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Theorem funcf1 14758
Description: The object part of a functor is a function on objects. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
funcf1.b  |-  B  =  ( Base `  D
)
funcf1.c  |-  C  =  ( Base `  E
)
funcf1.f  |-  ( ph  ->  F ( D  Func  E ) G )
Assertion
Ref Expression
funcf1  |-  ( ph  ->  F : B --> C )

Proof of Theorem funcf1
Dummy variables  m  n  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcf1.f . . 3  |-  ( ph  ->  F ( D  Func  E ) G )
2 funcf1.b . . . 4  |-  B  =  ( Base `  D
)
3 funcf1.c . . . 4  |-  C  =  ( Base `  E
)
4 eqid 2433 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
5 eqid 2433 . . . 4  |-  ( Hom  `  E )  =  ( Hom  `  E )
6 eqid 2433 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
7 eqid 2433 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
8 eqid 2433 . . . 4  |-  (comp `  D )  =  (comp `  D )
9 eqid 2433 . . . 4  |-  (comp `  E )  =  (comp `  E )
10 df-br 4281 . . . . . . 7  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
111, 10sylib 196 . . . . . 6  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
12 funcrcl 14755 . . . . . 6  |-  ( <. F ,  G >.  e.  ( D  Func  E
)  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
1311, 12syl 16 . . . . 5  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
1413simpld 456 . . . 4  |-  ( ph  ->  D  e.  Cat )
1513simprd 460 . . . 4  |-  ( ph  ->  E  e.  Cat )
162, 3, 4, 5, 6, 7, 8, 9, 14, 15isfunc 14756 . . 3  |-  ( ph  ->  ( F ( D 
Func  E ) G  <->  ( F : B --> C  /\  G  e.  X_ z  e.  ( B  X.  B ) ( ( ( F `
 ( 1st `  z
) ) ( Hom  `  E ) ( F `
 ( 2nd `  z
) ) )  ^m  ( ( Hom  `  D
) `  z )
)  /\  A. x  e.  B  ( (
( x G x ) `  ( ( Id `  D ) `
 x ) )  =  ( ( Id
`  E ) `  ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x ( Hom  `  D ) y ) A. n  e.  ( y ( Hom  `  D
) z ) ( ( x G z ) `  ( n ( <. x ,  y
>. (comp `  D )
z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  E )
( F `  z
) ) ( ( x G y ) `
 m ) ) ) ) ) )
171, 16mpbid 210 . 2  |-  ( ph  ->  ( F : B --> C  /\  G  e.  X_ z  e.  ( B  X.  B ) ( ( ( F `  ( 1st `  z ) ) ( Hom  `  E
) ( F `  ( 2nd `  z ) ) )  ^m  (
( Hom  `  D ) `
 z ) )  /\  A. x  e.  B  ( ( ( x G x ) `
 ( ( Id
`  D ) `  x ) )  =  ( ( Id `  E ) `  ( F `  x )
)  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x ( Hom  `  D ) y ) A. n  e.  ( y ( Hom  `  D
) z ) ( ( x G z ) `  ( n ( <. x ,  y
>. (comp `  D )
z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  E )
( F `  z
) ) ( ( x G y ) `
 m ) ) ) ) )
1817simp1d 993 1  |-  ( ph  ->  F : B --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755   A.wral 2705   <.cop 3871   class class class wbr 4280    X. cxp 4825   -->wf 5402   ` cfv 5406  (class class class)co 6080   1stc1st 6564   2ndc2nd 6565    ^m cmap 7202   X_cixp 7251   Basecbs 14156   Hom chom 14231  compcco 14232   Catccat 14584   Idccid 14585    Func cfunc 14746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-map 7204  df-ixp 7252  df-func 14750
This theorem is referenced by:  funcsect  14764  funcinv  14765  funciso  14766  funcoppc  14767  cofu1  14776  cofucl  14780  cofuass  14781  cofulid  14782  cofurid  14783  funcres  14788  funcres2  14790  wunfunc  14791  funcres2c  14793  fullpropd  14812  fthsect  14817  fthinv  14818  fthmon  14819  ffthiso  14821  cofull  14826  cofth  14827  fuccocl  14856  fucidcl  14857  fuclid  14858  fucrid  14859  fucass  14860  fucsect  14864  fucinv  14865  invfuc  14866  fuciso  14867  natpropd  14868  fucpropd  14869  catciso  14957  prfval  14991  prfcl  14995  prf1st  14996  prf2nd  14997  1st2ndprf  14998  evlfcllem  15013  evlfcl  15014  curf1cl  15020  curfcl  15024  uncf1  15028  uncf2  15029  curfuncf  15030  uncfcurf  15031  diag1cl  15034  curf2ndf  15039  yon1cl  15055  oyon1cl  15063  yonedalem3a  15066  yonedalem4c  15069  yonedalem3b  15071  yonedalem3  15072  yonedainv  15073  yonffthlem  15074  yoniso  15077
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