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Theorem prdstopn 21354
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 6441 . . . . . . 7 ((𝑅 Fn 𝐼𝐼𝑊) → 𝑅 ∈ V)
63, 4, 5syl2anc 692 . . . . . 6 (𝜑𝑅 ∈ V)
7 eqid 2621 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
8 eqidd 2622 . . . . . 6 (𝜑 → dom 𝑅 = dom 𝑅)
9 eqid 2621 . . . . . 6 (TopSet‘𝑌) = (TopSet‘𝑌)
101, 2, 6, 7, 8, 9prdstset 16058 . . . . 5 (𝜑 → (TopSet‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
11 topnfn 16018 . . . . . . . . . . 11 TopOpen Fn V
12 dffn2 6009 . . . . . . . . . . . 12 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
133, 12sylib 208 . . . . . . . . . . 11 (𝜑𝑅:𝐼⟶V)
14 fnfco 6031 . . . . . . . . . . 11 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
1511, 13, 14sylancr 694 . . . . . . . . . 10 (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
16 eqid 2621 . . . . . . . . . . 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 21296 . . . . . . . . . 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 692 . . . . . . . . 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 4417 . . . . . . . 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 simpl2 1063 . . . . . . . . . . . . . . . 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))
21 fvco2 6235 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 Fn 𝐼𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
223, 21sylan 488 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
23 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
24 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . 23 (TopSet‘(𝑅𝑦)) = (TopSet‘(𝑅𝑦))
2523, 24topnval 16027 . . . . . . . . . . . . . . . . . . . . . 22 ((TopSet‘(𝑅𝑦)) ↾t (Base‘(𝑅𝑦))) = (TopOpen‘(𝑅𝑦))
26 restsspw 16024 . . . . . . . . . . . . . . . . . . . . . 22 ((TopSet‘(𝑅𝑦)) ↾t (Base‘(𝑅𝑦))) ⊆ 𝒫 (Base‘(𝑅𝑦))
2725, 26eqsstr3i 3620 . . . . . . . . . . . . . . . . . . . . 21 (TopOpen‘(𝑅𝑦)) ⊆ 𝒫 (Base‘(𝑅𝑦))
2822, 27syl6eqss 3639 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ⊆ 𝒫 (Base‘(𝑅𝑦)))
2928sseld 3586 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦𝐼) → ((𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔𝑦) ∈ 𝒫 (Base‘(𝑅𝑦))))
30 fvex 6163 . . . . . . . . . . . . . . . . . . . 20 (𝑔𝑦) ∈ V
3130elpw 4141 . . . . . . . . . . . . . . . . . . 19 ((𝑔𝑦) ∈ 𝒫 (Base‘(𝑅𝑦)) ↔ (𝑔𝑦) ⊆ (Base‘(𝑅𝑦)))
3229, 31syl6ib 241 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝐼) → ((𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → (𝑔𝑦) ⊆ (Base‘(𝑅𝑦))))
3332ralimdva 2957 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) → ∀𝑦𝐼 (𝑔𝑦) ⊆ (Base‘(𝑅𝑦))))
3433imp 445 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦)) → ∀𝑦𝐼 (𝑔𝑦) ⊆ (Base‘(𝑅𝑦)))
3520, 34sylan2 491 . . . . . . . . . . . . . . 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)))
36 ss2ixp 7873 . . . . . . . . . . . . . . 15 (∀𝑦𝐼 (𝑔𝑦) ⊆ (Base‘(𝑅𝑦)) → X𝑦𝐼 (𝑔𝑦) ⊆ X𝑦𝐼 (Base‘(𝑅𝑦)))
3735, 36syl 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)))
38 simprr 795 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))) → 𝑥 = X𝑦𝐼 (𝑔𝑦))
391, 7, 2, 4, 3prdsbas2 16061 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘𝑌) = X𝑦𝐼 (Base‘(𝑅𝑦)))
4039adantr 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)))
4137, 38, 403sstr4d 3632 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦))) → 𝑥 ⊆ (Base‘𝑌))
4241ex 450 . . . . . . . . . . . 12 (𝜑 → (((𝑔 Fn 𝐼 ∧ ∀𝑦𝐼 (𝑔𝑦) ∈ ((TopOpen ∘ 𝑅)‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐼𝑧)(𝑔𝑦) = ((TopOpen ∘ 𝑅)‘𝑦)) ∧ 𝑥 = X𝑦𝐼 (𝑔𝑦)) → 𝑥 ⊆ (Base‘𝑌)))
4342exlimdv 1858 . . . . . . . . . . 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)))
44 selpw 4142 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 (Base‘𝑌) ↔ 𝑥 ⊆ (Base‘𝑌))
4543, 44syl6ibr 242 . . . . . . . . . 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)))
4645abssdv 3660 . . . . . . . . 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))
47 fvex 6163 . . . . . . . . . . 11 (Base‘𝑌) ∈ V
4847pwex 4813 . . . . . . . . . 10 𝒫 (Base‘𝑌) ∈ V
4948ssex 4767 . . . . . . . . 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)
50 unitg 20695 . . . . . . . . 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))})
5146, 49, 503syl 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))})
5219, 51eqtrd 2655 . . . . . . 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))})
53 sspwuni 4582 . . . . . . . 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))
5446, 53sylib 208 . . . . . . 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))
5552, 54eqsstrd 3623 . . . . . 6 (𝜑 (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
56 sspwuni 4582 . . . . . 6 ((∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌) ↔ (∏t‘(TopOpen ∘ 𝑅)) ⊆ (Base‘𝑌))
5755, 56sylibr 224 . . . . 5 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) ⊆ 𝒫 (Base‘𝑌))
5810, 57eqsstrd 3623 . . . 4 (𝜑 → (TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌))
597, 9topnid 16028 . . . 4 ((TopSet‘𝑌) ⊆ 𝒫 (Base‘𝑌) → (TopSet‘𝑌) = (TopOpen‘𝑌))
6058, 59syl 17 . . 3 (𝜑 → (TopSet‘𝑌) = (TopOpen‘𝑌))
61 prdstopn.o . . 3 𝑂 = (TopOpen‘𝑌)
6260, 61syl6eqr 2673 . 2 (𝜑 → (TopSet‘𝑌) = 𝑂)
6362, 10eqtr3d 2657 1 (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3189  cdif 3556  wss 3559  𝒫 cpw 4135   cuni 4407  dom cdm 5079  ccom 5083   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  Xcixp 7860  Fincfn 7907  Basecbs 15792  TopSetcts 15879  t crest 16013  TopOpenctopn 16014  topGenctg 16030  tcpt 16031  Xscprds 16038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-fz 12277  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-plusg 15886  df-mulr 15887  df-sca 15889  df-vsca 15890  df-ip 15891  df-tset 15892  df-ple 15893  df-ds 15896  df-hom 15898  df-cco 15899  df-rest 16015  df-topn 16016  df-topgen 16036  df-pt 16037  df-prds 16040
This theorem is referenced by:  xpstopnlem2  21537  prdstmdd  21850  prdstgpd  21851  prdsxmslem2  22257
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