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Theorem bastop1 22496
Description: A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom "(topGenβ€˜π΅) = 𝐽 " to express "𝐡 is a basis for topology 𝐽 " since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
bastop1 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ ((topGenβ€˜π΅) = 𝐽 ↔ βˆ€π‘₯ ∈ 𝐽 βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐽,𝑦
Proof of Theorem bastop1
StepHypRef Expression
1 tgss 22471 . . . . 5 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ (topGenβ€˜π΅) βŠ† (topGenβ€˜π½))
2 tgtop 22476 . . . . . 6 (𝐽 ∈ Top β†’ (topGenβ€˜π½) = 𝐽)
32adantr 482 . . . . 5 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ (topGenβ€˜π½) = 𝐽)
41, 3sseqtrd 4023 . . . 4 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ (topGenβ€˜π΅) βŠ† 𝐽)
5 eqss 3998 . . . . 5 ((topGenβ€˜π΅) = 𝐽 ↔ ((topGenβ€˜π΅) βŠ† 𝐽 ∧ 𝐽 βŠ† (topGenβ€˜π΅)))
65baib 537 . . . 4 ((topGenβ€˜π΅) βŠ† 𝐽 β†’ ((topGenβ€˜π΅) = 𝐽 ↔ 𝐽 βŠ† (topGenβ€˜π΅)))
74, 6syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ ((topGenβ€˜π΅) = 𝐽 ↔ 𝐽 βŠ† (topGenβ€˜π΅)))
8 dfss3 3971 . . 3 (𝐽 βŠ† (topGenβ€˜π΅) ↔ βˆ€π‘₯ ∈ 𝐽 π‘₯ ∈ (topGenβ€˜π΅))
97, 8bitrdi 287 . 2 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ ((topGenβ€˜π΅) = 𝐽 ↔ βˆ€π‘₯ ∈ 𝐽 π‘₯ ∈ (topGenβ€˜π΅)))
10 ssexg 5324 . . . . 5 ((𝐡 βŠ† 𝐽 ∧ 𝐽 ∈ Top) β†’ 𝐡 ∈ V)
1110ancoms 460 . . . 4 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ 𝐡 ∈ V)
12 eltg3 22465 . . . 4 (𝐡 ∈ V β†’ (π‘₯ ∈ (topGenβ€˜π΅) ↔ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
1311, 12syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ (π‘₯ ∈ (topGenβ€˜π΅) ↔ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
1413ralbidv 3178 . 2 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ (βˆ€π‘₯ ∈ 𝐽 π‘₯ ∈ (topGenβ€˜π΅) ↔ βˆ€π‘₯ ∈ 𝐽 βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
159, 14bitrd 279 1 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ ((topGenβ€˜π΅) = 𝐽 ↔ βˆ€π‘₯ ∈ 𝐽 βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475   βŠ† wss 3949  βˆͺ cuni 4909  β€˜cfv 6544  topGenctg 17383  Topctop 22395
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-topgen 17389  df-top 22396
This theorem is referenced by:  bastop2  22497
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