Theorem prdstopn 22230

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 6974	. . . . . . 7 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V)
6	3, 4, 5	syl2anc 586	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
7		eqid 2821	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
8		eqidd 2822	. . . . . 6 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
9		eqid 2821	. . . . . 6 ⊢ (TopSet‘𝑌) = (TopSet‘𝑌)
10	1, 2, 6, 7, 8, 9	prdstset 16733	. . . . 5 ⊢ (𝜑 → (TopSet‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
11		topnfn 16693	. . . . . . . . . . 11 ⊢ TopOpen Fn V
12		dffn2 6511	. . . . . . . . . . . 12 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
13	3, 12	sylib 220	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅:𝐼⟶V)
14		fnfco 6538	. . . . . . . . . . 11 ⊢ ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
15	11, 13, 14	sylancr 589	. . . . . . . . . 10 ⊢ (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
16		eqid 2821	. . . . . . . . . . 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 22172	. . . . . . . . . 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 586	. . . . . . . . 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 4842	. . . . . . . 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 6753	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 Fn 𝐼 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
21	3, 20	sylan 582	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
22		eqid 2821	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
23		eqid 2821	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (TopSet‘(𝑅‘𝑦)) = (TopSet‘(𝑅‘𝑦))
24	22, 23	topnval 16702	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((TopSet‘(𝑅‘𝑦)) ↾t (Base‘(𝑅‘𝑦))) = (TopOpen‘(𝑅‘𝑦))
25		restsspw 16699	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((TopSet‘(𝑅‘𝑦)) ↾t (Base‘(𝑅‘𝑦))) ⊆ 𝒫 (Base‘(𝑅‘𝑦))
26	24, 25	eqsstrri 4002	. . . . . . . . . . . . . . . . . . . 20 ⊢ (TopOpen‘(𝑅‘𝑦)) ⊆ 𝒫 (Base‘(𝑅‘𝑦))
27	21, 26	eqsstrdi 4021	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ⊆ 𝒫 (Base‘(𝑅‘𝑦)))
28	27	sseld 3966	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((𝑔‘𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔‘𝑦) ∈ 𝒫 (Base‘(𝑅‘𝑦))))
29		fvex 6678	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔‘𝑦) ∈ V
30	29	elpw 4546	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔‘𝑦) ∈ 𝒫 (Base‘(𝑅‘𝑦)) ↔ (𝑔‘𝑦) ⊆ (Base‘(𝑅‘𝑦)))
31	28, 30	syl6ib 253	. . . . . . . . . . . . . . . . 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 1188	. . . . . . . . . . . . . . . 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 508	. . . . . . . . . . . . . . 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 8468	. . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 16736	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝑌) = X𝑦 ∈ 𝐼 (Base‘(𝑅‘𝑦)))
39	38	adantr 483	. . . . . . . . . . . . . 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 4014	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 1930	. . . . . . . . . . 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 4547	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 (Base‘𝑌) ↔ 𝑥 ⊆ (Base‘𝑌))
44	42, 43	syl6ibr 254	. . . . . . . . . 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 4045	. . . . . . . . 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 6678	. . . . . . . . . . 11 ⊢ (Base‘𝑌) ∈ V
47	46	pwex 5274	. . . . . . . . . 10 ⊢ 𝒫 (Base‘𝑌) ∈ V
48	47	ssex 5218	. . . . . . . . 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 21569	. . . . . . . . 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 2856	. . . . . . 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 5015	. . . . . . . 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 220	. . . . . . 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 4005	. . . . . 6 ⊢ (𝜑 → ∪ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
55		sspwuni 5015	. . . . . 6 ⊢ ((∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌) ↔ ∪ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
56	54, 55	sylibr 236	. . . . 5 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌))
57	10, 56	eqsstrd 4005	. . . 4 ⊢ (𝜑 → (TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌))
58	7, 9	topnid 16703	. . . 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	syl6eqr 2874	. 2 ⊢ (𝜑 → (TopSet‘𝑌) = 𝑂)
62	61, 10	eqtr3d 2858	1 ⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {cab 2799  ∀wral 3138  ∃wrex 3139  Vcvv 3495   ∖ cdif 3933   ⊆ wss 3936  𝒫 cpw 4539  ∪ cuni 4832  dom cdm 5550   ∘ ccom 5554   Fn wfn 6345  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150  Xcixp 8455  Fincfn 8503  Basecbs 16477  TopSetcts 16565   ↾_t crest 16688  TopOpenctopn 16689  topGenctg 16705  ∏_tcpt 16706  X_scprds 16713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12887  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-plusg 16572  df-mulr 16573  df-sca 16575  df-vsca 16576  df-ip 16577  df-tset 16578  df-ple 16579  df-ds 16581  df-hom 16583  df-cco 16584  df-rest 16690  df-topn 16691  df-topgen 16711  df-pt 16712  df-prds 16715

This theorem is referenced by:  xpstopnlem2  22413  prdstmdd  22726  prdstgpd  22727  prdsxmslem2  23133