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Theorem funcf1 15282
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 2457 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
5 eqid 2457 . . . 4  |-  ( Hom  `  E )  =  ( Hom  `  E )
6 eqid 2457 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
7 eqid 2457 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
8 eqid 2457 . . . 4  |-  (comp `  D )  =  (comp `  D )
9 eqid 2457 . . . 4  |-  (comp `  E )  =  (comp `  E )
10 df-br 4457 . . . . . . 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 15279 . . . . . 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 459 . . . 4  |-  ( ph  ->  D  e.  Cat )
1513simprd 463 . . . 4  |-  ( ph  ->  E  e.  Cat )
162, 3, 4, 5, 6, 7, 8, 9, 14, 15isfunc 15280 . . 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 1008 1  |-  ( ph  ->  F : B --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   <.cop 4038   class class class wbr 4456    X. cxp 5006   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798    ^m cmap 7438   X_cixp 7488   Basecbs 14644   Hom chom 14723  compcco 14724   Catccat 15081   Idccid 15082    Func cfunc 15270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-ixp 7489  df-func 15274
This theorem is referenced by:  funcsect  15288  funcinv  15289  funciso  15290  funcoppc  15291  cofu1  15300  cofucl  15304  cofuass  15305  cofulid  15306  cofurid  15307  funcres  15312  funcres2  15314  wunfunc  15315  funcres2c  15317  fullpropd  15336  fthsect  15341  fthinv  15342  fthmon  15343  ffthiso  15345  cofull  15350  cofth  15351  fuccocl  15380  fucidcl  15381  fuclid  15382  fucrid  15383  fucass  15384  fucsect  15388  fucinv  15389  invfuc  15390  fuciso  15391  natpropd  15392  fucpropd  15393  catciso  15513  prfval  15595  prfcl  15599  prf1st  15600  prf2nd  15601  1st2ndprf  15602  evlfcllem  15617  evlfcl  15618  curf1cl  15624  curfcl  15628  uncf1  15632  uncf2  15633  curfuncf  15634  uncfcurf  15635  diag1cl  15638  curf2ndf  15643  yon1cl  15659  oyon1cl  15667  yonedalem3a  15670  yonedalem4c  15673  yonedalem3b  15675  yonedalem3  15676  yonedainv  15677  yonffthlem  15678  yoniso  15681
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