Theorem natfval 18153

Description: Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)

Hypotheses

Ref	Expression
natfval.1	⊢ 𝑁 = (𝐶 Nat 𝐷)
natfval.b	⊢ 𝐵 = (Base‘𝐶)
natfval.h	⊢ 𝐻 = (Hom ‘𝐶)
natfval.j	⊢ 𝐽 = (Hom ‘𝐷)
natfval.o	⊢ · = (comp‘𝐷)

Assertion

Ref	Expression
natfval	⊢ 𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))})

Distinct variable groups:   𝑓,𝑎,𝑔,ℎ,𝑟,𝑠,𝑥,𝑦   𝐵,𝑎,𝑓,𝑔,𝑟,𝑠,𝑥,𝑦   𝐶,𝑎,𝑓,𝑔,ℎ,𝑟,𝑠,𝑥,𝑦   𝐽,𝑎,𝑓,𝑔,𝑟,𝑠   𝐻,𝑎,𝑓,𝑔,ℎ,𝑟,𝑠   · ,𝑎,𝑓,𝑔,𝑟,𝑠   𝐷,𝑎,𝑓,𝑔,ℎ,𝑟,𝑠,𝑥,𝑦

Allowed substitution hints:   𝐵(ℎ)   · (𝑥, 𝑦, ℎ)   𝐻(𝑥, 𝑦)   𝐽(𝑥, 𝑦, ℎ)   𝑁(𝑥, 𝑦, 𝑓, 𝑔, ℎ, 𝑠, 𝑟, 𝑎)

Proof of Theorem natfval

Dummy variables 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		natfval.1	. 2 ⊢ 𝑁 = (𝐶 Nat 𝐷)
2		oveq12 7452	. . . . 5 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (𝑡 Func 𝑢) = (𝐶 Func 𝐷))
3		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → 𝑡 = 𝐶)
4	3	fveq2d 6912	. . . . . . . . . . 11 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (Base‘𝑡) = (Base‘𝐶))
5		natfval.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐶)
6	4, 5	eqtr4di 2816	. . . . . . . . . 10 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (Base‘𝑡) = 𝐵)
7	6	ixpeq1d 8965	. . . . . . . . 9 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → X𝑥 ∈ (Base‘𝑡)((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) = X𝑥 ∈ 𝐵 ((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)))
8		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → 𝑢 = 𝐷)
9	8	fveq2d 6912	. . . . . . . . . . . 12 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (Hom ‘𝑢) = (Hom ‘𝐷))
10		natfval.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (Hom ‘𝐷)
11	9, 10	eqtr4di 2816	. . . . . . . . . . 11 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (Hom ‘𝑢) = 𝐽)
12	11	oveqd 7460	. . . . . . . . . 10 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → ((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) = ((𝑟‘𝑥)𝐽(𝑠‘𝑥)))
13	12	ixpeq2dv 8969	. . . . . . . . 9 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → X𝑥 ∈ 𝐵 ((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) = X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)))
14	7, 13	eqtrd 2798	. . . . . . . 8 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → X𝑥 ∈ (Base‘𝑡)((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) = X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)))
15	3	fveq2d 6912	. . . . . . . . . . . . 13 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (Hom ‘𝑡) = (Hom ‘𝐶))
16		natfval.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (Hom ‘𝐶)
17	15, 16	eqtr4di 2816	. . . . . . . . . . . 12 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (Hom ‘𝑡) = 𝐻)
18	17	oveqd 7460	. . . . . . . . . . 11 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (𝑥(Hom ‘𝑡)𝑦) = (𝑥𝐻𝑦))
19	8	fveq2d 6912	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (comp‘𝑢) = (comp‘𝐷))
20		natfval.o	. . . . . . . . . . . . . . 15 ⊢ · = (comp‘𝐷)
21	19, 20	eqtr4di 2816	. . . . . . . . . . . . . 14 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (comp‘𝑢) = · )
22	21	oveqd 7460	. . . . . . . . . . . . 13 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦)) = (⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦)))
23	22	oveqd 7460	. . . . . . . . . . . 12 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → ((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = ((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)))
24	21	oveqd 7460	. . . . . . . . . . . . 13 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦)) = (⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦)))
25	24	oveqd 7460	. . . . . . . . . . . 12 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥)))
26	23, 25	eqeq12d 2779	. . . . . . . . . . 11 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥)) ↔ ((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))))
27	18, 26	raleqbidv 3338	. . . . . . . . . 10 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (∀ℎ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥)) ↔ ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))))
28	6, 27	raleqbidv 3338	. . . . . . . . 9 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (∀𝑦 ∈ (Base‘𝑡)∀ℎ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥)) ↔ ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))))
29	6, 28	raleqbidv 3338	. . . . . . . 8 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ℎ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))))
30	14, 29	rabeqbidv 3433	. . . . . . 7 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → {𝑎 ∈ X𝑥 ∈ (Base‘𝑡)((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ℎ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥))} = {𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))})
31	30	csbeq2dv 3858	. . . . . 6 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝑡)((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ℎ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥))} = ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))})
32	31	csbeq2dv 3858	. . . . 5 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝑡)((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ℎ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥))} = ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))})
33	2, 2, 32	mpoeq123dv 7518	. . . 4 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (𝑓 ∈ (𝑡 Func 𝑢), 𝑔 ∈ (𝑡 Func 𝑢) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝑡)((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ℎ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥))}) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))}))
34		df-nat 18150	. . . 4 ⊢ Nat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ (𝑓 ∈ (𝑡 Func 𝑢), 𝑔 ∈ (𝑡 Func 𝑢) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝑡)((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ℎ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥))}))
35		ovex 7476	. . . . 5 ⊢ (𝐶 Func 𝐷) ∈ V
36	35, 35	mpoex 8117	. . . 4 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))}) ∈ V
37	33, 34, 36	ovmpoa 7599	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Nat 𝐷) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))}))
38	34	mpondm0 7685	. . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Nat 𝐷) = ∅)
39		funcrcl 18067	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
40	39	con3i 155	. . . . . . 7 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ¬ 𝑓 ∈ (𝐶 Func 𝐷))
41	40	eq0rdv 4368	. . . . . 6 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = ∅)
42	41	olcd 888	. . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) = ∅ ∨ (𝐶 Func 𝐷) = ∅))
43		0mpo0 7526	. . . . 5 ⊢ (((𝐶 Func 𝐷) = ∅ ∨ (𝐶 Func 𝐷) = ∅) → (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))}) = ∅)
44	42, 43	syl 18	. . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))}) = ∅)
45	38, 44	eqtr4d 2801	. . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Nat 𝐷) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))}))
46	37, 45	pm2.61i 184	. 2 ⊢ (𝐶 Nat 𝐷) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))})
47	1, 46	eqtri 2786	1 ⊢ 𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))})

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  ⦋csb 3851  ∅c0 4281  ⟨cop 4838  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  1^st c1st 8023  2^nd c2nd 8024  Xcixp 8953  Basecbs 17416  Hom chom 17468  compcco 17469  Catccat 17867   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:  isnat  18154  natffn  18156  natrcl  18157  wunnat  18163  natpropd  18183