Theorem isnat 17579

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 17578	. . . . 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 6771	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st ‘𝑓) ∈ V)
9		simprl 767	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → 𝑓 = ⟨𝐹, 𝐺⟩)
10	9	fveq2d 6760	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st ‘𝑓) = (1st ‘⟨𝐹, 𝐺⟩))
11		relfunc 17493	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
12		isnat.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
13		brrelex12 5630	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
14	11, 12, 13	sylancr 586	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
15		op1stg 7816	. . . . . . . 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 2778	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) → (1st ‘𝑓) = 𝐹)
19		fvexd 6771	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st ‘𝑔) ∈ V)
20		simplrr 774	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → 𝑔 = ⟨𝐾, 𝐿⟩)
21	20	fveq2d 6760	. . . . . . 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 5630	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐾(𝐶 Func 𝐷)𝐿) → (𝐾 ∈ V ∧ 𝐿 ∈ V))
24	11, 22, 23	sylancr 586	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ∈ V ∧ 𝐿 ∈ V))
25		op1stg 7816	. . . . . . . . 9 ⊢ ((𝐾 ∈ V ∧ 𝐿 ∈ V) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
26	24, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
27	26	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
28	21, 27	eqtrd 2778	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) → (1st ‘𝑔) = 𝐾)
29		simplr 765	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑟 = 𝐹)
30	29	fveq1d 6758	. . . . . . . . 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 6758	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑠‘𝑥) = (𝐾‘𝑥))
33	30, 32	oveq12d 7273	. . . . . . . 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 8659	. . . . . . 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 6758	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑟‘𝑦) = (𝐹‘𝑦))
36	30, 35	opeq12d 4809	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
37	31	fveq1d 6758	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑠‘𝑦) = (𝐾‘𝑦))
38	36, 37	oveq12d 7273	. . . . . . . . . . 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 2739	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑎‘𝑦) = (𝑎‘𝑦))
40	9	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → 𝑓 = ⟨𝐹, 𝐺⟩)
41	40	fveq2d 6760	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘𝑓) = (2nd ‘⟨𝐹, 𝐺⟩))
42		op2ndg 7817	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
43	14, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
44	43	ad3antrrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
45	41, 44	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘𝑓) = 𝐺)
46	45	oveqd 7272	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑥(2nd ‘𝑓)𝑦) = (𝑥𝐺𝑦))
47	46	fveq1d 6758	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑥(2nd ‘𝑓)𝑦)‘ℎ) = ((𝑥𝐺𝑦)‘ℎ))
48	38, 39, 47	oveq123d 7276	. . . . . . . . . 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 4809	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ = ⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩)
50	49, 37	oveq12d 7273	. . . . . . . . . . 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 6760	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘𝑔) = (2nd ‘⟨𝐾, 𝐿⟩))
53		op2ndg 7817	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ V ∧ 𝐿 ∈ V) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
54	24, 53	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
55	54	ad3antrrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
56	52, 55	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (2nd ‘𝑔) = 𝐿)
57	56	oveqd 7272	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑥(2nd ‘𝑔)𝑦) = (𝑥𝐿𝑦))
58	57	fveq1d 6758	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → ((𝑥(2nd ‘𝑔)𝑦)‘ℎ) = ((𝑥𝐿𝑦)‘ℎ))
59		eqidd 2739	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓 = ⟨𝐹, 𝐺⟩ ∧ 𝑔 = ⟨𝐾, 𝐿⟩)) ∧ 𝑟 = 𝐹) ∧ 𝑠 = 𝐾) → (𝑎‘𝑥) = (𝑎‘𝑥))
60	50, 58, 59	oveq123d 7276	. . . . . . . . . 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 2754	. . . . . . . . 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 3120	. . . . . . . 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 3122	. . . . . . 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 3410	. . . . . 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 3868	. . . . 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 3868	. . . 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 5071	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
68	12, 67	sylib 217	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
69		df-br 5071	. . . . 5 ⊢ (𝐾(𝐶 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
70	22, 69	sylib 217	. . . 4 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
71		ovex 7288	. . . . . . . 8 ⊢ ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∈ V
72	71	rgenw 3075	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∈ V
73		ixpexg 8668	. . . . . . 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 5251	. . . . 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 7403	. . 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 2824	. 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 6755	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑦) = (𝐴‘𝑦))
80	79	oveq1d 7270	. . . . . 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 6755	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑥) = (𝐴‘𝑥))
82	81	oveq2d 7271	. . . . . 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 2754	. . . . 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 3120	. . . 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 3122	. . 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 3617	. 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 286	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 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067  Vcvv 3422  ⦋csb 3828  ⟨cop 4564   class class class wbr 5070  Rel wrel 5585  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  1^st c1st 7802  2^nd c2nd 7803  Xcixp 8643  Basecbs 16840  Hom chom 16899  compcco 16900   Func cfunc 17485   Nat cnat 17573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-ixp 8644  df-func 17489  df-nat 17575

This theorem is referenced by:  isnat2  17580  natixp  17584  nati  17587