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

Theorem funcixp 13757
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 2296 . . . 4  |-  ( Base `  E )  =  (
Base `  E )
4 funcixp.h . . . 4  |-  H  =  (  Hom  `  D
)
5 funcixp.j . . . 4  |-  J  =  (  Hom  `  E
)
6 eqid 2296 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
7 eqid 2296 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
8 eqid 2296 . . . 4  |-  (comp `  D )  =  (comp `  D )
9 eqid 2296 . . . 4  |-  (comp `  E )  =  (comp `  E )
10 df-br 4040 . . . . . . 7  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
111, 10sylib 188 . . . . . 6  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
12 funcrcl 13753 . . . . . 6  |-  ( <. F ,  G >.  e.  ( D  Func  E
)  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
1311, 12syl 15 . . . . 5  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
1413simpld 445 . . . 4  |-  ( ph  ->  D  e.  Cat )
1513simprd 449 . . . 4  |-  ( ph  ->  E  e.  Cat )
162, 3, 4, 5, 6, 7, 8, 9, 14, 15isfunc 13754 . . 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 201 . 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 968 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656   class class class wbr 4039    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137    ^m cmap 6788   X_cixp 6833   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582   Idccid 13583    Func cfunc 13744
This theorem is referenced by:  funcf2  13758  funcfn2  13759  wunfunc  13789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-ixp 6834  df-func 13748
  Copyright terms: Public domain W3C validator