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Theorem prdstopn 22877
Description: Topology of a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
Hypotheses
Ref Expression
prdstopn.y 𝑌 = (𝑆Xs𝑅)
prdstopn.s (𝜑𝑆𝑉)
prdstopn.i (𝜑𝐼𝑊)
prdstopn.r (𝜑𝑅 Fn 𝐼)
prdstopn.o 𝑂 = (TopOpen‘𝑌)
Assertion
Ref Expression
prdstopn (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))

Proof of Theorem prdstopn
Dummy variables 𝑥 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdstopn.y . . . . . 6 𝑌 = (𝑆Xs𝑅)
2 prdstopn.s . . . . . 6 (𝜑𝑆𝑉)
3 prdstopn.r . . . . . . 7 (𝜑𝑅 Fn 𝐼)
4 prdstopn.i . . . . . . 7 (𝜑𝐼𝑊)
5 fnex 7143 . . . . . . 7 ((𝑅 Fn 𝐼𝐼𝑊) → 𝑅 ∈ V)
63, 4, 5syl2anc 584 . . . . . 6 (𝜑𝑅 ∈ V)
7 eqid 2736 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
8 eqidd 2737 . . . . . 6 (𝜑 → dom 𝑅 = dom 𝑅)
9 eqid 2736 . . . . . 6 (TopSet‘𝑌) = (TopSet‘𝑌)
101, 2, 6, 7, 8, 9prdstset 17266 . . . . 5 (𝜑 → (TopSet‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
11 topnfn 17225 . . . . . . . . . . 11 TopOpen Fn V
12 dffn2 6647 . . . . . . . . . . . 12 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
133, 12sylib 217 . . . . . . . . . . 11 (𝜑𝑅:𝐼⟶V)
14 fnfco 6684 . . . . . . . . . . 11 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
1511, 13, 14sylancr 587 . . . . . . . . . 10 (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
16 eqid 2736 . . . . . . . . . . 11 {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))}
1716ptval 22819 . . . . . . . . . 10 ((𝐼𝑊 ∧ (TopOpen ∘ 𝑅) Fn 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))}))
184, 15, 17syl2anc 584 . . . . . . . . 9 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))}))
1918unieqd 4865 . . . . . . . 8 (𝜑 (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))}))
20 fvco2 6915 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 Fn 𝐼𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
213, 20sylan 580 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
22 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
23 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (TopSet‘(𝑅𝑦)) = (TopSet‘(𝑅𝑦))
2422, 23topnval 17234 . . . . . . . . . . . . . . . . . . . . 21 ((TopSet‘(𝑅𝑦)) ↾t (Base‘(𝑅𝑦))) = (TopOpen‘(𝑅𝑦))
25 restsspw 17231 . . . . . . . . . . . . . . . . . . . . 21 ((TopSet‘(𝑅𝑦)) ↾t (Base‘(𝑅𝑦))) ⊆ 𝒫 (Base‘(𝑅𝑦))
2624, 25eqsstrri 3966 . . . . . . . . . . . . . . . . . . . 20 (TopOpen‘(𝑅𝑦)) ⊆ 𝒫 (Base‘(𝑅𝑦))
2721, 26eqsstrdi 3985 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ⊆ 𝒫 (Base‘(𝑅𝑦)))
2827sseld 3930 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝐼) → ((𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔𝑦) ∈ 𝒫 (Base‘(𝑅𝑦))))
29 fvex 6832 . . . . . . . . . . . . . . . . . . 19 (𝑔𝑦) ∈ V
3029elpw 4550 . . . . . . . . . . . . . . . . . 18 ((𝑔𝑦) ∈ 𝒫 (Base‘(𝑅𝑦)) ↔ (𝑔𝑦) ⊆ (Base‘(𝑅𝑦)))
3128, 30syl6ib 250 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦𝐼) → ((𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔𝑦) ⊆ (Base‘(𝑅𝑦))))
3231ralimdva 3160 . . . . . . . . . . . . . . . 16 (𝜑 → (∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → ∀𝑦𝐼 (𝑔𝑦) ⊆ (Base‘(𝑅𝑦))))
33 simpl2 1191 . . . . . . . . . . . . . . . 16 (((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦)) → ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦))
3432, 33impel 506 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))) → ∀𝑦𝐼 (𝑔𝑦) ⊆ (Base‘(𝑅𝑦)))
35 ss2ixp 8761 . . . . . . . . . . . . . . 15 (∀𝑦𝐼 (𝑔𝑦) ⊆ (Base‘(𝑅𝑦)) → X𝑦𝐼 (𝑔𝑦) ⊆ X𝑦𝐼 (Base‘(𝑅𝑦)))
3634, 35syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))) → X𝑦𝐼 (𝑔𝑦) ⊆ X𝑦𝐼 (Base‘(𝑅𝑦)))
37 simprr 770 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))) → 𝑥 = X𝑦𝐼 (𝑔𝑦))
381, 7, 2, 4, 3prdsbas2 17269 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘𝑌) = X𝑦𝐼 (Base‘(𝑅𝑦)))
