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Theorem fucidcl 13839
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 13737 . . . . . . . 8  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
31, 2syl 15 . . . . . . 7  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
43simprd 449 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
5 eqid 2283 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
6 fucidcl.x . . . . . . 7  |-  .1.  =  ( Id `  D )
75, 6cidfn 13581 . . . . . 6  |-  ( D  e.  Cat  ->  .1.  Fn  ( Base `  D
) )
84, 7syl 15 . . . . 5  |-  ( ph  ->  .1.  Fn  ( Base `  D ) )
9 dffn2 5390 . . . . 5  |-  (  .1. 
Fn  ( Base `  D
)  <->  .1.  : ( Base `  D ) --> _V )
108, 9sylib 188 . . . 4  |-  ( ph  ->  .1.  : ( Base `  D ) --> _V )
11 eqid 2283 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
12 relfunc 13736 . . . . . 6  |-  Rel  ( C  Func  D )
13 1st2ndbr 6169 . . . . . 6  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1412, 1, 13sylancr 644 . . . . 5  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
1511, 5, 14funcf1 13740 . . . 4  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
16 fcompt 5694 . . . 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 642 . . 3  |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  =  ( x  e.  ( Base `  C
)  |->  (  .1.  `  ( ( 1st `  F
) `  x )
) ) )
18 eqid 2283 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
194adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
2015ffvelrnda 5665 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
215, 18, 6, 19, 20catidcl 13584 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  (  .1.  `  ( ( 1st `  F
) `  x )
)  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) )
2221ralrimiva 2626 . . . 4  |-  ( ph  ->  A. x  e.  (
Base `  C )
(  .1.  `  (
( 1st `  F
) `  x )
)  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  x )
) )
23 fvex 5539 . . . . 5  |-  ( Base `  C )  e.  _V
24 mptelixpg 6853 . . . . 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 203 . . 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 2357 . 2  |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  e.  X_ x  e.  (
Base `  C )
( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  x
) ) )
284adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  D  e.  Cat )
29 simpr1 961 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  x  e.  ( Base `  C )
)
3029, 20syldan 456 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
31 eqid 2283 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
3215adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( 1st `  F ) : (
Base `  C ) --> ( Base `  D )
)
33 simpr2 962 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  y  e.  ( Base `  C )
)
3432, 33ffvelrnd 5666 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
35 eqid 2283 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
3614adantr 451 . . . . . . . 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 13742 . . . . . . 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 963 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  f  e.  ( x (  Hom  `  C ) y ) )
3937, 38ffvelrnd 5666 . . . . . 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 13585 . . . . 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 13586 . . . . 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 2318 . . . 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 5596 . . . . . 6  |-  ( ( ( 1st `  F
) : ( Base `  C ) --> ( Base `  D )  /\  y  e.  ( Base `  C
) )  ->  (
(  .1.  o.  ( 1st `  F ) ) `
 y )  =  (  .1.  `  (
( 1st `  F
) `  y )
) )
4432, 33, 43syl2anc 642 . . . . 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 5873 . . . 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 5596 . . . . . 6  |-  ( ( ( 1st `  F
) : ( Base `  C ) --> ( Base `  D )  /\  x  e.  ( Base `  C
) )  ->  (
(  .1.  o.  ( 1st `  F ) ) `
 x )  =  (  .1.  `  (
( 1st `  F
) `  x )
) )
4732, 29, 46syl2anc 642 . . . . 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 5874 . . . 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 2325 . . 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 2636 . 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 13822 . 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 888 1  |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  e.  ( F N F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   <.cop 3643   class class class wbr 4023    e. cmpt 4077    o. ccom 4693   Rel wrel 4694    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   X_cixp 6817   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567    Func cfunc 13728   Nat cnat 13815   FuncCat cfuc 13816
This theorem is referenced by:  fuclid  13840  fucrid  13841  fuccatid  13843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-map 6774  df-ixp 6818  df-cat 13570  df-cid 13571  df-func 13732  df-nat 13817
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