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Theorem natfn 17698
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 2733 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
51, 2, 3, 4natixp 17696 . 2 (𝜑𝐴X𝑥𝐵 ((𝐹𝑥)(Hom ‘𝐷)(𝐾𝑥)))
6 ixpfn 8711 . 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 1537  wcel 2101  cop 4570   Fn wfn 6442  cfv 6447  (class class class)co 7295  Xcixp 8705  Basecbs 16940  Hom chom 17001   Nat cnat 17685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-ov 7298  df-oprab 7299  df-mpo 7300  df-1st 7851  df-2nd 7852  df-ixp 8706  df-func 17601  df-nat 17687
This theorem is referenced by:  fuclid  17712  fucrid  17713  curfuncf  17984  yonedainv  18027  yonffthlem  18028
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