Theorem isnatd 49338

Description: Property of being a natural transformation; deduction form. (Contributed by Zhi Wang, 29-Sep-2025.)

Hypotheses

Ref	Expression
isnatd.1	⊢ 𝑁 = (𝐶 Nat 𝐷)
isnatd.b	⊢ 𝐵 = (Base‘𝐶)
isnatd.h	⊢ 𝐻 = (Hom ‘𝐶)
isnatd.j	⊢ 𝐽 = (Hom ‘𝐷)
isnatd.o	⊢ · = (comp‘𝐷)
isnatd.f	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
isnatd.g	⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿)
isnatd.a	⊢ (𝜑 → 𝐴 Fn 𝐵)
isnatd.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐴‘𝑥) ∈ ((𝐹‘𝑥)𝐽(𝐾‘𝑥)))
isnatd.3	⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ℎ ∈ (𝑥𝐻𝑦)) → ((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)))

Assertion

Ref	Expression
isnatd	⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))

Distinct variable groups:   𝐴,ℎ,𝑥,𝑦   𝐵,ℎ,𝑥,𝑦   𝐶,ℎ,𝑥,𝑦   𝐷,ℎ,𝑥,𝑦   ℎ,𝐹,𝑥,𝑦   ℎ,𝐺,𝑥,𝑦   ℎ,𝐻   ℎ,𝐾,𝑥,𝑦   ℎ,𝐿,𝑥,𝑦   𝜑,ℎ,𝑥,𝑦

Allowed substitution hints:   · (𝑥,𝑦,ℎ)   𝐻(𝑥,𝑦)   𝐽(𝑥,𝑦,ℎ)   𝑁(𝑥,𝑦,ℎ)

Proof of Theorem isnatd

Step	Hyp	Ref	Expression
1		isnatd.a	. . . . 5 ⊢ (𝜑 → 𝐴 Fn 𝐵)
2		dffn5 6889	. . . . 5 ⊢ (𝐴 Fn 𝐵 ↔ 𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
3	1, 2	sylib 218	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
4		isnatd.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	4	fvexi 6845	. . . . 5 ⊢ 𝐵 ∈ V
6	5	mptex 7166	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∈ V
7	3, 6	eqeltrdi 2841	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
8		isnatd.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐴‘𝑥) ∈ ((𝐹‘𝑥)𝐽(𝐾‘𝑥)))
9	8	ralrimiva 3126	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ ((𝐹‘𝑥)𝐽(𝐾‘𝑥)))
10		elixp2 8834	. . 3 ⊢ (𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ↔ (𝐴 ∈ V ∧ 𝐴 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ ((𝐹‘𝑥)𝐽(𝐾‘𝑥))))
11	7, 1, 9, 10	syl3anbrc 1344	. 2 ⊢ (𝜑 → 𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)))
12		isnatd.3	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ℎ ∈ (𝑥𝐻𝑦)) → ((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)))
13	12	ralrimiva 3126	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)))
14	13	ralrimivva 3177	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)))
15		isnatd.1	. . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷)
16		isnatd.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
17		isnatd.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐷)
18		isnatd.o	. . 3 ⊢ · = (comp‘𝐷)
19		isnatd.f	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
20		isnatd.g	. . 3 ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿)
21	15, 4, 16, 17, 18, 19, 20	isnat 17867	. 2 ⊢ (𝜑 → (𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)))))
22	11, 14, 21	mpbir2and 713	1 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  Vcvv 3438  ⟨cop 4583   class class class wbr 5095   ↦ cmpt 5176   Fn wfn 6484  ‘cfv 6489  (class class class)co 7355  Xcixp 8830  Basecbs 17130  Hom chom 17182  compcco 17183   Func cfunc 17771   Nat cnat 17861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-ixp 8831  df-func 17775  df-nat 17863

This theorem is referenced by:  natoppf  49344  fuco22natlem  49460