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Theorem isnat2 16431
 Description: Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
natfval.1 𝑁 = (𝐶 Nat 𝐷)
natfval.b 𝐵 = (Base‘𝐶)
natfval.h 𝐻 = (Hom ‘𝐶)
natfval.j 𝐽 = (Hom ‘𝐷)
natfval.o · = (comp‘𝐷)
isnat2.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
isnat2.g (𝜑𝐺 ∈ (𝐶 Func 𝐷))
Assertion
Ref Expression
isnat2 (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴X𝑥𝐵 (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ∧ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩ · ((1st𝐺)‘𝑦))((𝑥(2nd𝐹)𝑦)‘)) = (((𝑥(2nd𝐺)𝑦)‘)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐺)‘𝑦))(𝐴𝑥)))))
Distinct variable groups:   𝑥,,𝑦,𝐴   𝑥,𝐵,𝑦   𝐶,,𝑥,𝑦   ,𝐹,𝑥,𝑦   ,𝐺,𝑥,𝑦   ,𝐻   𝜑,,𝑥,𝑦   𝐷,,𝑥,𝑦
Allowed substitution hints:   𝐵()   · (𝑥,𝑦,)   𝐻(𝑥,𝑦)   𝐽(𝑥,𝑦,)   𝑁(𝑥,𝑦,)

Proof of Theorem isnat2
StepHypRef Expression
1 relfunc 16345 . . . . 5 Rel (𝐶 Func 𝐷)
2 isnat2.f . . . . 5 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
3 1st2nd 7105 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
41, 2, 3sylancr 694 . . . 4 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
5 isnat2.g . . . . 5 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
6 1st2nd 7105 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
71, 5, 6sylancr 694 . . . 4 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
84, 7oveq12d 6567 . . 3 (𝜑 → (𝐹𝑁𝐺) = (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
98eleq2d 2673 . 2 (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ 𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩)))
10 natfval.1 . . 3 𝑁 = (𝐶 Nat 𝐷)
11 natfval.b . . 3 𝐵 = (Base‘𝐶)
12 natfval.h . . 3 𝐻 = (Hom ‘𝐶)
13 natfval.j . . 3 𝐽 = (Hom ‘𝐷)
14 natfval.o . . 3 · = (comp‘𝐷)
15 1st2ndbr 7108 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
161, 2, 15sylancr 694 . . 3 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
17 1st2ndbr 7108 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
181, 5, 17sylancr 694 . . 3 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
1910, 11, 12, 13, 14, 16, 18isnat 16430 . 2 (𝜑 → (𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩) ↔ (𝐴X𝑥𝐵 (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ∧ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩ · ((1st𝐺)‘𝑦))((𝑥(2nd𝐹)𝑦)‘)) = (((𝑥(2nd𝐺)𝑦)‘)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐺)‘𝑦))(𝐴𝑥)))))
209, 19bitrd 267 1 (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴X𝑥𝐵 (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ∧ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩ · ((1st𝐺)‘𝑦))((𝑥(2nd𝐹)𝑦)‘)) = (((𝑥(2nd𝐺)𝑦)‘)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐺)‘𝑦))(𝐴𝑥)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟨cop 4131   class class class wbr 4583  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Xcixp 7794  Basecbs 15695  Hom chom 15779  compcco 15780   Func cfunc 16337   Nat cnat 16424 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-ixp 7795  df-func 16341  df-nat 16426 This theorem is referenced by:  fuccocl  16447  fucidcl  16448  invfuc  16457  curf2cl  16694  yonedalem4c  16740  yonedalem3  16743
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