Theorem isnatd 49127

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 6926	. . . . 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 6879	. . . . 5 ⊢ 𝐵 ∈ V
6	5	mptex 7204	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∈ V
7	3, 6	eqeltrdi 2837	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
8		isnatd.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐴‘𝑥) ∈ ((𝐹‘𝑥)𝐽(𝐾‘𝑥)))
9	8	ralrimiva 3127	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ ((𝐹‘𝑥)𝐽(𝐾‘𝑥)))
10		elixp2 8878	. . 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 3127	. . 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 3182	. 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 17918	. 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 1540   ∈ wcel 2109  ∀wral 3046  Vcvv 3455  ⟨cop 4603   class class class wbr 5115   ↦ cmpt 5196   Fn wfn 6514  ‘cfv 6519  (class class class)co 7394  Xcixp 8874  Basecbs 17185  Hom chom 17237  compcco 17238   Func cfunc 17822   Nat cnat 17912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-ixp 8875  df-func 17826  df-nat 17914

This theorem is referenced by:  fuco22natlem  49240