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Theorem natixp 18007
Description: A natural transformation is a function from the objects of 𝐶 to homomorphisms from 𝐹(𝑥) to 𝐺(𝑥). (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)
natixp.j 𝐽 = (Hom ‘𝐷)
Assertion
Ref Expression
natixp (𝜑𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝑥,𝐶   𝑥,𝐾   𝜑,𝑥   𝑥,𝐷   𝑥,𝐿   𝑥,𝐵   𝑥,𝐽
Allowed substitution hint:   𝑁(𝑥)

Proof of Theorem natixp
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 natixp.2 . . 3 (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))
2 natrcl.1 . . . 4 𝑁 = (𝐶 Nat 𝐷)
3 natixp.b . . . 4 𝐵 = (Base‘𝐶)
4 eqid 2735 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
5 natixp.j . . . 4 𝐽 = (Hom ‘𝐷)
6 eqid 2735 . . . 4 (comp‘𝐷) = (comp‘𝐷)
72natrcl 18005 . . . . . . 7 (𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) → (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) ∧ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷)))
81, 7syl 17 . . . . . 6 (𝜑 → (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) ∧ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷)))
98simpld 494 . . . . 5 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
10 df-br 5149 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylibr 234 . . . 4 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
128simprd 495 . . . . 5 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
13 df-br 5149 . . . . 5 (𝐾(𝐶 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
1412, 13sylibr 234 . . . 4 (𝜑𝐾(𝐶 Func 𝐷)𝐿)
152, 3, 4, 5, 6, 11, 14isnat 18002 . . 3 (𝜑 → (𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) ↔ (𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐾𝑦))((𝑥𝐺𝑦)‘𝑧)) = (((𝑥𝐿𝑦)‘𝑧)(⟨(𝐹𝑥), (𝐾𝑥)⟩(comp‘𝐷)(𝐾𝑦))(𝐴𝑥)))))
161, 15mpbid 232 . 2 (𝜑 → (𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐾𝑦))((𝑥𝐺𝑦)‘𝑧)) = (((𝑥𝐿𝑦)‘𝑧)(⟨(𝐹𝑥), (𝐾𝑥)⟩(comp‘𝐷)(𝐾𝑦))(𝐴𝑥))))
1716simpld 494 1 (𝜑𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  cop 4637   class class class wbr 5148  cfv 6563  (class class class)co 7431  Xcixp 8936  Basecbs 17245  Hom chom 17309  compcco 17310   Func cfunc 17905   Nat cnat 17996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-ixp 8937  df-func 17909  df-nat 17998
This theorem is referenced by:  natcl  18008  natfn  18009
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