3938adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))) → (Base‘𝑌) = X𝑦𝐼 (Base‘(𝑅𝑦)))
4036, 37, 393sstr4d 3978 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))) → 𝑥 ⊆ (Base‘𝑌))
4140ex 413 . . . . . . . . . . . 12 (𝜑 → (((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦)) → 𝑥 ⊆ (Base‘𝑌)))
4241exlimdv 1935 . . . . . . . . . . 11 (𝜑 → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦)) → 𝑥 ⊆ (Base‘𝑌)))
43 velpw 4551 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 (Base‘𝑌) ↔ 𝑥 ⊆ (Base‘𝑌))
4442, 43syl6ibr 251 . . . . . . . . . 10 (𝜑 → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦)) → 𝑥 ∈ 𝒫 (Base‘𝑌)))
4544abssdv 4012 . . . . . . . . 9 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ⊆ 𝒫 (Base‘𝑌))
46 fvex 6832 . . . . . . . . . . 11 (Base‘𝑌) ∈ V
4746pwex 5320 . . . . . . . . . 10 𝒫 (Base‘𝑌) ∈ V
4847ssex 5262 . . . . . . . . 9 ({𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ⊆ 𝒫 (Base‘𝑌) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ∈ V)
49 unitg 22215 . . . . . . . . 9 ({𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ∈ V → (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))}) = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))})
5045, 48, 493syl 18 . . . . . . . 8 (𝜑 (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))}) = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))})
5119, 50eqtrd 2776 . . . . . . 7 (𝜑 (∏t‘(TopOpen ∘ 𝑅)) = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))})
52 sspwuni 5044 . . . . . . . 8 ({𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ⊆ 𝒫 (Base‘𝑌) ↔ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ⊆ (Base‘𝑌))
5345, 52sylib 217 . . . . . . 7 (𝜑 {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))} ⊆ (Base‘𝑌))
5451, 53eqsstrd 3969 . . . . . 6 (𝜑 (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
55 sspwuni 5044 . . . . . 6 ((∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌) ↔ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
5654, 55sylibr 233 . . . . 5 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌))
5710, 56eqsstrd 3969 . . . 4 (𝜑 → (TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌))
587, 9topnid 17235 . . . 4 ((TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌) → (TopSet‘𝑌) = (TopOpen‘𝑌))
5957, 58syl 17 . . 3 (𝜑 → (TopSet‘𝑌) = (TopOpen‘𝑌))
60 prdstopn.o . . 3 𝑂 = (TopOpen‘𝑌)
6159, 60eqtr4di 2794 . 2 (𝜑 → (TopSet‘𝑌) = 𝑂)
6261, 10eqtr3d 2778 1 (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1540  wex 1780  wcel 2105  {cab 2713  wral 3061  wrex 3070  Vcvv 3441  cdif 3894  wss 3897  𝒫 cpw 4546   cuni 4851  dom cdm 5614  ccom 5618   Fn wfn 6468  wf 6469  cfv 6473  (class class class)co 7329  Xcixp 8748  Fincfn 8796  Basecbs 17001  TopSetcts 17057  t crest 17220  TopOpenctopn 17221  topGenctg 17237  tcpt 17238  Xscprds 17245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-1st 7891  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-1o 8359  df-er 8561  df-map 8680  df-ixp 8749  df-en 8797  df-dom 8798  df-sdom 8799  df-fin 8800  df-sup 9291  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-2 12129  df-3 12130  df-4 12131  df-5 12132  df-6 12133  df-7 12134  df-8 12135  df-9 12136  df-n0 12327  df-z 12413  df-dec 12531  df-uz 12676  df-fz 13333  df-struct 16937  df-slot 16972  df-ndx 16984  df-base 17002  df-plusg 17064  df-mulr 17065  df-sca 17067  df-vsca 17068  df-ip 17069  df-tset 17070  df-ple 17071  df-ds 17073  df-hom 17075  df-cco 17076  df-rest 17222  df-topn 17223  df-topgen 17243  df-pt 17244  df-prds 17247
This theorem is referenced by:  xpstopnlem2  23060  prdstmdd  23373  prdstgpd  23374  prdsxmslem2  23783
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