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Theorem prdstopn 23002
Description: Topology of a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
Hypotheses
Ref Expression
prdstopn.y π‘Œ = (𝑆Xs𝑅)
prdstopn.s (πœ‘ β†’ 𝑆 ∈ 𝑉)
prdstopn.i (πœ‘ β†’ 𝐼 ∈ π‘Š)
prdstopn.r (πœ‘ β†’ 𝑅 Fn 𝐼)
prdstopn.o 𝑂 = (TopOpenβ€˜π‘Œ)
Assertion
Ref Expression
prdstopn (πœ‘ β†’ 𝑂 = (∏tβ€˜(TopOpen ∘ 𝑅)))

Proof of Theorem prdstopn
Dummy variables π‘₯ 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdstopn.y . . . . . 6 π‘Œ = (𝑆Xs𝑅)
2 prdstopn.s . . . . . 6 (πœ‘ β†’ 𝑆 ∈ 𝑉)
3 prdstopn.r . . . . . . 7 (πœ‘ β†’ 𝑅 Fn 𝐼)
4 prdstopn.i . . . . . . 7 (πœ‘ β†’ 𝐼 ∈ π‘Š)
5 fnex 7171 . . . . . . 7 ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ π‘Š) β†’ 𝑅 ∈ V)
63, 4, 5syl2anc 585 . . . . . 6 (πœ‘ β†’ 𝑅 ∈ V)
7 eqid 2733 . . . . . 6 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
8 eqidd 2734 . . . . . 6 (πœ‘ β†’ dom 𝑅 = dom 𝑅)
9 eqid 2733 . . . . . 6 (TopSetβ€˜π‘Œ) = (TopSetβ€˜π‘Œ)
101, 2, 6, 7, 8, 9prdstset 17356 . . . . 5 (πœ‘ β†’ (TopSetβ€˜π‘Œ) = (∏tβ€˜(TopOpen ∘ 𝑅)))
11 topnfn 17315 . . . . . . . . . . 11 TopOpen Fn V
12 dffn2 6674 . . . . . . . . . . . 12 (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟢V)
133, 12sylib 217 . . . . . . . . . . 11 (πœ‘ β†’ 𝑅:𝐼⟢V)
14 fnfco 6711 . . . . . . . . . . 11 ((TopOpen Fn V ∧ 𝑅:𝐼⟢V) β†’ (TopOpen ∘ 𝑅) Fn 𝐼)
1511, 13, 14sylancr 588 . . . . . . . . . 10 (πœ‘ β†’ (TopOpen ∘ 𝑅) Fn 𝐼)
16 eqid 2733 . . . . . . . . . . 11 {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))} = {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))}
1716ptval 22944 . . . . . . . . . 10 ((𝐼 ∈ π‘Š ∧ (TopOpen ∘ 𝑅) Fn 𝐼) β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) = (topGenβ€˜{π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))}))
184, 15, 17syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) = (topGenβ€˜{π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))}))
1918unieqd 4883 . . . . . . . 8 (πœ‘ β†’ βˆͺ (∏tβ€˜(TopOpen ∘ 𝑅)) = βˆͺ (topGenβ€˜{π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))}))
20 fvco2 6942 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 Fn 𝐼 ∧ 𝑦 ∈ 𝐼) β†’ ((TopOpen ∘ 𝑅)β€˜π‘¦) = (TopOpenβ€˜(π‘…β€˜π‘¦)))
213, 20sylan 581 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ ((TopOpen ∘ 𝑅)β€˜π‘¦) = (TopOpenβ€˜(π‘…β€˜π‘¦)))
22 eqid 2733 . . . . . . . . . . . . . . . . . . . . . 22 (Baseβ€˜(π‘…β€˜π‘¦)) = (Baseβ€˜(π‘…β€˜π‘¦))
23 eqid 2733 . . . . . . . . . . . . . . . . . . . . . 22 (TopSetβ€˜(π‘…β€˜π‘¦)) = (TopSetβ€˜(π‘…β€˜π‘¦))
2422, 23topnval 17324 . . . . . . . . . . . . . . . . . . . . 21 ((TopSetβ€˜(π‘…β€˜π‘¦)) β†Ύt (Baseβ€˜(π‘…β€˜π‘¦))) = (TopOpenβ€˜(π‘…β€˜π‘¦))
25 restsspw 17321 . . . . . . . . . . . . . . . . . . . . 21 ((TopSetβ€˜(π‘…β€˜π‘¦)) β†Ύt (Baseβ€˜(π‘…β€˜π‘¦))) βŠ† 𝒫 (Baseβ€˜(π‘…β€˜π‘¦))
2624, 25eqsstrri 3983 . . . . . . . . . . . . . . . . . . . 20 (TopOpenβ€˜(π‘…β€˜π‘¦)) βŠ† 𝒫 (Baseβ€˜(π‘…β€˜π‘¦))
