Theorem prdstopn 23618

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 7168	. . . . . . 7 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V)
6	3, 4, 5	syl2anc 590	. . . . . 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 17427	. . . . 5 ⊢ (𝜑 → (TopSet‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
11		topnfn 17386	. . . . . . . . . . 11 ⊢ TopOpen Fn V
12		dffn2 6664	. . . . . . . . . . . 12 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
13	3, 12	sylib 219	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅:𝐼⟶V)
14		fnfco 6699	. . . . . . . . . . 11 ⊢ ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
15	11, 13, 14	sylancr 593	. . . . . . . . . 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 23560	. . . . . . . . . 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 590	. . . . . . . . 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 4858	. . . . . . . 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 6931	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 Fn 𝐼 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
21	3, 20	sylan 586	. . . . . . . . . . . . . . . . . . . 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 17395	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((TopSet‘(𝑅‘𝑦)) ↾t (Base‘(𝑅‘𝑦))) = (TopOpen‘(𝑅‘𝑦))
25		restsspw 17392	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((TopSet‘(𝑅‘𝑦)) ↾t (Base‘(𝑅‘𝑦))) ⊆ 𝒫 (Base‘(𝑅‘𝑦))
26	24, 25	eqsstrri 3969	. . . . . . . . . . . . . . . . . . . 20 ⊢ (TopOpen‘(𝑅‘𝑦)) ⊆ 𝒫 (Base‘(𝑅‘𝑦))
27	21, 26	eqsstrdi 3966	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ⊆ 𝒫 (Base‘(𝑅‘𝑦)))
28	27	sseld 3921	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔‘𝑦) ∈ 𝒫 (Base‘(𝑅‘𝑦))))
29		fvex 6847	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔‘𝑦) ∈ V
30	29	elpw 4540	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔‘𝑦) ∈ 𝒫 (Base‘(𝑅‘𝑦)) ↔ (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦)))
31	28, 30	imbitrdi 252	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦))))
32	31	ralimdva 3152	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → ∀𝑦 ∈ 𝐼 (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦))))
33		simpl2 1199	. . . . . . . . . . . . . . . 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 510	. . . . . . . . . . . . . . 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 8855	. . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . 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 17430	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝑌) = X𝑦 ∈ 𝐼 (Base‘(𝑅‘𝑦)))
39	38	adantr 481	. . . . . . . . . . . . . 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 3977	. . . . . . . . . . . . 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 413	. . . . . . . . . . . 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 1940	. . . . . . . . . . 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 4541	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 (Base‘𝑌) ↔ 𝑥 ⊆ (Base‘𝑌))
44	42, 43	imbitrrdi 253	. . . . . . . . . 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 4005	. . . . . . . . 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 6847	. . . . . . . . . . 11 ⊢ (Base‘𝑌) ∈ V
47	46	pwex 5316	. . . . . . . . . 10 ⊢ 𝒫 (Base‘𝑌) ∈ V
48	47	ssex 5256	. . . . . . . . 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 22957	. . . . . . . . 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 2775	. . . . . . 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 5036	. . . . . . . 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 219	. . . . . . 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 3956	. . . . . 6 ⊢ (𝜑 → ∪ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
55		sspwuni 5036	. . . . . 6 ⊢ ((∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌) ↔ ∪ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
56	54, 55	sylibr 235	. . . . 5 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌))
57	10, 56	eqsstrd 3956	. . . 4 ⊢ (𝜑 → (TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌))
58	7, 9	topnid 17396	. . . 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 2793	. 2 ⊢ (𝜑 → (TopSet‘𝑌) = 𝑂)
62	61, 10	eqtr3d 2777	1 ⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2718  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ∖ cdif 3887   ⊆ wss 3890  𝒫 cpw 4536  ∪ cuni 4845  dom cdm 5625   ∘ ccom 5629   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  Xcixp 8842  Fincfn 8890  Basecbs 17177  TopSetcts 17224   ↾_t crest 17381  TopOpenctopn 17382  topGenctg 17398  ∏_tcpt 17399  X_scprds 17406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-map 8772  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9352  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-n0 12436  df-z 12523  df-dec 12643  df-uz 12787  df-fz 13460  df-struct 17115  df-slot 17150  df-ndx 17162  df-base 17178  df-plusg 17231  df-mulr 17232  df-sca 17234  df-vsca 17235  df-ip 17236  df-tset 17237  df-ple 17238  df-ds 17240  df-hom 17242  df-cco 17243  df-rest 17383  df-topn 17384  df-topgen 17404  df-pt 17405  df-prds 17408

This theorem is referenced by:  xpstopnlem2  23801  prdstmdd  24114  prdstgpd  24115  prdsxmslem2  24519