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Theorem basgen2 22355
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 3933 . . . 4 (𝐽 βŠ† (topGenβ€˜π΅) ↔ βˆ€π‘₯ ∈ 𝐽 π‘₯ ∈ (topGenβ€˜π΅))
2 ssexg 5281 . . . . . . 7 ((𝐡 βŠ† 𝐽 ∧ 𝐽 ∈ Top) β†’ 𝐡 ∈ V)
32ancoms 460 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ 𝐡 ∈ V)
4 eltg2b 22325 . . . . . 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 1471 . 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 22354 . 2 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽 ∧ 𝐽 βŠ† (topGenβ€˜π΅)) β†’ (topGenβ€˜π΅) = 𝐽)
108, 9syld3an3 1410 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3444   βŠ† wss 3911  β€˜cfv 6497  topGenctg 17324  Topctop 22258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-topgen 17330  df-top 22259
This theorem is referenced by:  pptbas  22374  2ndcctbss  22822  2ndcomap  22825  dis2ndc  22827  met2ndci  23894
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