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Theorem funcfn2 14021
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 2404 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
3 eqid 2404 . . 3  |-  (  Hom  `  E )  =  (  Hom  `  E )
4 funcfn2.f . . 3  |-  ( ph  ->  F ( D  Func  E ) G )
51, 2, 3, 4funcixp 14019 . 2  |-  ( ph  ->  G  e.  X_ x  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  x ) ) (  Hom  `  E )
( F `  ( 2nd `  x ) ) )  ^m  ( (  Hom  `  D ) `  x ) ) )
6 ixpfn 7027 . 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 1649    e. wcel 1721   class class class wbr 4172    X. cxp 4835    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307    ^m cmap 6977   X_cixp 7022   Basecbs 13424    Hom chom 13495    Func cfunc 14006
This theorem is referenced by:  funcoppc  14027  cofuval  14034  cofulid  14042  cofurid  14043  prf1st  14256  prf2nd  14257  1st2ndprf  14258  curfuncf  14290  uncfcurf  14291  curf2ndf  14299
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-map 6979  df-ixp 7023  df-func 14010
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