Structured version   Unicode version

Theorem fucidcl 15313
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 15211 . . . . . . . 8  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
43simprd 463 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
5 eqid 2443 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
6 fucidcl.x . . . . . . 7  |-  .1.  =  ( Id `  D )
75, 6cidfn 15058 . . . . . 6  |-  ( D  e.  Cat  ->  .1.  Fn  ( Base `  D
) )
84, 7syl 16 . . . . 5  |-  ( ph  ->  .1.  Fn  ( Base `  D ) )
9 dffn2 5722 . . . . 5  |-  (  .1. 
Fn  ( Base `  D
)  <->  .1.  : ( Base `  D ) --> _V )
108, 9sylib 196 . . . 4  |-  ( ph  ->  .1.  : ( Base `  D ) --> _V )
11 eqid 2443 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
12 relfunc 15210 . . . . . 6  |-  Rel  ( C  Func  D )
13 1st2ndbr 6834 . . . . . 6  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1412, 1, 13sylancr 663 . . . . 5  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
1511, 5, 14funcf1 15214 . . . 4  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
16 fcompt 6052 . . . 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 661 . . 3  |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  =  ( x  e.  ( Base `  C
)  |->  (  .1.  `  ( ( 1st `  F
) `  x )
) ) )
18 eqid 2443 . . . . . 6  |-  ( Hom  `  D )  =  ( Hom  `  D )
194adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
2015ffvelrnda 6016 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
215, 18, 6, 19, 20catidcl 15061 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  (  .1.  `  ( ( 1st `  F
) `  x )
)  e.  ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  x )
) )
2221ralrimiva 2857 . . . 4  |-  ( ph  ->  A. x  e.  (
Base `  C )
(  .1.  `  (
( 1st `  F
) `  x )
)  e.  ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  x )
) )
23 fvex 5866 . . . . 5  |-  ( Base `  C )  e.  _V
24 mptelixpg 7508 . . . . 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 5 . . . 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 212 . . 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 2531 . 2  |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  e.  X_ x  e.  (
Base `  C )
( ( ( 1st `  F ) `  x
) ( Hom  `  D
) ( ( 1st `  F ) `  x
) ) )
284adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  D  e.  Cat )
29 simpr1 1003 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  x  e.  ( Base `  C )
)
3029, 20syldan 470 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
31 eqid 2443 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
3215adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( 1st `  F ) : (
Base `  C ) --> ( Base `  D )
)
33 simpr2 1004 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  y  e.  ( Base `  C )
)
3432, 33ffvelrnd 6017 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
35 eqid 2443 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
3614adantr 465 . . . . . . . 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 15216 . . . . . . 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 1005 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  f  e.  ( x ( Hom  `  C ) y ) )
3937, 38ffvelrnd 6017 . . . . . 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 15062 . . . . 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 15063 . . . . 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 2487 . . . 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 5935 . . . . . 6  |-  ( ( ( 1st `  F
) : ( Base `  C ) --> ( Base `  D )  /\  y  e.  ( Base `  C
) )  ->  (
(  .1.  o.  ( 1st `  F ) ) `
 y )  =  (  .1.  `  (
( 1st `  F
) `  y )
) )
4432, 33, 43syl2anc 661 . . . . 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 6296 . . . 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 5935 . . . . . 6  |-  ( ( ( 1st `  F
) : ( Base `  C ) --> ( Base `  D )  /\  x  e.  ( Base `  C
) )  ->  (
(  .1.  o.  ( 1st `  F ) ) `
 x )  =  (  .1.  `  (
( 1st `  F
) `  x )
) )
4732, 29, 46syl2anc 661 . . . . 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 6297 . . . 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 2494 . . 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 2865 . 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 15296 . 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 922 1  |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  e.  ( F N F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095   <.cop 4020   class class class wbr 4437    |-> cmpt 4495    o. ccom 4993   Rel wrel 4994    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   X_cixp 7471   Basecbs 14614   Hom chom 14690  compcco 14691   Catccat 15043   Idccid 15044    Func cfunc 15202   Nat cnat 15289   FuncCat cfuc 15290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-map 7424  df-ixp 7472  df-cat 15047  df-cid 15048  df-func 15206  df-nat 15291
This theorem is referenced by:  fuclid  15314  fucrid  15315  fuccatid  15317
  Copyright terms: Public domain W3C validator