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Theorem funcfn2 13743
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 2283 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
3 eqid 2283 . . 3  |-  (  Hom  `  E )  =  (  Hom  `  E )
4 funcfn2.f . . 3  |-  ( ph  ->  F ( D  Func  E ) G )
51, 2, 3, 4funcixp 13741 . 2  |-  ( ph  ->  G  e.  X_ x  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  x ) ) (  Hom  `  E )
( F `  ( 2nd `  x ) ) )  ^m  ( (  Hom  `  D ) `  x ) ) )
6 ixpfn 6822 . 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 15 1  |-  ( ph  ->  G  Fn  ( B  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   class class class wbr 4023    X. cxp 4687    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772   X_cixp 6817   Basecbs 13148    Hom chom 13219    Func cfunc 13728
This theorem is referenced by:  funcoppc  13749  cofuval  13756  cofulid  13764  cofurid  13765  prf1st  13978  prf2nd  13979  1st2ndprf  13980  curfuncf  14012  uncfcurf  14013  curf2ndf  14021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-ixp 6818  df-func 13732
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