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Theorem basgen2 21600
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 3959 . . . 4 (𝐽 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐽 𝑥 ∈ (topGen‘𝐵))
2 ssexg 5230 . . . . . . 7 ((𝐵𝐽𝐽 ∈ Top) → 𝐵 ∈ V)
32ancoms 461 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐵𝐽) → 𝐵 ∈ V)
4 eltg2b 21570 . . . . . 6 (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
53, 4syl 17 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
65ralbidv 3200 . . . 4 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (∀𝑥𝐽 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐽𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
71, 6syl5bb 285 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝐽 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐽𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)))
87biimp3ar 1466 . 2 ((𝐽 ∈ Top ∧ 𝐵𝐽 ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)) → 𝐽 ⊆ (topGen‘𝐵))
9 basgen 21599 . 2 ((𝐽 ∈ Top ∧ 𝐵𝐽𝐽 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = 𝐽)
108, 9syld3an3 1405 1 ((𝐽 ∈ Top ∧ 𝐵𝐽 ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥)) → (topGen‘𝐵) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1536  wcel 2113  wral 3141  wrex 3142  Vcvv 3497  wss 3939  cfv 6358  topGenctg 16714  Topctop 21504
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 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fv 6366  df-topgen 16720  df-top 21505
This theorem is referenced by:  pptbas  21619  2ndcctbss  22066  2ndcomap  22069  dis2ndc  22071  met2ndci  23135
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