2721, 26eqsstrdi 4002 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ ((TopOpen ∘ 𝑅)β€˜π‘¦) βŠ† 𝒫 (Baseβ€˜(π‘…β€˜π‘¦)))
2827sseld 3947 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ ((π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) β†’ (π‘”β€˜π‘¦) ∈ 𝒫 (Baseβ€˜(π‘…β€˜π‘¦))))
29 fvex 6859 . . . . . . . . . . . . . . . . . . 19 (π‘”β€˜π‘¦) ∈ V
3029elpw 4568 . . . . . . . . . . . . . . . . . 18 ((π‘”β€˜π‘¦) ∈ 𝒫 (Baseβ€˜(π‘…β€˜π‘¦)) ↔ (π‘”β€˜π‘¦) βŠ† (Baseβ€˜(π‘…β€˜π‘¦)))
3128, 30syl6ib 251 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ ((π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) β†’ (π‘”β€˜π‘¦) βŠ† (Baseβ€˜(π‘…β€˜π‘¦))))
3231ralimdva 3161 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) β†’ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) βŠ† (Baseβ€˜(π‘…β€˜π‘¦))))
33 simpl2 1193 . . . . . . . . . . . . . . . 16 (((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦)) β†’ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦))
3432, 33impel 507 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ ((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))) β†’ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) βŠ† (Baseβ€˜(π‘…β€˜π‘¦)))
35 ss2ixp 8854 . . . . . . . . . . . . . . 15 (βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) βŠ† (Baseβ€˜(π‘…β€˜π‘¦)) β†’ X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦) βŠ† X𝑦 ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘¦)))
3634, 35syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ ((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))) β†’ X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦) βŠ† X𝑦 ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘¦)))
37 simprr 772 . . . . . . . . . . . . . 14 ((πœ‘ ∧ ((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))) β†’ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))
381, 7, 2, 4, 3prdsbas2 17359 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (Baseβ€˜π‘Œ) = X𝑦 ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘¦)))
3938adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ ((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))) β†’ (Baseβ€˜π‘Œ) = X𝑦 ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘¦)))
4036, 37, 393sstr4d 3995 . . . . . . . . . . . . 13 ((πœ‘ ∧ ((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))) β†’ π‘₯ βŠ† (Baseβ€˜π‘Œ))
4140ex 414 . . . . . . . . . . . 12 (πœ‘ β†’ (((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦)) β†’ π‘₯ βŠ† (Baseβ€˜π‘Œ)))
4241exlimdv 1937 . . . . . . . . . . 11 (πœ‘ β†’ (βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦)) β†’ π‘₯ βŠ† (Baseβ€˜π‘Œ)))
43 velpw 4569 . . . . . . . . . . 11 (π‘₯ ∈ 𝒫 (Baseβ€˜π‘Œ) ↔ π‘₯ βŠ† (Baseβ€˜π‘Œ))
4442, 43syl6ibr 252 . . . . . . . . . 10 (πœ‘ β†’ (βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦)) β†’ π‘₯ ∈ 𝒫 (Baseβ€˜π‘Œ)))
4544abssdv 4029 . . . . . . . . 9 (πœ‘ β†’ {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))} βŠ† 𝒫 (Baseβ€˜π‘Œ))
46 fvex 6859 . . . . . . . . . . 11 (Baseβ€˜π‘Œ) ∈ V
