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

Theorem fuccatid 15199
Description: The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fuccat.q  |-  Q  =  ( C FuncCat  D )
fuccat.r  |-  ( ph  ->  C  e.  Cat )
fuccat.s  |-  ( ph  ->  D  e.  Cat )
fuccatid.1  |-  .1.  =  ( Id `  D )
Assertion
Ref Expression
fuccatid  |-  ( ph  ->  ( Q  e.  Cat  /\  ( Id `  Q
)  =  ( f  e.  ( C  Func  D )  |->  (  .1.  o.  ( 1st `  f ) ) ) ) )
Distinct variable groups:    C, f    ph, f    D, f    Q, f
Allowed substitution hint:    .1. ( f)

Proof of Theorem fuccatid
Dummy variables  e 
g  h  r  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fuccat.q . . . 4  |-  Q  =  ( C FuncCat  D )
21fucbas 15190 . . 3  |-  ( C 
Func  D )  =  (
Base `  Q )
32a1i 11 . 2  |-  ( ph  ->  ( C  Func  D
)  =  ( Base `  Q ) )
4 eqid 2467 . . . 4  |-  ( C Nat 
D )  =  ( C Nat  D )
51, 4fuchom 15191 . . 3  |-  ( C Nat 
D )  =  ( Hom  `  Q )
65a1i 11 . 2  |-  ( ph  ->  ( C Nat  D )  =  ( Hom  `  Q
) )
7 eqidd 2468 . 2  |-  ( ph  ->  (comp `  Q )  =  (comp `  Q )
)
8 ovex 6310 . . . 4  |-  ( C FuncCat  D )  e.  _V
91, 8eqeltri 2551 . . 3  |-  Q  e. 
_V
109a1i 11 . 2  |-  ( ph  ->  Q  e.  _V )
11 biid 236 . 2  |-  ( ( ( e  e.  ( C  Func  D )  /\  f  e.  ( C  Func  D ) )  /\  ( g  e.  ( C  Func  D
)  /\  h  e.  ( C  Func  D ) )  /\  ( r  e.  ( e ( C Nat  D ) f )  /\  s  e.  ( f ( C Nat 
D ) g )  /\  t  e.  ( g ( C Nat  D
) h ) ) )  <->  ( ( e  e.  ( C  Func  D )  /\  f  e.  ( C  Func  D
) )  /\  (
g  e.  ( C 
Func  D )  /\  h  e.  ( C  Func  D
) )  /\  (
r  e.  ( e ( C Nat  D ) f )  /\  s  e.  ( f ( C Nat 
D ) g )  /\  t  e.  ( g ( C Nat  D
) h ) ) ) )
12 fuccatid.1 . . 3  |-  .1.  =  ( Id `  D )
13 simpr 461 . . 3  |-  ( (
ph  /\  f  e.  ( C  Func  D ) )  ->  f  e.  ( C  Func  D ) )
141, 4, 12, 13fucidcl 15195 . 2  |-  ( (
ph  /\  f  e.  ( C  Func  D ) )  ->  (  .1.  o.  ( 1st `  f
) )  e.  ( f ( C Nat  D
) f ) )
15 eqid 2467 . . 3  |-  (comp `  Q )  =  (comp `  Q )
16 simpr31 1086 . . 3  |-  ( (
ph  /\  ( (
e  e.  ( C 
Func  D )  /\  f  e.  ( C  Func  D
) )  /\  (
g  e.  ( C 
Func  D )  /\  h  e.  ( C  Func  D
) )  /\  (
r  e.  ( e ( C Nat  D ) f )  /\  s  e.  ( f ( C Nat 
D ) g )  /\  t  e.  ( g ( C Nat  D
) h ) ) ) )  ->  r  e.  ( e ( C Nat 
D ) f ) )
171, 4, 15, 12, 16fuclid 15196 . 2  |-  ( (
ph  /\  ( (
e  e.  ( C 
Func  D )  /\  f  e.  ( C  Func  D
) )  /\  (
g  e.  ( C 
Func  D )  /\  h  e.  ( C  Func  D
) )  /\  (
r  e.  ( e ( C Nat  D ) f )  /\  s  e.  ( f ( C Nat 
D ) g )  /\  t  e.  ( g ( C Nat  D
) h ) ) ) )  ->  (
(  .1.  o.  ( 1st `  f ) ) ( <. e ,  f
>. (comp `  Q )
f ) r )  =  r )
18 simpr32 1087 . . 3  |-  ( (
ph  /\  ( (
