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Theorem basgen2 22842
Description: Given a topology 𝐽, show that a subset 𝐡 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 ∧ 𝐡 βŠ† 𝐽 ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝐡 (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† π‘₯)) β†’ (topGenβ€˜π΅) = 𝐽)
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐡   π‘₯,𝐽,𝑦,𝑧
Proof of Theorem basgen2
StepHypRef Expression
1 dfss3 3965 . . . 4 (𝐽 βŠ† (topGenβ€˜π΅) ↔ βˆ€π‘₯ ∈ 𝐽 π‘₯ ∈ (topGenβ€˜π΅))
2 ssexg 5316 . . . . . . 7 ((𝐡 βŠ† 𝐽 ∧ 𝐽 ∈ Top) β†’ 𝐡 ∈ V)
32ancoms 458 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ 𝐡 ∈ V)
4 eltg2b 22812 . . . . . 6 (𝐡 ∈ V β†’ (π‘₯ ∈ (topGenβ€˜π΅) ↔ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝐡 (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† π‘₯)))
53, 4syl 17 . . . . 5 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ (π‘₯ ∈ (topGenβ€˜π΅) ↔ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝐡 (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† π‘₯)))
65ralbidv 3171 . . . 4 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ (βˆ€π‘₯ ∈ 𝐽 π‘₯ ∈ (topGenβ€˜π΅) ↔ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝐡 (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† π‘₯)))
71, 6bitrid 283 . . 3 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ (𝐽 βŠ† (topGenβ€˜π΅) ↔ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝐡 (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† π‘₯)))
87biimp3ar 1466 . 2 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽 ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝐡 (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† π‘₯)) β†’ 𝐽 βŠ† (topGenβ€˜π΅))
9 basgen 22841 . 2 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽 ∧ 𝐽 βŠ† (topGenβ€˜π΅)) β†’ (topGenβ€˜π΅) = 𝐽)
108, 9syld3an3 1406 1 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽 ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝐡 (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† π‘₯)) β†’ (topGenβ€˜π΅) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064  Vcvv 3468   βŠ† wss 3943  β€˜cfv 6536  topGenctg 17389  Topctop 22745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-topgen 17395  df-top 22746
This theorem is referenced by:  pptbas  22861  2ndcctbss  23309  2ndcomap  23312  dis2ndc  23314  met2ndci  24381
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