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Theorem prdstopn 23652
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 7237 . . . . . . 7 ((𝑅 Fn 𝐼𝐼𝑊) → 𝑅 ∈ V)
63, 4, 5syl2anc 584 . . . . . 6 (𝜑𝑅 ∈ V)
7 eqid 2735 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
8 eqidd 2736 . . . . . 6 (𝜑 → dom 𝑅 = dom 𝑅)
9 eqid 2735 . . . . . 6 (TopSet‘𝑌) = (TopSet‘𝑌)
101, 2, 6, 7, 8, 9prdstset 17513 . . . . 5 (𝜑 → (TopSet‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
11 topnfn 17472 . . . . . . . . . . 11 TopOpen Fn V
12 dffn2 6739 . . . . . . . . . . . 12 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
133, 12sylib 218 . . . . . . . . . . 11 (𝜑𝑅:𝐼⟶V)
14 fnfco 6774 . . . . . . . . . . 11 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
1511, 13, 14sylancr 587 . . . . . . . . . 10 (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
16 eqid 2735 . . . . . . . . . . 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 23594 . . . . . . . . . 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 4925 . . . . . . . 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 7006 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 Fn 𝐼𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
213, 20sylan 580 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
22 eqid 2735 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
23 eqid 2735 . . . . . . . . . . . . . . . . . . . . . 22 (TopSet‘(𝑅𝑦)) = (TopSet‘(𝑅𝑦))
2422, 23topnval 17481 . . . . . . . . . . . . . . . . . . . . 21 ((TopSet‘(𝑅𝑦)) ↾t (Base‘(𝑅𝑦))) = (TopOpen‘(𝑅𝑦))
25 restsspw 17478 . . . . . . . . . . . . . . . . . . . . 21 ((TopSet‘(𝑅𝑦)) ↾t (Base‘(𝑅𝑦))) ⊆ 𝒫 (Base‘(𝑅𝑦))
2624, 25eqsstrri 4031 . . . . . . . . . . . . . . . . . . . 20 (TopOpen‘(𝑅𝑦)) ⊆ 𝒫 (Base‘(𝑅𝑦))
2721, 26eqsstrdi 4050 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ⊆ 𝒫 (Base‘(𝑅𝑦)))
2827sseld 3994 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝐼) → ((𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔𝑦) ∈ 𝒫 (Base‘(𝑅𝑦))))
29 fvex 6920 . . . . . . . . . . . . . . . . . . 19 (𝑔𝑦) ∈ V
3029elpw 4609 . . . . . . . . . . . . . . . . . 18 ((𝑔𝑦) ∈ 𝒫 (Base‘(𝑅𝑦)) ↔ (𝑔𝑦) ⊆ (Base‘(𝑅𝑦)))
3128, 30imbitrdi 251 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦𝐼) → ((𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔𝑦) ⊆ (Base‘(𝑅𝑦))))
3231ralimdva 3165 . . . . . . . . . . . . . . . 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 505 . . . . . . . . . . . . . . 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 8949 . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . 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 17516 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘𝑌) = X𝑦𝐼 (Base‘(𝑅𝑦)))
3938adantr 480 . . . . . . . . . . . . . 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 4043 . . . . . . . . . . . . 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 412 . . . . . . . . . . . 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 1931 . . . . . . . . . . 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 4610 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 (Base‘𝑌) ↔ 𝑥 ⊆ (Base‘𝑌))
4442, 43imbitrrdi 252 . . . . . . . . . 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 4078 . . . . . . . . 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 6920 . . . . . . . . . . 11 (Base‘𝑌) ∈ V
4746pwex 5386 . . . . . . . . . 10 𝒫 (Base‘𝑌) ∈ V
4847ssex 5327 . . . . . . . . 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 22990 . . . . . . . . 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 2775 . . . . . . 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 5105 . . . . . . . 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 218 . . . . . . 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 4034 . . . . . 6 (𝜑 (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
55 sspwuni 5105 . . . . . 6 ((∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌) ↔ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
5654, 55sylibr 234 . . . . 5 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌))
5710, 56eqsstrd 4034 . . . 4 (𝜑 → (TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌))
587, 9topnid 17482 . . . 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 2793 . 2 (𝜑 → (TopSet‘𝑌) = 𝑂)
6261, 10eqtr3d 2777 1 (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  wss 3963  𝒫 cpw 4605   cuni 4912  dom cdm 5689  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Xcixp 8936  Fincfn 8984  Basecbs 17245  TopSetcts 17304  t crest 17467  TopOpenctopn 17468  topGenctg 17484  tcpt 17485  Xscprds 17492
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-topgen 17490  df-pt 17491  df-prds 17494
This theorem is referenced by:  xpstopnlem2  23835  prdstmdd  24148  prdstgpd  24149  prdsxmslem2  24558
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