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Theorem topbas 22695
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 22621 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑦𝐽) → (𝑥𝑦) ∈ 𝐽)
213expb 1118 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) → (𝑥𝑦) ∈ 𝐽)
3 simpr 483 . . . . . . 7 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → 𝑧 ∈ (𝑥𝑦))
4 ssid 4003 . . . . . . 7 (𝑥𝑦) ⊆ (𝑥𝑦)
53, 4jctir 519 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
6 eleq2 2820 . . . . . . . 8 (𝑤 = (𝑥𝑦) → (𝑧𝑤𝑧 ∈ (𝑥𝑦)))
7 sseq1 4006 . . . . . . . 8 (𝑤 = (𝑥𝑦) → (𝑤 ⊆ (𝑥𝑦) ↔ (𝑥𝑦) ⊆ (𝑥𝑦)))
86, 7anbi12d 629 . . . . . . 7 (𝑤 = (𝑥𝑦) → ((𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))))
98rspcev 3611 . . . . . 6 (((𝑥𝑦) ∈ 𝐽 ∧ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
102, 5, 9syl2an2r 681 . . . . 5 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
1110exp31 418 . . . 4 (𝐽 ∈ Top → ((𝑥𝐽𝑦𝐽) → (𝑧 ∈ (𝑥𝑦) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))))
1211ralrimdv 3150 . . 3 (𝐽 ∈ Top → ((𝑥𝐽𝑦𝐽) → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1312ralrimivv 3196 . 2 (𝐽 ∈ Top → ∀𝑥𝐽𝑦𝐽𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
14 isbasis2g 22671 . 2 (𝐽 ∈ Top → (𝐽 ∈ TopBases ↔ ∀𝑥𝐽𝑦𝐽𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1513, 14mpbird 256 1 (𝐽 ∈ Top → 𝐽 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  wral 3059  wrex 3068  cin 3946  wss 3947  Topctop 22615  TopBasesctb 22668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1087  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-in 3954  df-ss 3964  df-pw 4603  df-uni 4908  df-top 22616  df-bases 22669
This theorem is referenced by:  resttop  22884  dis1stc  23223  txtop  23293  onpsstopbas  35618
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