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Theorem natfn 17212
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 2818 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
51, 2, 3, 4natixp 17210 . 2 (𝜑𝐴X𝑥𝐵 ((𝐹𝑥)(Hom ‘𝐷)(𝐾𝑥)))
6 ixpfn 8455 . 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 1528  wcel 2105  cop 4563   Fn wfn 6343  cfv 6348  (class class class)co 7145  Xcixp 8449  Basecbs 16471  Hom chom 16564   Nat cnat 17199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-ixp 8450  df-func 17116  df-nat 17201
This theorem is referenced by:  fuclid  17224  fucrid  17225  curfuncf  17476  yonedainv  17519  yonffthlem  17520
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