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Theorem natfn 17651
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 (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))
natixp.b 𝐵 = (Base‘𝐶)
Assertion
Ref Expression
natfn (𝜑𝐴 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 (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))
3 natixp.b . . 3 𝐵 = (Base‘𝐶)
4 eqid 2739 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
51, 2, 3, 4natixp 17649 . 2 (𝜑𝐴X𝑥𝐵 ((𝐹𝑥)(Hom ‘𝐷)(𝐾𝑥)))
6 ixpfn 8665 . 2 (𝐴X𝑥𝐵 ((𝐹𝑥)(Hom ‘𝐷)(𝐾𝑥)) → 𝐴 Fn 𝐵)
75, 6syl 17 1 (𝜑𝐴 Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2109  cop 4572   Fn wfn 6425  cfv 6430  (class class class)co 7268  Xcixp 8659  Basecbs 16893  Hom chom 16954   Nat cnat 17638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-ixp 8660  df-func 17554  df-nat 17640
This theorem is referenced by:  fuclid  17665  fucrid  17666  curfuncf  17937  yonedainv  17980  yonffthlem  17981
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