Theorem prdstopn 21652

Description: Topology of a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)

Hypotheses

Ref	Expression
prdstopn.y	⊢ 𝑌 = (𝑆Xs𝑅)
prdstopn.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdstopn.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
prdstopn.r	⊢ (𝜑 → 𝑅 Fn 𝐼)
prdstopn.o	⊢ 𝑂 = (TopOpen‘𝑌)

Assertion

Ref	Expression
prdstopn	⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))

Proof of Theorem prdstopn

Dummy variables 𝑥 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdstopn.y	. . . . . 6 ⊢ 𝑌 = (𝑆Xs𝑅)
2		prdstopn.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
3		prdstopn.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 Fn 𝐼)
4		prdstopn.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
5		fnex 6625	. . . . . . 7 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V)
6	3, 4, 5	syl2anc 573	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
7		eqid 2771	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
8		eqidd 2772	. . . . . 6 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
9		eqid 2771	. . . . . 6 ⊢ (TopSet‘𝑌) = (TopSet‘𝑌)
10	1, 2, 6, 7, 8, 9	prdstset 16334	. . . . 5 ⊢ (𝜑 → (TopSet‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
11		topnfn 16294	. . . . . . . . . . 11 ⊢ TopOpen Fn V
12		dffn2 6187	. . . . . . . . . . . 12 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
13	3, 12	sylib 208	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅:𝐼⟶V)
14		fnfco 6209	. . . . . . . . . . 11 ⊢ ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
15	11, 13, 14	sylancr 575	. . . . . . . . . 10 ⊢ (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
16		eqid 2771	. . . . . . . . . . 11 ⊢ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))}
17	16	ptval 21594	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑊 ∧ (TopOpen ∘ 𝑅) Fn 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))}))
18	4, 15, 17	syl2anc 573	. . . . . . . . 9 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))}))
19	18	unieqd 4584	. . . . . . . 8 ⊢ (𝜑 → ∪ (∏t‘(TopOpen ∘ 𝑅)) = ∪ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))}))
20		simpl2 1229	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦)) → ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦))
21		fvco2 6415	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 Fn 𝐼 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
22	3, 21	sylan 569	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
23		eqid 2771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
24		eqid 2771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (TopSet‘(𝑅‘𝑦)) = (TopSet‘(𝑅‘𝑦))
25	23, 24	topnval 16303	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((TopSet‘(𝑅‘𝑦)) ↾t (Base‘(𝑅‘𝑦))) = (TopOpen‘(𝑅‘𝑦))
26		restsspw 16300	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((TopSet‘(𝑅‘𝑦)) ↾t (Base‘(𝑅‘𝑦))) ⊆ 𝒫 (Base‘(𝑅‘𝑦))
27	25, 26	eqsstr3i 3785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (TopOpen‘(𝑅‘𝑦)) ⊆ 𝒫 (Base‘(𝑅‘𝑦))
28	22, 27	syl6eqss 3804	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ⊆ 𝒫 (Base‘(𝑅‘𝑦)))
29	28	sseld 3751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔‘𝑦) ∈ 𝒫 (Base‘(𝑅‘𝑦))))
30		fvex 6342	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔‘𝑦) ∈ V
31	30	elpw 4303	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔‘𝑦) ∈ 𝒫 (Base‘(𝑅‘𝑦)) ↔ (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦)))
32	29, 31	syl6ib 241	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦))))
33	32	ralimdva 3111	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦))))
34	33	imp 393	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦)) → ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦)))
35	20, 34	sylan2 580	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))) → ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦)))
