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Theorem funcixp 14899
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 2454 . . . 4  |-  ( Base `  E )  =  (
Base `  E )
4 funcixp.h . . . 4  |-  H  =  ( Hom  `  D
)
5 funcixp.j . . . 4  |-  J  =  ( Hom  `  E
)
6 eqid 2454 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
7 eqid 2454 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
8 eqid 2454 . . . 4  |-  (comp `  D )  =  (comp `  D )
9 eqid 2454 . . . 4  |-  (comp `  E )  =  (comp `  E )
10 df-br 4404 . . . . . . 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 14895 . . . . . 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 14896 . . 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 210 . 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 1001 1  |-  ( ph  ->  G  e.  X_ z  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  z ) ) J ( F `  ( 2nd `  z ) ) )  ^m  ( H `
 z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   <.cop 3994   class class class wbr 4403    X. cxp 4949   -->wf 5525   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689    ^m cmap 7327   X_cixp 7376   Basecbs 14295   Hom chom 14371  compcco 14372   Catccat 14724   Idccid 14725    Func cfunc 14886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329  df-ixp 7377  df-func 14890
This theorem is referenced by:  funcf2  14900  funcfn2  14901  wunfunc  14931
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