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Theorem natfn 14071
Description: A natural transformation is a function on the objects of  C. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
natrcl.1  |-  N  =  ( C Nat  D )
natixp.2  |-  ( ph  ->  A  e.  ( <. F ,  G >. N
<. K ,  L >. ) )
natixp.b  |-  B  =  ( Base `  C
)
Assertion
Ref Expression
natfn  |-  ( ph  ->  A  Fn  B )

Proof of Theorem natfn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 natrcl.1 . . 3  |-  N  =  ( C Nat  D )
2 natixp.2 . . 3  |-  ( ph  ->  A  e.  ( <. F ,  G >. N
<. K ,  L >. ) )
3 natixp.b . . 3  |-  B  =  ( Base `  C
)
4 eqid 2380 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
51, 2, 3, 4natixp 14069 . 2  |-  ( ph  ->  A  e.  X_ x  e.  B  ( ( F `  x )
(  Hom  `  D ) ( K `  x
) ) )
6 ixpfn 6997 . 2  |-  ( A  e.  X_ x  e.  B  ( ( F `  x ) (  Hom  `  D ) ( K `
 x ) )  ->  A  Fn  B
)
75, 6syl 16 1  |-  ( ph  ->  A  Fn  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   <.cop 3753    Fn wfn 5382   ` cfv 5387  (class class class)co 6013   X_cixp 6992   Basecbs 13389    Hom chom 13460   Nat cnat 14058
This theorem is referenced by:  fuclid  14083  fucrid  14084  curfuncf  14255  yonedainv  14298  yonffthlem  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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-ixp 6993  df-func 13975  df-nat 14060
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