isnat3 - Mathbox for Zhi Wang

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Theorem isnat3 50547

Description: A natural transformation is an exact component function with typed values satisfying naturality. (Contributed by Zhi Wang, 14-Sep-2026.)

**Hypotheses**
Ref	Expression
isnat3.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
isnat3.b	⊢ 𝐵 = (Base‘𝐶)
isnat3.h	⊢ 𝐻 = (Hom ‘𝐶)
isnat3.j	⊢ 𝐽 = (Hom ‘𝐷)
isnat3.o	⊢ · = (comp‘𝐷)
isnat3.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
isnat3.g	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))

**Assertion**
Ref	Expression
isnat3	⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥)))))

Distinct variable groups: 𝑥,𝐴,𝑦,ℎ 𝑥,𝐵,𝑦,ℎ 𝑥,𝐶,𝑦,ℎ 𝑥,𝐷,𝑦,ℎ 𝑥,𝐹,𝑦,ℎ 𝑥,𝐺,𝑦,ℎ 𝑥,𝐻,𝑦,ℎ 𝑥,𝐽,𝑦,ℎ 𝑥,𝑁,𝑦,ℎ 𝜑,𝑥,𝑦,ℎ 𝑥, · ,𝑦,ℎ

**Proof of Theorem isnat3**
Step	Hyp	Ref	Expression
1		isnat3.n	. . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷)
2		isnat3.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		isnat3.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
4		isnat3.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐷)
5		isnat3.o	. . 3 ⊢ · = (comp‘𝐷)
6		isnat3.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		isnat3.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
8	1, 2, 3, 4, 5, 6, 7	isnat2 18155	. 2 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥)))))
9		anass 474	. . 3 ⊢ (((𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥))) ↔ (𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∧ (∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥)))))
10	2	fvexi 6922	. . . . 5 ⊢ 𝐵 ∈ V
11	10	elixpmpt 50209	. . . 4 ⊢ (𝐴 ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ↔ (𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))))
12	11	anbi1i 636	. . 3 ⊢ ((𝐴 ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥))) ↔ ((𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥))))
13		3anass 1111	. . 3 ⊢ ((𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥))) ↔ (𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∧ (∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥)))))
14	9, 12, 13	3bitr4i 306	. 2 ⊢ ((𝐴 ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥))) ↔ (𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥))))
15	8, 14	bitrdi 290	1 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ⟨cop 4838 ↦ cmpt 5189 ‘cfv 6558 (class class class)co 7443 1^st c1st 8023 2^nd c2nd 8024 Xcixp 8953 Basecbs 17416 Hom chom 17468 compcco 17469 Func cfunc 18058 Nat cnat 18148

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-ov 7446 df-oprab 7447 df-mpo 7448 df-1st 8025 df-2nd 8026 df-ixp 8954 df-func 18062 df-nat 18150

This theorem is used by: fucoisfunc2lem1 50693

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