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Theorem funcixp 14095
Description: The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
funcixp.b  |-  B  =  ( Base `  D
)
funcixp.h  |-  H  =  (  Hom  `  D
)
funcixp.j  |-  J  =  (  Hom  `  E
)
funcixp.f  |-  ( ph  ->  F ( D  Func  E ) G )
Assertion
Ref Expression
funcixp  |-  ( ph  ->  G  e.  X_ z  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  z ) ) J ( F `  ( 2nd `  z ) ) )  ^m  ( H `
 z ) ) )
Distinct variable groups:    z, B    z, D    z, E    ph, z    z, F    z, G    z, J    z, H

Proof of Theorem funcixp
Dummy variables  m  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcixp.f . . 3  |-  ( ph  ->  F ( D  Func  E ) G )
2 funcixp.b . . . 4  |-  B  =  ( Base `  D
)
3 eqid 2442 . . . 4  |-  ( Base `  E )  =  (
Base `  E )
4 funcixp.h . . . 4  |-  H  =  (  Hom  `  D
)
5 funcixp.j . . . 4  |-  J  =  (  Hom  `  E
)
6 eqid 2442 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
7 eqid 2442 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
8 eqid 2442 . . . 4  |-  (comp `  D )  =  (comp `  D )
9 eqid 2442 . . . 4  |-  (comp `  E )  =  (comp `  E )
10 df-br 4238 . . . . . . 7  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
111, 10sylib 190 . . . . . 6  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
12 funcrcl 14091 . . . . . 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 447 . . . 4  |-  ( ph  ->  D  e.  Cat )
1513simprd 451 . . . 4  |-  ( ph  ->  E  e.  Cat )
162, 3, 4, 5, 6, 7, 8, 9, 14, 15isfunc 14092 . . 3  |-  ( ph  ->  ( F ( D 
Func  E ) G  <->  ( F : B --> ( Base `  E
)  /\  G  e.  X_ z  e.  ( B  X.  B ) ( ( ( F `  ( 1st `  z ) ) J ( F `
 ( 2nd `  z
) ) )  ^m  ( H `  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 H y ) A. n  e.  ( y H 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 203 . 2  |-  ( ph  ->  ( F : B --> ( Base `  E )  /\  G  e.  X_ z  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  z ) ) J ( F `  ( 2nd `  z ) ) )  ^m  ( H `
 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 H y ) A. n  e.  ( y H 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 ) ) ) ) )
1817simp2d 971 1  |-  ( ph  ->  G  e.  X_ z  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  z ) ) J ( F `  ( 2nd `  z ) ) )  ^m  ( H `
 z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   A.wral 2711   <.cop 3841   class class class wbr 4237    X. cxp 4905   -->wf 5479   ` cfv 5483  (class class class)co 6110   1stc1st 6376   2ndc2nd 6377    ^m cmap 7047   X_cixp 7092   Basecbs 13500    Hom chom 13571  compcco 13572   Catccat 13920   Idccid 13921    Func cfunc 14082
This theorem is referenced by:  funcf2  14096  funcfn2  14097  wunfunc  14127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-map 7049  df-ixp 7093  df-func 14086
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