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Theorem fuccatid 15826
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 15817 . . 3  |-  ( C 
Func  D )  =  (
Base `  Q )
32a1i 11 . 2  |-  ( ph  ->  ( C  Func  D
)  =  ( Base `  Q ) )
4 eqid 2420 . . . 4  |-  ( C Nat 
D )  =  ( C Nat  D )
51, 4fuchom 15818 . . 3  |-  ( C Nat 
D )  =  ( Hom  `  Q )
65a1i 11 . 2  |-  ( ph  ->  ( C Nat  D )  =  ( Hom  `  Q
) )
7 eqidd 2421 . 2  |-  ( ph  ->  (comp `  Q )  =  (comp `  Q )
)
8 ovex 6324 . . . 4  |-  ( C FuncCat  D )  e.  _V
91, 8eqeltri 2504 . . 3  |-  Q  e. 
_V
109a1i 11 . 2  |-  ( ph  ->  Q  e.  _V )
11 biid 239 . 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 462 . . 3  |-  ( (
ph  /\  f  e.  ( C  Func  D ) )  ->  f  e.  ( C  Func  D ) )
141, 4, 12, 13fucidcl 15822 . 2  |-  ( (
ph  /\  f  e.  ( C  Func  D ) )  ->  (  .1.  o.  ( 1st `  f
) )  e.  ( f ( C Nat  D
) f ) )
15 eqid 2420 . . 3  |-  (comp `  Q )  =  (comp `  Q )
16 simpr31 1095 . . 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 15823 . 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 1096 . . 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 15824 . 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 15821 . 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 1097 . . 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 15825 . 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 15539 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 370    /\ w3a 982    = wceq 1437    e. wcel 1867   _Vcvv 3078    |-> cmpt 4475    o. ccom 4849   ` cfv 5592  (class class class)co 6296   1stc1st 6796   Basecbs 15081   Hom chom 15161  compcco 15162   Catccat 15522   Idccid 15523    Func cfunc 15711   Nat cnat 15798   FuncCat cfuc 15799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-fz 11772  df-struct 15083  df-ndx 15084  df-slot 15085  df-base 15086  df-hom 15174  df-cco 15175  df-cat 15526  df-cid 15527  df-func 15715  df-nat 15800  df-fuc 15801
This theorem is referenced by:  fuccat  15827  fucid  15828
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