Theorem isnatd 49692

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 6899	. . . . 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 6855	. . . . 5 ⊢ 𝐵 ∈ V
6	5	mptex 7178	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∈ V
7	3, 6	eqeltrdi 2845	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
8		isnatd.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐴‘𝑥) ∈ ((𝐹‘𝑥)𝐽(𝐾‘𝑥)))
9	8	ralrimiva 3130	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ ((𝐹‘𝑥)𝐽(𝐾‘𝑥)))
10		elixp2 8849	. . 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 1345	. 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 3130	. . 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 3181	. 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 17917	. 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 714	1 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430  ⟨cop 4574   class class class wbr 5086   ↦ cmpt 5167   Fn wfn 6494  ‘cfv 6499  (class class class)co 7367  Xcixp 8845  Basecbs 17179  Hom chom 17231  compcco 17232   Func cfunc 17821   Nat cnat 17911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-ixp 8846  df-func 17825  df-nat 17913

This theorem is referenced by:  natoppf  49698  fuco22natlem  49814