36		ss2ixp 8075	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦)) → X𝑦 ∈ 𝐼 (𝑔‘𝑦) ⊆ X𝑦 ∈ 𝐼 (Base‘(𝑅‘𝑦)))
37	35, 36	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))) → X𝑦 ∈ 𝐼 (𝑔‘𝑦) ⊆ X𝑦 ∈ 𝐼 (Base‘(𝑅‘𝑦)))
38		simprr 756	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))) → 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))
39	1, 7, 2, 4, 3	prdsbas2 16337	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝑌) = X𝑦 ∈ 𝐼 (Base‘(𝑅‘𝑦)))
40	39	adantr 466	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))) → (Base‘𝑌) = X𝑦 ∈ 𝐼 (Base‘(𝑅‘𝑦)))
41	37, 38, 40	3sstr4d 3797	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))) → 𝑥 ⊆ (Base‘𝑌))
42	41	ex 397	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦)) → 𝑥 ⊆ (Base‘𝑌)))
43	42	exlimdv 2013	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦)) → 𝑥 ⊆ (Base‘𝑌)))
44		selpw 4304	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 (Base‘𝑌) ↔ 𝑥 ⊆ (Base‘𝑌))
45	43, 44	syl6ibr 242	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦)) → 𝑥 ∈ 𝒫 (Base‘𝑌)))
46	45	abssdv 3825	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))} ⊆ 𝒫 (Base‘𝑌))
47		fvex 6342	. . . . . . . . . . 11 ⊢ (Base‘𝑌) ∈ V
48	47	pwex 4981	. . . . . . . . . 10 ⊢ 𝒫 (Base‘𝑌) ∈ V
49	48	ssex 4936	. . . . . . . . 9 ⊢ ({𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))} ⊆ 𝒫 (Base‘𝑌) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))} ∈ V)
50		unitg 20992	. . . . . . . . 9 ⊢ ({𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))} ∈ V → ∪ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))}) = ∪ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))})
51	46, 49, 50	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ∪ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))}) = ∪ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))})
52	19, 51	eqtrd 2805	. . . . . . 7 ⊢ (𝜑 → ∪ (∏t‘(TopOpen ∘ 𝑅)) = ∪ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))})
53		sspwuni 4745	. . . . . . . 8 ⊢ ({𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))} ⊆ 𝒫 (Base‘𝑌) ↔ ∪ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))} ⊆ (Base‘𝑌))
54	46, 53	sylib 208	. . . . . . 7 ⊢ (𝜑 → ∪ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑦) = ∪ ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐼 (𝑔‘𝑦))} ⊆ (Base‘𝑌))
55	52, 54	eqsstrd 3788	. . . . . 6 ⊢ (𝜑 → ∪ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
56		sspwuni 4745	. . . . . 6 ⊢ ((∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌) ↔ ∪ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
57	55, 56	sylibr 224	. . . . 5 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌))
58	10, 57	eqsstrd 3788	. . . 4 ⊢ (𝜑 → (TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌))
59	7, 9	topnid 16304	. . . 4 ⊢ ((TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌) → (TopSet‘𝑌) = (TopOpen‘𝑌))
60	58, 59	syl 17	. . 3 ⊢ (𝜑 → (TopSet‘𝑌) = (TopOpen‘𝑌))
61		prdstopn.o	. . 3 ⊢ 𝑂 = (TopOpen‘𝑌)
62	60, 61	syl6eqr 2823	. 2 ⊢ (𝜑 → (TopSet‘𝑌) = 𝑂)
63	62, 10	eqtr3d 2807	1 ⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))

Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631  ∃wex 1852   ∈ wcel 2145  {cab 2757  ∀wral 3061  ∃wrex 3062  Vcvv 3351   ∖ cdif 3720   ⊆ wss 3723  𝒫 cpw 4297  ∪ cuni 4574  dom cdm 5249   ∘ ccom 5253   Fn wfn 6026  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793  Xcixp 8062  Fincfn 8109  Basecbs 16064  TopSetcts 16155   ↾_t crest 16289  TopOpenctopn 16290  topGenctg 16306  ∏_tcpt 16307  X_scprds 16314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-fz 12534  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-plusg 16162  df-mulr 16163  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16172  df-hom 16174  df-cco 16175  df-rest 16291  df-topn 16292  df-topgen 16312  df-pt 16313  df-prds 16316

This theorem is referenced by:  xpstopnlem2  21835  prdstmdd  22147  prdstgpd  22148  prdsxmslem2  22554