MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prdstopn Structured version   Visualization version   GIF version

Theorem prdstopn 23637
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 7238 . . . . . . 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 17512 . . . . 5 (𝜑 → (TopSet‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
11 topnfn 17471 . . . . . . . . . . 11 TopOpen Fn V
12 dffn2 6737 . . . . . . . . . . . 12 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
133, 12sylib 218 . . . . . . . . . . 11 (𝜑𝑅:𝐼⟶V)
14 fnfco 6772 . . . . . . . . . . 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 23579 . . . . . . . . . 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 4919 . . . . . . . 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 7005 . . . . . . . . . . . . . . . . . . . . 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 17480 . . . . . . . . . . . . . . . . . . . . 21 ((TopSet‘(𝑅𝑦)) ↾t (Base‘(𝑅𝑦))) = (TopOpen‘(𝑅𝑦))
25 restsspw 17477 . . . . . . . . . . . . . . . . . . . . 21 ((TopSet‘(𝑅𝑦)) ↾t (Base‘(𝑅𝑦))) ⊆ 𝒫 (Base‘(𝑅𝑦))
2624, 25eqsstrri 4030 . . . . . . . . . . . . . . . . . . . 20 (TopOpen‘(𝑅𝑦)) ⊆ 𝒫 (Base‘(𝑅𝑦))
2721, 26eqsstrdi 4027 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ⊆ 𝒫 (Base‘(𝑅𝑦)))
2827sseld 3981 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝐼) → ((𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔𝑦) ∈ 𝒫 (Base‘(𝑅𝑦))))
29 fvex 6918 . . . . . . . . . . . . . . . . . . 19 (𝑔𝑦) ∈ V
3029elpw 4603 . . . . . . . . . . . . . . . . . 18 ((𝑔𝑦) ∈ 𝒫 (Base‘(𝑅𝑦)) ↔ (𝑔𝑦) ⊆ (Base‘(𝑅𝑦)))
3128, 30imbitrdi 251 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦𝐼) → ((𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔𝑦) ⊆ (Base‘(𝑅𝑦))))
3231ralimdva 3166 . . . . . . . . . . . . . . . 16 (𝜑 → (∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → ∀𝑦𝐼 (𝑔𝑦) ⊆ (Base‘(𝑅𝑦))))
33 simpl2 1192 . . . . . . . . . . . . . . . 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 8951 . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . 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 17515 . . . . . . . . . . . . . . 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 4038 . . . . . . . . . . . . 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 1932 . . . . . . . . . . 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 4604 . . . . . . . . . . 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 4067 . . . . . . . . 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 6918 . . . . . . . . . . 11 (Base‘𝑌) ∈ V
4746pwex 5379 . . . . . . . . . 10 𝒫 (Base‘𝑌) ∈ V
4847ssex 5320 . . . . . . . . 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 22975 . . . . . . . . 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 5099 . . . . . . . 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 4017 . . . . . 6 (𝜑 (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
55 sspwuni 5099 . . . . . 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 4017 . . . 4 (𝜑 → (TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌))
587, 9topnid 17481 . . . 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 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wral 3060  wrex 3069  Vcvv 3479  cdif 3947  wss 3950  𝒫 cpw 4599   cuni 4906  dom cdm 5684  ccom 5688   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  Xcixp 8938  Fincfn 8986  Basecbs 17248  TopSetcts 17304  t crest 17466  TopOpenctopn 17467  topGenctg 17483  tcpt 17484  Xscprds 17491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17249  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 17468  df-topn 17469  df-topgen 17489  df-pt 17490  df-prds 17493
This theorem is referenced by:  xpstopnlem2  23820  prdstmdd  24133  prdstgpd  24134  prdsxmslem2  24543
  Copyright terms: Public domain W3C validator