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Theorem basgen2 21525
Description: Given a topology 𝐽, show that a subset 𝐵 satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
basgen2 ((𝐽 ∈ Top ∧ 𝐵𝐽 ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)) → (topGen‘𝐵) = 𝐽)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐽,𝑦,𝑧

Proof of Theorem basgen2
StepHypRef Expression
1 dfss3 3953 . . . 4 (𝐽 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐽 𝑥 ∈ (topGen‘𝐵))
2 ssexg 5218 . . . . . . 7 ((𝐵𝐽𝐽 ∈ Top) → 𝐵 ∈ V)
32ancoms 459 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐵𝐽) → 𝐵 ∈ V)
4 eltg2b 21495 . . . . . 6 (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
53, 4syl 17 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
65ralbidv 3194 . . . 4 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (∀𝑥𝐽 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐽𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
71, 6syl5bb 284 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝐽 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐽𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
87biimp3ar 1461 . 2 ((𝐽 ∈ Top ∧ 𝐵𝐽 ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)) → 𝐽 ⊆ (topGen‘𝐵))
9 basgen 21524 . 2 ((𝐽 ∈ Top ∧ 𝐵𝐽𝐽 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = 𝐽)
108, 9syld3an3 1401 1 ((𝐽 ∈ Top ∧ 𝐵𝐽 ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)) → (topGen‘𝐵) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  wss 3933  cfv 6348  topGenctg 16699  Topctop 21429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-topgen 16705  df-top 21430
This theorem is referenced by:  pptbas  21544  2ndcctbss  21991  2ndcomap  21994  dis2ndc  21996  met2ndci  23059
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