Theorem isnat 17854

Description: Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)

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)
isnat.f	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
isnat.g	⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿)

Assertion

Ref	Expression
isnat	⊢ (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)))))

Distinct variable groups:   𝑥,ℎ,𝑦,𝐴   𝑥,𝐵,𝑦   𝐶,ℎ,𝑥,𝑦   ℎ,𝐹,𝑥,𝑦   ℎ,𝐺,𝑥,𝑦   ℎ,𝐻   𝜑,ℎ,𝑥,𝑦   ℎ,𝐾,𝑥,𝑦   ℎ,𝐿,𝑥,𝑦   𝐷,ℎ,𝑥,𝑦

Allowed substitution hints:   𝐵(ℎ)   · (𝑥,𝑦,ℎ)   𝐻(𝑥,𝑦)   𝐽(𝑥,𝑦,ℎ)   𝑁(𝑥,𝑦,ℎ)

Proof of Theorem isnat

Dummy variables 𝑎 𝑓 𝑔 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		natfval.1	. . . . . 6 ⊢ 𝑁 = (𝐶 Nat 𝐷)
2		natfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
3		natfval.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
4		natfval.j	. . . . . 6 ⊢ 𝐽 = (Hom ‘𝐷)
5		natfval.o	. . . . . 6 ⊢ · = (comp‘𝐷)
6	1, 2, 3, 4, 5	natfval 17853	. . . . 5 ⊢ 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))})
7	6	a1i 11	. . . 4 ⊢ (𝜑 → 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))}))
8		fvexd 6837	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st ‘𝑓) ∈ V)
9		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → 𝑓 = ⟨𝐹, 𝐺⟩)
10	9	fveq2d 6826	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st ‘𝑓) = (1st ‘⟨𝐹, 𝐺⟩))
11		relfunc 17766	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
12		isnat.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
13		brrelex12 5668	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
14	11, 12, 13	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
15		op1stg 7933	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
17	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
18	10, 17	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st ‘𝑓) = 𝐹)
19		fvexd 6837	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st ‘𝑔) ∈ V)
20		simplrr 777	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → 𝑔 = ⟨𝐾, 𝐿⟩)
21	20	fveq2d 6826	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st ‘𝑔) = (1st ‘⟨𝐾, 𝐿⟩))
22		isnat.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿)
23		brrelex12 5668	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐾(𝐶 Func 𝐷)𝐿) → (𝐾 ∈ V ∧ 𝐿 ∈ V))
24	11, 22, 23	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ∈ V ∧ 𝐿 ∈ V))
25		op1stg 7933	. . . . . . . . 9 ⊢ ((𝐾 ∈ V ∧ 𝐿 ∈ V) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
26	24, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
27	26	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
28	21, 27	eqtrd 2766	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st ‘𝑔) = 𝐾)
29		simplr 768	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑟 = 𝐹)
30	29	fveq1d 6824	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑟‘𝑥) = (𝐹‘𝑥))
31		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑠 = 𝐾)
32	31	fveq1d 6824	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑠‘𝑥) = (𝐾‘𝑥))
33	30, 32	oveq12d 7364	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) = ((𝐹‘𝑥)𝐽(𝐾‘𝑥)))
34	33	ixpeq2dv 8837	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) = X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)))
35	29	fveq1d 6824	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑟‘𝑦) = (𝐹‘𝑦))
36	30, 35	opeq12d 4833	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
37	31	fveq1d 6824	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑠‘𝑦) = (𝐾‘𝑦))
38	36, 37	oveq12d 7364	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦)) = (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦)))
39		eqidd 2732	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑎‘𝑦) = (𝑎‘𝑦))
40	9	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑓 = ⟨𝐹, 𝐺⟩)
41	40	fveq2d 6826	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘𝑓) = (2nd ‘⟨𝐹, 𝐺⟩))
42		op2ndg 7934	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
43	14, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
44	43	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
45	41, 44	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘𝑓) = 𝐺)
46	45	oveqd 7363	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑥(2nd ‘𝑓)𝑦) = (𝑥𝐺𝑦))
47	46	fveq1d 6824	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑥(2nd ‘𝑓)𝑦)‘ℎ) = ((𝑥𝐺𝑦)‘ℎ))
48	38, 39, 47	oveq123d 7367	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = ((𝑎‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)))
49	30, 32	opeq12d 4833	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ = ⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩)
50	49, 37	oveq12d 7364	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦)) = (⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦)))
51	20	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑔 = ⟨𝐾, 𝐿⟩)
