Theorem natfval 17885

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 7377	. . . . 5 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (𝑡 Func 𝑢) = (𝐶 Func 𝐷))
3		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → 𝑡 = 𝐶)
4	3	fveq2d 6846	. . . . . . . . . . 11 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (Base‘𝑡) = (Base‘𝐶))
5		natfval.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐶)
6	4, 5	eqtr4di 2790	. . . . . . . . . 10 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (Base‘𝑡) = 𝐵)
7	6	ixpeq1d 8859	. . . . . . . . 9 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → X𝑥 ∈ (Base‘𝑡)((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) = X𝑥 ∈ 𝐵 ((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)))
8		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → 𝑢 = 𝐷)
9	8	fveq2d 6846	. . . . . . . . . . . 12 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (Hom ‘𝑢) = (Hom ‘𝐷))
10		natfval.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (Hom ‘𝐷)
11	9, 10	eqtr4di 2790	. . . . . . . . . . 11 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (Hom ‘𝑢) = 𝐽)
12	11	oveqd 7385	. . . . . . . . . 10 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → ((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) = ((𝑟‘𝑥)𝐽(𝑠‘𝑥)))
13	12	ixpeq2dv 8863	. . . . . . . . 9 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → X𝑥 ∈ 𝐵 ((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) = X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)))
14	7, 13	eqtrd 2772	. . . . . . . 8 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → X𝑥 ∈ (Base‘𝑡)((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) = X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)))
15	3	fveq2d 6846	. . . . . . . . . . . . 13 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (Hom ‘𝑡) = (Hom ‘𝐶))
16		natfval.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (Hom ‘𝐶)
17	15, 16	eqtr4di 2790	. . . . . . . . . . . 12 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (Hom ‘𝑡) = 𝐻)
18	17	oveqd 7385	. . . . . . . . . . 11 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (𝑥(Hom ‘𝑡)𝑦) = (𝑥𝐻𝑦))
19	8	fveq2d 6846	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (comp‘𝑢) = (comp‘𝐷))
20		natfval.o	. . . . . . . . . . . . . . 15 ⊢ · = (comp‘𝐷)
21	19, 20	eqtr4di 2790	. . . . . . . . . . . . . 14 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (comp‘𝑢) = · )
22	21	oveqd 7385	. . . . . . . . . . . . 13 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦)) = (⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦)))
23	22	oveqd 7385	. . . . . . . . . . . 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 7385	. . . . . . . . . . . . 13 ⊢ ((𝑡 = 𝐶 ∧ 𝑢 = 𝐷) → (⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦)) = (⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦)))
25	24	oveqd 7385	. . . . . . . . . . . 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 2753	. . . . . . . . . . 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 3318	. . . . . . . . . 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 3318	. . . . . . . . 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 3318	. . . . . . . 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 3419	. . . . . . 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 7443	. . . 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 17882	. . . 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 7401	. . . . 5 ⊢ (𝐶 Func 𝐷) ∈ V
36	35, 35	mpoex 8033	. . . 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 7523	. . 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 7608	. . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Nat 𝐷) = ∅)
39		funcrcl 17799	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
40	39	con3i 154	. . . . . . 7 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ¬ 𝑓 ∈ (𝐶 Func 𝐷))
41	40	eq0rdv 4361	. . . . . 6 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = ∅)
42	41	olcd 875	. . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) = ∅ ∨ (𝐶 Func 𝐷) = ∅))
43		0mpo0 7451	. . . . 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 17	. . . 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 2775	. . 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 182	. 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 2760	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))})

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401  ⦋csb 3851  ∅c0 4287  ⟨cop 4588  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1^st c1st 7941  2^nd c2nd 7942  Xcixp 8847  Basecbs 17148  Hom chom 17200  compcco 17201  Catccat 17599   Func cfunc 17790   Nat cnat 17880

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-ixp 8848  df-func 17794  df-nat 17882

This theorem is referenced by:  isnat  17886  natffn  17888  natrcl  17889  wunnat  17895  natpropd  17915