Theorem prdstopn 23657

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 7254	. . . . . . 7 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V)
6	3, 4, 5	syl2anc 583	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
7		eqid 2740	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
8		eqidd 2741	. . . . . 6 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
9		eqid 2740	. . . . . 6 ⊢ (TopSet‘𝑌) = (TopSet‘𝑌)
10	1, 2, 6, 7, 8, 9	prdstset 17526	. . . . 5 ⊢ (𝜑 → (TopSet‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
11		topnfn 17485	. . . . . . . . . . 11 ⊢ TopOpen Fn V
12		dffn2 6749	. . . . . . . . . . . 12 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
13	3, 12	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅:𝐼⟶V)
14		fnfco 6786	. . . . . . . . . . 11 ⊢ ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
15	11, 13, 14	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
16		eqid 2740	. . . . . . . . . . 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 23599	. . . . . . . . . 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 583	. . . . . . . . 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 4944	. . . . . . . 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		fvco2 7019	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 Fn 𝐼 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
21	3, 20	sylan 579	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
22		eqid 2740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
23		eqid 2740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (TopSet‘(𝑅‘𝑦)) = (TopSet‘(𝑅‘𝑦))
24	22, 23	topnval 17494	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((TopSet‘(𝑅‘𝑦)) ↾t (Base‘(𝑅‘𝑦))) = (TopOpen‘(𝑅‘𝑦))
25		restsspw 17491	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((TopSet‘(𝑅‘𝑦)) ↾t (Base‘(𝑅‘𝑦))) ⊆ 𝒫 (Base‘(𝑅‘𝑦))
26	24, 25	eqsstrri 4044	. . . . . . . . . . . . . . . . . . . 20 ⊢ (TopOpen‘(𝑅‘𝑦)) ⊆ 𝒫 (Base‘(𝑅‘𝑦))
27	21, 26	eqsstrdi 4063	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ⊆ 𝒫 (Base‘(𝑅‘𝑦)))
28	27	sseld 4007	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔‘𝑦) ∈ 𝒫 (Base‘(𝑅‘𝑦))))
29		fvex 6933	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔‘𝑦) ∈ V
30	29	elpw 4626	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔‘𝑦) ∈ 𝒫 (Base‘(𝑅‘𝑦)) ↔ (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦)))
31	28, 30	imbitrdi 251	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦))))
32	31	ralimdva 3173	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦))))
33		simpl2 1192	. . . . . . . . . . . . . . . 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))
34	32, 33	impel 505	. . . . . . . . . . . . . . 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)))
35		ss2ixp 8968	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦)) → X𝑦 ∈ 𝐼 (𝑔‘𝑦) ⊆ X𝑦 ∈ 𝐼 (Base‘(𝑅‘𝑦)))
36	34, 35	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)))
37		simprr 772	. . . . . . . . . . . . . 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))
38	1, 7, 2, 4, 3	prdsbas2 17529	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝑌) = X𝑦 ∈ 𝐼 (Base‘(𝑅‘𝑦)))
39	38	adantr 480	. . . . . . . . . . . . . 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)))
40	36, 37, 39	3sstr4d 4056	. . . . . . . . . . . . 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))
41	40	ex 412	. . . . . . . . . . . 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)))
42	41	exlimdv 1932	. . . . . . . . . . 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)))
43		velpw 4627	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 (Base‘𝑌) ↔ 𝑥 ⊆ (Base‘𝑌))
44	42, 43	imbitrrdi 252	. . . . . . . . . 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)))
45	44	abssdv 4091	. . . . . . . . 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))
46		fvex 6933	. . . . . . . . . . 11 ⊢ (Base‘𝑌) ∈ V
47	46	pwex 5398	. . . . . . . . . 10 ⊢ 𝒫 (Base‘𝑌) ∈ V
48	47	ssex 5339	. . . . . . . . 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)
49		unitg 22995	. . . . . . . . 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))})
50	45, 48, 49	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))})
51	19, 50	eqtrd 2780	. . . . . . 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))})
52		sspwuni 5123	. . . . . . . 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))
53	45, 52	sylib 218	. . . . . . 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))
54	51, 53	eqsstrd 4047	. . . . . 6 ⊢ (𝜑 → ∪ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
55		sspwuni 5123	. . . . . 6 ⊢ ((∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌) ↔ ∪ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
56	54, 55	sylibr 234	. . . . 5 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌))
57	10, 56	eqsstrd 4047	. . . 4 ⊢ (𝜑 → (TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌))
58	7, 9	topnid 17495	. . . 4 ⊢ ((TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌) → (TopSet‘𝑌) = (TopOpen‘𝑌))
59	57, 58	syl 17	. . 3 ⊢ (𝜑 → (TopSet‘𝑌) = (TopOpen‘𝑌))
60		prdstopn.o	. . 3 ⊢ 𝑂 = (TopOpen‘𝑌)
61	59, 60	eqtr4di 2798	. 2 ⊢ (𝜑 → (TopSet‘𝑌) = 𝑂)
62	61, 10	eqtr3d 2782	1 ⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931  dom cdm 5700   ∘ ccom 5704   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Xcixp 8955  Fincfn 9003  Basecbs 17258  TopSetcts 17317   ↾_t crest 17480  TopOpenctopn 17481  topGenctg 17497  ∏_tcpt 17498  X_scprds 17505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-hom 17335  df-cco 17336  df-rest 17482  df-topn 17483  df-topgen 17503  df-pt 17504  df-prds 17507

This theorem is referenced by:  xpstopnlem2  23840  prdstmdd  24153  prdstgpd  24154  prdsxmslem2  24563