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Theorem fuchom 14160
Description: The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fucbas.q  |-  Q  =  ( C FuncCat  D )
fuchom.n  |-  N  =  ( C Nat  D )
Assertion
Ref Expression
fuchom  |-  N  =  (  Hom  `  Q
)

Proof of Theorem fuchom
Dummy variables  a 
b  f  g  h  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucbas.q . . . . 5  |-  Q  =  ( C FuncCat  D )
2 eqid 2438 . . . . 5  |-  ( C 
Func  D )  =  ( C  Func  D )
3 fuchom.n . . . . 5  |-  N  =  ( C Nat  D )
4 eqid 2438 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
5 eqid 2438 . . . . 5  |-  (comp `  D )  =  (comp `  D )
6 simpl 445 . . . . 5  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  C  e.  Cat )
7 simpr 449 . . . . 5  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  D  e.  Cat )
8 eqid 2438 . . . . . 6  |-  (comp `  Q )  =  (comp `  Q )
91, 2, 3, 4, 5, 6, 7, 8fuccofval 14158 . . . . 5  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  (comp `  Q )  =  ( v  e.  ( ( C  Func  D )  X.  ( C 
Func  D ) ) ,  h  e.  ( C 
Func  D )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g N h ) ,  a  e.  ( f N g ) 
|->  ( x  e.  (
Base `  C )  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  D
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) )
101, 2, 3, 4, 5, 6, 7, 9fucval 14157 . . . 4  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  Q  =  { <. (
Base `  ndx ) ,  ( C  Func  D
) >. ,  <. (  Hom  `  ndx ) ,  N >. ,  <. (comp ` 
ndx ) ,  (comp `  Q ) >. } )
11 catstr 14156 . . . 4  |-  { <. (
Base `  ndx ) ,  ( C  Func  D
) >. ,  <. (  Hom  `  ndx ) ,  N >. ,  <. (comp ` 
ndx ) ,  (comp `  Q ) >. } Struct  <. 1 , ; 1 5 >.
12 homid 13645 . . . 4  |-  Hom  = Slot  (  Hom  `  ndx )
13 snsstp2 3952 . . . 4  |-  { <. (  Hom  `  ndx ) ,  N >. }  C_  { <. (
Base `  ndx ) ,  ( C  Func  D
) >. ,  <. (  Hom  `  ndx ) ,  N >. ,  <. (comp ` 
ndx ) ,  (comp `  Q ) >. }
14 ovex 6108 . . . . . 6  |-  ( C Nat 
D )  e.  _V
153, 14eqeltri 2508 . . . . 5  |-  N  e. 
_V
1615a1i 11 . . . 4  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  N  e.  _V )
17 eqid 2438 . . . 4  |-  (  Hom  `  Q )  =  (  Hom  `  Q )
1810, 11, 12, 13, 16, 17strfv3 13504 . . 3  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  (  Hom  `  Q
)  =  N )
1918eqcomd 2443 . 2  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  N  =  (  Hom  `  Q ) )
20 df-hom 13555 . . . 4  |-  Hom  = Slot ; 1 4
2120str0 13507 . . 3  |-  (/)  =  (  Hom  `  (/) )
223natffn 14148 . . . . 5  |-  N  Fn  ( ( C  Func  D )  X.  ( C 
Func  D ) )
23 funcrcl 14062 . . . . . . . . . 10  |-  ( f  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
2423con3i 130 . . . . . . . . 9  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  -.  f  e.  ( C  Func  D )
)
2524eq0rdv 3664 . . . . . . . 8  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C  Func  D
)  =  (/) )
2625xpeq2d 4904 . . . . . . 7  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( ( C  Func  D )  X.  ( C 
Func  D ) )  =  ( ( C  Func  D )  X.  (/) ) )
27 xp0 5293 . . . . . . 7  |-  ( ( C  Func  D )  X.  (/) )  =  (/)
2826, 27syl6eq 2486 . . . . . 6  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( ( C  Func  D )  X.  ( C 
Func  D ) )  =  (/) )
2928fneq2d 5539 . . . . 5  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( N  Fn  (
( C  Func  D
)  X.  ( C 
Func  D ) )  <->  N  Fn  (/) ) )
3022, 29mpbii 204 . . . 4  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  N  Fn  (/) )
31 fn0 5566 . . . 4  |-  ( N  Fn  (/)  <->  N  =  (/) )
3230, 31sylib 190 . . 3  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  N  =  (/) )
33 fnfuc 14144 . . . . . . 7  |- FuncCat  Fn  ( Cat  X.  Cat )
34 fndm 5546 . . . . . . 7  |-  ( FuncCat  Fn  ( Cat  X.  Cat )  ->  dom FuncCat  =  ( Cat  X. 
Cat ) )
3533, 34ax-mp 8 . . . . . 6  |-  dom FuncCat  =  ( Cat  X.  Cat )
3635ndmov 6233 . . . . 5  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C FuncCat  D )  =  (/) )
371, 36syl5eq 2482 . . . 4  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  Q  =  (/) )
3837fveq2d 5734 . . 3  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  (  Hom  `  Q
)  =  (  Hom  `  (/) ) )
3921, 32, 383eqtr4a 2496 . 2  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  N  =  (  Hom  `  Q ) )
4019, 39pm2.61i 159 1  |-  N  =  (  Hom  `  Q
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   {ctp 3818   <.cop 3819    X. cxp 4878   dom cdm 4880    Fn wfn 5451   ` cfv 5456  (class class class)co 6083   1c1 8993   4c4 10053   5c5 10054  ;cdc 10384   ndxcnx 13468   Basecbs 13471    Hom chom 13542  compcco 13543   Catccat 13891    Func cfunc 14053   Nat cnat 14140   FuncCat cfuc 14141
This theorem is referenced by:  fuccatid  14168  fucsect  14171  fuciso  14174  evlfcllem  14320  evlfcl  14321  curfcl  14331  uncf2  14336  curfuncf  14337  diag2cl  14345  curf2ndf  14346  yonedalem21  14372  yonedalem22  14377  yonedalem3b  14378  yonedalem3  14379  yonffthlem  14381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-hom 13555  df-cco 13556  df-func 14057  df-nat 14142  df-fuc 14143
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