MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcf1 Structured version   Unicode version

Theorem funcf1 14063
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 2436 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
5 eqid 2436 . . . 4  |-  (  Hom  `  E )  =  (  Hom  `  E )
6 eqid 2436 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
7 eqid 2436 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
8 eqid 2436 . . . 4  |-  (comp `  D )  =  (comp `  D )
9 eqid 2436 . . . 4  |-  (comp `  E )  =  (comp `  E )
10 df-br 4213 . . . . . . 7  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
111, 10sylib 189 . . . . . 6  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
12 funcrcl 14060 . . . . . 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 446 . . . 4  |-  ( ph  ->  D  e.  Cat )
1513simprd 450 . . . 4  |-  ( ph  ->  E  e.  Cat )
162, 3, 4, 5, 6, 7, 8, 9, 14, 15isfunc 14061 . . 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 202 . 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 969 1  |-  ( ph  ->  F : B --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   <.cop 3817   class class class wbr 4212    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348    ^m cmap 7018   X_cixp 7063   Basecbs 13469    Hom chom 13540  compcco 13541   Catccat 13889   Idccid 13890    Func cfunc 14051
This theorem is referenced by:  funcsect  14069  funcinv  14070  funciso  14071  funcoppc  14072  cofu1  14081  cofucl  14085  cofuass  14086  cofulid  14087  cofurid  14088  funcres  14093  funcres2  14095  wunfunc  14096  funcres2c  14098  fullpropd  14117  fthsect  14122  fthinv  14123  fthmon  14124  ffthiso  14126  cofull  14131  cofth  14132  fuccocl  14161  fucidcl  14162  fuclid  14163  fucrid  14164  fucass  14165  fucsect  14169  fucinv  14170  invfuc  14171  fuciso  14172  natpropd  14173  fucpropd  14174  catciso  14262  prfval  14296  prfcl  14300  prf1st  14301  prf2nd  14302  1st2ndprf  14303  evlfcllem  14318  evlfcl  14319  curf1cl  14325  curfcl  14329  uncf1  14333  uncf2  14334  curfuncf  14335  uncfcurf  14336  diag1cl  14339  curf2ndf  14344  yon1cl  14360  oyon1cl  14368  yonedalem3a  14371  yonedalem4c  14374  yonedalem3b  14376  yonedalem3  14377  yonedainv  14378  yonffthlem  14379  yoniso  14382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-ixp 7064  df-func 14055
  Copyright terms: Public domain W3C validator