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Theorem topbas 22862
Description: A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
topbas (𝐽 ∈ Top → 𝐽 ∈ TopBases)

Proof of Theorem topbas
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inopn 22788 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑦𝐽) → (𝑥𝑦) ∈ 𝐽)
213expb 1118 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) → (𝑥𝑦) ∈ 𝐽)
3 simpr 484 . . . . . . 7 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → 𝑧 ∈ (𝑥𝑦))
4 ssid 4000 . . . . . . 7 (𝑥𝑦) ⊆ (𝑥𝑦)
53, 4jctir 520 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
6 eleq2 2817 . . . . . . . 8 (𝑤 = (𝑥𝑦) → (𝑧𝑤𝑧 ∈ (𝑥𝑦)))
7 sseq1 4003 . . . . . . . 8 (𝑤 = (𝑥𝑦) → (𝑤 ⊆ (𝑥𝑦) ↔ (𝑥𝑦) ⊆ (𝑥𝑦)))
86, 7anbi12d 630 . . . . . . 7 (𝑤 = (𝑥𝑦) → ((𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))))
98rspcev 3607 . . . . . 6 (((𝑥𝑦) ∈ 𝐽 ∧ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
102, 5, 9syl2an2r 684 . . . . 5 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
1110exp31 419 . . . 4 (𝐽 ∈ Top → ((𝑥𝐽𝑦𝐽) → (𝑧 ∈ (𝑥𝑦) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))))
1211ralrimdv 3147 . . 3 (𝐽 ∈ Top → ((𝑥𝐽𝑦𝐽) → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1312ralrimivv 3193 . 2 (𝐽 ∈ Top → ∀𝑥𝐽𝑦𝐽𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
14 isbasis2g 22838 . 2 (𝐽 ∈ Top → (𝐽 ∈ TopBases ↔ ∀𝑥𝐽𝑦𝐽𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1513, 14mpbird 257 1 (𝐽 ∈ Top → 𝐽 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3056  wrex 3065  cin 3943  wss 3944  Topctop 22782  TopBasesctb 22835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-in 3951  df-ss 3961  df-pw 4600  df-uni 4904  df-top 22783  df-bases 22836
This theorem is referenced by:  resttop  23051  dis1stc  23390  txtop  23460  onpsstopbas  35850
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