Theorem prdstopn 23766

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 7217	. . . . . . 7 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V)
6	3, 4, 5	syl2anc 595	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
7		eqid 2763	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
8		eqidd 2764	. . . . . 6 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
9		eqid 2763	. . . . . 6 ⊢ (TopSet‘𝑌) = (TopSet‘𝑌)
10	1, 2, 6, 7, 8, 9	prdstset 17520	. . . . 5 ⊢ (𝜑 → (TopSet‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
11		topnfn 17479	. . . . . . . . . . 11 ⊢ TopOpen Fn V
12		dffn2 6709	. . . . . . . . . . . 12 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
13	3, 12	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅:𝐼⟶V)
14		fnfco 6745	. . . . . . . . . . 11 ⊢ ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
15	11, 13, 14	sylancr 598	. . . . . . . . . 10 ⊢ (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
16		eqid 2763	. . . . . . . . . . 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 23708	. . . . . . . . . 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 595	. . . . . . . . 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 4886	. . . . . . . 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 6980	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 Fn 𝐼 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
21	3, 20	sylan 591	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
22		eqid 2763	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
23		eqid 2763	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (TopSet‘(𝑅‘𝑦)) = (TopSet‘(𝑅‘𝑦))
24	22, 23	topnval 17488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((TopSet‘(𝑅‘𝑦)) ↾t (Base‘(𝑅‘𝑦))) = (TopOpen‘(𝑅‘𝑦))
25		restsspw 17485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((TopSet‘(𝑅‘𝑦)) ↾t (Base‘(𝑅‘𝑦))) ⊆ 𝒫 (Base‘(𝑅‘𝑦))
26	24, 25	eqsstrri 3985	. . . . . . . . . . . . . . . . . . . 20 ⊢ (TopOpen‘(𝑅‘𝑦)) ⊆ 𝒫 (Base‘(𝑅‘𝑦))
27	21, 26	eqsstrdi 3982	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ⊆ 𝒫 (Base‘(𝑅‘𝑦)))
28	27	sseld 3937	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔‘𝑦) ∈ 𝒫 (Base‘(𝑅‘𝑦))))
29		fvex 6896	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔‘𝑦) ∈ V
30	29	elpw 4567	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔‘𝑦) ∈ 𝒫 (Base‘(𝑅‘𝑦)) ↔ (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦)))
31	28, 30	imbitrdi 254	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦))))
32	31	ralimdva 3177	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦))))
33		simpl2 1211	. . . . . . . . . . . . . . . 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 514	. . . . . . . . . . . . . . 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 8909	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦)) → X𝑦 ∈ 𝐼 (𝑔‘𝑦) ⊆ X𝑦 ∈ 𝐼 (Base‘(𝑅‘𝑦)))
36	34, 35	syl 18	. . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . 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 17523	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝑌) = X𝑦 ∈ 𝐼 (Base‘(𝑅‘𝑦)))
39	38	adantr 485	. . . . . . . . . . . . . 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 3993	. . . . . . . . . . . . 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 417	. . . . . . . . . . . 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 1963	. . . . . . . . . . 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 4568	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 (Base‘𝑌) ↔ 𝑥 ⊆ (Base‘𝑌))
44	42, 43	imbitrrdi 255	. . . . . . . . . 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 4022	. . . . . . . . 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 6896	. . . . . . . . . . 11 ⊢ (Base‘𝑌) ∈ V
47	46	pwex 5353	. . . . . . . . . 10 ⊢ 𝒫 (Base‘𝑌) ∈ V
48	47	ssex 5292	. . . . . . . . 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 23105	. . . . . . . . 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 19	. . . . . . . 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 2798	. . . . . . 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 5067	. . . . . . . 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 221	. . . . . . 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 3972	. . . . . 6 ⊢ (𝜑 → ∪ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
55		sspwuni 5067	. . . . . 6 ⊢ ((∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌) ↔ ∪ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
56	54, 55	sylibr 237	. . . . 5 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌))
57	10, 56	eqsstrd 3972	. . . 4 ⊢ (𝜑 → (TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌))
58	7, 9	topnid 17489	. . . 4 ⊢ ((TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌) → (TopSet‘𝑌) = (TopOpen‘𝑌))
59	57, 58	syl 18	. . 3 ⊢ (𝜑 → (TopSet‘𝑌) = (TopOpen‘𝑌))
60		prdstopn.o	. . 3 ⊢ 𝑂 = (TopOpen‘𝑌)
61	59, 60	eqtr4di 2816	. 2 ⊢ (𝜑 → (TopSet‘𝑌) = 𝑂)
62	61, 10	eqtr3d 2800	1 ⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906  𝒫 cpw 4563  ∪ cuni 4873  dom cdm 5663   ∘ ccom 5667   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412  Xcixp 8896  Fincfn 8944  Basecbs 17270  TopSetcts 17317   ↾_t crest 17474  TopOpenctopn 17475  topGenctg 17491  ∏_tcpt 17492  X_scprds 17499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-map 8827  df-ixp 8897  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-sup 9403  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-z 12593  df-dec 12713  df-uz 12864  df-fz 13537  df-struct 17208  df-slot 17243  df-ndx 17255  df-base 17271  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 17476  df-topn 17477  df-topgen 17497  df-pt 17498  df-prds 17501

This theorem is referenced by:  xpstopnlem2  23949  prdstmdd  24262  prdstgpd  24263  prdsxmslem2  24667