4746pwex 5339 . . . . . . . . . 10 𝒫 (Baseβ€˜π‘Œ) ∈ V
4847ssex 5282 . . . . . . . . 9 ({π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))} βŠ† 𝒫 (Baseβ€˜π‘Œ) β†’ {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))} ∈ V)
49 unitg 22340 . . . . . . . . 9 ({π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))} ∈ V β†’ βˆͺ (topGenβ€˜{π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))}) = βˆͺ {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))})
5045, 48, 493syl 18 . . . . . . . 8 (πœ‘ β†’ βˆͺ (topGenβ€˜{π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))}) = βˆͺ {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))})
5119, 50eqtrd 2773 . . . . . . 7 (πœ‘ β†’ βˆͺ (∏tβ€˜(TopOpen ∘ 𝑅)) = βˆͺ {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))})
52 sspwuni 5064 . . . . . . . 8 ({π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))} βŠ† 𝒫 (Baseβ€˜π‘Œ) ↔ βˆͺ {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))} βŠ† (Baseβ€˜π‘Œ))
5345, 52sylib 217 . . . . . . 7 (πœ‘ β†’ βˆͺ {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐼 ∧ βˆ€π‘¦ ∈ 𝐼 (π‘”β€˜π‘¦) ∈ ((TopOpen ∘ 𝑅)β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐼 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ ((TopOpen ∘ 𝑅)β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐼 (π‘”β€˜π‘¦))} βŠ† (Baseβ€˜π‘Œ))
5451, 53eqsstrd 3986 . . . . . 6 (πœ‘ β†’ βˆͺ (∏tβ€˜(TopOpen ∘ 𝑅)) βŠ† (Baseβ€˜π‘Œ))
55 sspwuni 5064 . . . . . 6 ((∏tβ€˜(TopOpen ∘ 𝑅)) βŠ† 𝒫 (Baseβ€˜π‘Œ) ↔ βˆͺ (∏tβ€˜(TopOpen ∘ 𝑅)) βŠ† (Baseβ€˜π‘Œ))
5654, 55sylibr 233 . . . . 5 (πœ‘ β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) βŠ† 𝒫 (Baseβ€˜π‘Œ))
5710, 56eqsstrd 3986 . . . 4 (πœ‘ β†’ (TopSetβ€˜π‘Œ) βŠ† 𝒫 (Baseβ€˜π‘Œ))
587, 9topnid 17325 . . . 4 ((TopSetβ€˜π‘Œ) βŠ† 𝒫 (Baseβ€˜π‘Œ) β†’ (TopSetβ€˜π‘Œ) = (TopOpenβ€˜π‘Œ))
5957, 58syl 17 . . 3 (πœ‘ β†’ (TopSetβ€˜π‘Œ) = (TopOpenβ€˜π‘Œ))
60 prdstopn.o . . 3 𝑂 = (TopOpenβ€˜π‘Œ)
6159, 60eqtr4di 2791 . 2 (πœ‘ β†’ (TopSetβ€˜π‘Œ) = 𝑂)
6261, 10eqtr3d 2775 1 (πœ‘ β†’ 𝑂 = (∏tβ€˜(TopOpen ∘ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447   βˆ– cdif 3911   βŠ† wss 3914  π’« cpw 4564  βˆͺ cuni 4869  dom cdm 5637   ∘ ccom 5641   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  Xcixp 8841  Fincfn 8889  Basecbs 17091  TopSetcts 17147   β†Ύt crest 17310  TopOpenctopn 17311  topGenctg 17327  βˆtcpt 17328  Xscprds 17335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-ixp 8842  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-sup 9386  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-fz 13434  df-struct 17027  df-slot 17062  df-ndx 17074  df-base 17092  df-plusg 17154  df-mulr 17155  df-sca 17157  df-vsca 17158  df-ip 17159  df-tset 17160  df-ple 17161  df-ds 17163  df-hom 17165  df-cco 17166  df-rest 17312  df-topn 17313  df-topgen 17333  df-pt 17334  df-prds 17337
This theorem is referenced by:  xpstopnlem2  23185  prdstmdd  23498  prdstgpd  23499  prdsxmslem2  23908
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