Theorem isnatd 50275

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 6935	. . . . 5 ⊢ (𝐴 Fn 𝐵 ↔ 𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
3	1, 2	sylib 221	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
4		isnatd.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	4	fvexi 6891	. . . . 5 ⊢ 𝐵 ∈ V
6	5	mptex 7221	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∈ V
7	3, 6	eqeltrdi 2869	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
8		isnatd.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐴‘𝑥) ∈ ((𝐹‘𝑥)𝐽(𝐾‘𝑥)))
9	8	ralrimiva 3155	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ ((𝐹‘𝑥)𝐽(𝐾‘𝑥)))
10		elixp2 8913	. . 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 1362	. 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 3155	. . 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 3206	. 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 18105	. 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 726	1 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  ⟨cop 4590   class class class wbr 5103   ↦ cmpt 5186   Fn wfn 6526  ‘cfv 6531  (class class class)co 7412  Xcixp 8909  Basecbs 17367  Hom chom 17419  compcco 17420   Func cfunc 18009   Nat cnat 18099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-ixp 8910  df-func 18013  df-nat 18101

This theorem is used by:  natoppf  50281  fuco22natlem  50397