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Theorem fucidcl 15188
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 15086 . . . . . . . 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 2467 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
6 fucidcl.x . . . . . . 7  |-  .1.  =  ( Id `  D )
75, 6cidfn 14930 . . . . . 6  |-  ( D  e.  Cat  ->  .1.  Fn  ( Base `  D
) )
84, 7syl 16 . . . . 5  |-  ( ph  ->  .1.  Fn  ( Base `  D ) )
9 dffn2 5730 . . . . 5  |-  (  .1. 
Fn  ( Base `  D
)  <->  .1.  : ( Base `  D ) --> _V )
108, 9sylib 196 . . . 4  |-  ( ph  ->  .1.  : ( Base `  D ) --> _V )
11 eqid 2467 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
12 relfunc 15085 . . . . . 6  |-  Rel  ( C  Func  D )
13 1st2ndbr 6830 . . . . . 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 15089 . . . 4  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
16 fcompt 6055 . . . 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 2467 . . . . . 6  |-  ( Hom  `  D )  =  ( Hom  `  D )
194adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
2015ffvelrnda 6019 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
215, 18, 6, 19, 20catidcl 14933 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  (  .1.  `  ( ( 1st `  F
) `  x )
)  e.  ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  x )
) )
2221ralrimiva 2878 . . . 4  |-  ( ph  ->  A. x  e.  (
Base `  C )
(  .1.  `  (
( 1st `  F
) `  x )
)  e.  ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  x )
) )
23 fvex 5874 . . . . 5  |-  ( Base `  C )  e.  _V
24 mptelixpg 7503 . . . . 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 2555 . 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 1002 . . . . . . 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 2467 . . . . . 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 1003 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  y  e.  ( Base `  C )
)
3432, 33ffvelrnd 6020 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
35 eqid 2467 . . . . . . . 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 15091 . . . . . . 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 1004 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  f  e.  ( x ( Hom  `  C ) y ) )
3937, 38ffvelrnd 6020 . . . . . 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 14934 . . . . 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 14935 . . . . 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 2511 . . . 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 5942 . . . . . 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 6297 . . . 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 5942 . . . . . 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 6298 . . . 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 2518 . . 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 2886 . 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 15171 . 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 920 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    o. ccom 5003   Rel wrel 5004    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   X_cixp 7466   Basecbs 14486   Hom chom 14562  compcco 14563   Catccat 14915   Idccid 14916    Func cfunc 15077   Nat cnat 15164   FuncCat cfuc 15165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419  df-ixp 7467  df-cat 14919  df-cid 14920  df-func 15081  df-nat 15166
This theorem is referenced by:  fuclid  15189  fucrid  15190  fuccatid  15192
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