e  e.  ( C 
Func  D )  /\  f  e.  ( C  Func  D
) )  /\  (
g  e.  ( C 
Func  D )  /\  h  e.  ( C  Func  D
) )  /\  (
r  e.  ( e ( C Nat  D ) f )  /\  s  e.  ( f ( C Nat 
D ) g )  /\  t  e.  ( g ( C Nat  D
) h ) ) ) )  ->  s  e.  ( f ( C Nat 
D ) g ) )
191, 4, 15, 12, 18fucrid 15197 . 2  |-  ( (
ph  /\  ( (
e  e.  ( C 
Func  D )  /\  f  e.  ( C  Func  D
) )  /\  (
g  e.  ( C 
Func  D )  /\  h  e.  ( C  Func  D
) )  /\  (
r  e.  ( e ( C Nat  D ) f )  /\  s  e.  ( f ( C Nat 
D ) g )  /\  t  e.  ( g ( C Nat  D
) h ) ) ) )  ->  (
s ( <. f ,  f >. (comp `  Q ) g ) (  .1.  o.  ( 1st `  f ) ) )  =  s )
201, 4, 15, 16, 18fuccocl 15194 . 2  |-  ( (
ph  /\  ( (
e  e.  ( C 
Func  D )  /\  f  e.  ( C  Func  D
) )  /\  (
g  e.  ( C 
Func  D )  /\  h  e.  ( C  Func  D
) )  /\  (
r  e.  ( e ( C Nat  D ) f )  /\  s  e.  ( f ( C Nat 
D ) g )  /\  t  e.  ( g ( C Nat  D
) h ) ) ) )  ->  (
s ( <. e ,  f >. (comp `  Q ) g ) r )  e.  ( e ( C Nat  D
) g ) )
21 simpr33 1088 . . 3  |-  ( (
ph  /\  ( (
e  e.  ( C 
Func  D )  /\  f  e.  ( C  Func  D
) )  /\  (
g  e.  ( C 
Func  D )  /\  h  e.  ( C  Func  D
) )  /\  (
r  e.  ( e ( C Nat  D ) f )  /\  s  e.  ( f ( C Nat 
D ) g )  /\  t  e.  ( g ( C Nat  D
) h ) ) ) )  ->  t  e.  ( g ( C Nat 
D ) h ) )
221, 4, 15, 16, 18, 21fucass 15198 . 2  |-  ( (
ph  /\  ( (
e  e.  ( C 
Func  D )  /\  f  e.  ( C  Func  D
) )  /\  (
g  e.  ( C 
Func  D )  /\  h  e.  ( C  Func  D
) )  /\  (
r  e.  ( e ( C Nat  D ) f )  /\  s  e.  ( f ( C Nat 
D ) g )  /\  t  e.  ( g ( C Nat  D
) h ) ) ) )  ->  (
( t ( <.
f ,  g >.
(comp `  Q )
h ) s ) ( <. e ,  f
>. (comp `  Q )
h ) r )  =  ( t (
<. e ,  g >.
(comp `  Q )
h ) ( s ( <. e ,  f
>. (comp `  Q )
g ) r ) ) )
233, 6, 7, 10, 11, 14, 17, 19, 20, 22iscatd2 14939 1  |-  ( ph  ->  ( Q  e.  Cat  /\  ( Id `  Q
)  =  ( f  e.  ( C  Func  D )  |->  (  .1.  o.  ( 1st `  f ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    |-> cmpt 4505    o. ccom 5003   ` cfv 5588  (class class class)co 6285   1stc1st 6783   Basecbs 14493   Hom chom 14569  compcco 14570   Catccat 14922   Idccid 14923    Func cfunc 15084   Nat cnat 15171   FuncCat cfuc 15172
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 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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-nel 2665  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-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-fz 11674  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-hom 14582  df-cco 14583  df-cat 14926  df-cid 14927  df-func 15088  df-nat 15173  df-fuc 15174
This theorem is referenced by:  fuccat  15200  fucid  15201
  Copyright terms: Public domain W3C validator