52	51	fveq2d 6826	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘𝑔) = (2nd ‘⟨𝐾, 𝐿⟩))
53		op2ndg 7934	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ V ∧ 𝐿 ∈ V) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
54	24, 53	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
55	54	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
56	52, 55	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘𝑔) = 𝐿)
57	56	oveqd 7363	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑥(2nd ‘𝑔)𝑦) = (𝑥𝐿𝑦))
58	57	fveq1d 6824	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑥(2nd ‘𝑔)𝑦)‘ℎ) = ((𝑥𝐿𝑦)‘ℎ))
59		eqidd 2732	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑎‘𝑥) = (𝑎‘𝑥))
60	50, 58, 59	oveq123d 7367	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝑎‘𝑥)))
61	48, 60	eqeq12d 2747	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (((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)) ↔ ((𝑎‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝑎‘𝑥))))
62	61	ralbidv 3155	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (∀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)) ↔ ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝑎‘𝑥))))
63	62	2ralbidv 3196	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (∀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)) ↔ ∀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))))
64	34, 63	rabeqbidv 3413	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → {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))} = {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))})
65	19, 28, 64	csbied2 3887	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → ⦋(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))} = {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))})
66	8, 18, 65	csbied2 3887	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → ⦋(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))} = {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))})
67		df-br 5092	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
68	12, 67	sylib 218	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
69		df-br 5092	. . . . 5 ⊢ (𝐾(𝐶 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
70	22, 69	sylib 218	. . . 4 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
71		ovex 7379	. . . . . . . 8 ⊢ ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∈ V
72	71	rgenw 3051	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∈ V
73		ixpexg 8846	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∈ V → X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∈ V)
74	72, 73	ax-mp 5	. . . . . 6 ⊢ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∈ V
75	74	rabex 5277	. . . . 5 ⊢ {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))} ∈ V
76	75	a1i 11	. . . 4 ⊢ (𝜑 → {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))} ∈ V)
77	7, 66, 68, 70, 76	ovmpod 7498	. . 3 ⊢ (𝜑 → (⟨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))})
78	77	eleq2d 2817	. 2 ⊢ (𝜑 → (𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩) ↔ 𝐴 ∈ {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))}))
79		fveq1 6821	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑦) = (𝐴‘𝑦))
80	79	oveq1d 7361	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = ((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)))
81		fveq1 6821	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑥) = (𝐴‘𝑥))
82	81	oveq2d 7362	. . . . . 6 ⊢ (𝑎 = 𝐴 → (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝑎‘𝑥)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥)))
83	80, 82	eqeq12d 2747	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝑎‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝑎‘𝑥)) ↔ ((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥))))
84	83	ralbidv 3155	. . . 4 ⊢ (𝑎 = 𝐴 → (∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝑎‘𝑥)) ↔ ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥))))
85	84	2ralbidv 3196	. . 3 ⊢ (𝑎 = 𝐴 → (∀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)) ↔ ∀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))))
86	85	elrab 3647	. 2 ⊢ (𝐴 ∈ {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))} ↔ (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))))
87	78, 86	bitrdi 287	1 ⊢ (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)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395  Vcvv 3436  ⦋csb 3850  ⟨cop 4582   class class class wbr 5091  Rel wrel 5621  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920  Xcixp 8821  Basecbs 17117  Hom chom 17169  compcco 17170   Func cfunc 17758   Nat cnat 17848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-ixp 8822  df-func 17762  df-nat 17850

This theorem is referenced by:  isnat2  17855  natixp  17859  nati  17862  isnatd  49254