MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcfn2 Structured version   Unicode version

Theorem funcfn2 14058
Description: The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
funcfn2.b  |-  B  =  ( Base `  D
)
funcfn2.f  |-  ( ph  ->  F ( D  Func  E ) G )
Assertion
Ref Expression
funcfn2  |-  ( ph  ->  G  Fn  ( B  X.  B ) )

Proof of Theorem funcfn2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 funcfn2.b . . 3  |-  B  =  ( Base `  D
)
2 eqid 2435 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
3 eqid 2435 . . 3  |-  (  Hom  `  E )  =  (  Hom  `  E )
4 funcfn2.f . . 3  |-  ( ph  ->  F ( D  Func  E ) G )
51, 2, 3, 4funcixp 14056 . 2  |-  ( ph  ->  G  e.  X_ x  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  x ) ) (  Hom  `  E )
( F `  ( 2nd `  x ) ) )  ^m  ( (  Hom  `  D ) `  x ) ) )
6 ixpfn 7060 . 2  |-  ( G  e.  X_ x  e.  ( B  X.  B ) ( ( ( F `
 ( 1st `  x
) ) (  Hom  `  E ) ( F `
 ( 2nd `  x
) ) )  ^m  ( (  Hom  `  D
) `  x )
)  ->  G  Fn  ( B  X.  B
) )
75, 6syl 16 1  |-  ( ph  ->  G  Fn  ( B  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   class class class wbr 4204    X. cxp 4868    Fn wfn 5441   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340    ^m cmap 7010   X_cixp 7055   Basecbs 13461    Hom chom 13532    Func cfunc 14043
This theorem is referenced by:  funcoppc  14064  cofuval  14071  cofulid  14079  cofurid  14080  prf1st  14293  prf2nd  14294  1st2ndprf  14295  curfuncf  14327  uncfcurf  14328  curf2ndf  14336
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-ixp 7056  df-func 14047
  Copyright terms: Public domain W3C validator