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Theorem bastop1 22131
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 "(topGen‘𝐵) = 𝐽 " to express "𝐵 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 ∧ 𝐵𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐽,𝑦

Proof of Theorem bastop1
StepHypRef Expression
1 tgss 22106 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (topGen‘𝐵) ⊆ (topGen‘𝐽))
2 tgtop 22111 . . . . . 6 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
32adantr 481 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (topGen‘𝐽) = 𝐽)
41, 3sseqtrd 3961 . . . 4 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (topGen‘𝐵) ⊆ 𝐽)
5 eqss 3936 . . . . 5 ((topGen‘𝐵) = 𝐽 ↔ ((topGen‘𝐵) ⊆ 𝐽𝐽 ⊆ (topGen‘𝐵)))
65baib 536 . . . 4 ((topGen‘𝐵) ⊆ 𝐽 → ((topGen‘𝐵) = 𝐽𝐽 ⊆ (topGen‘𝐵)))
74, 6syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → ((topGen‘𝐵) = 𝐽𝐽 ⊆ (topGen‘𝐵)))
8 dfss3 3909 . . 3 (𝐽 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐽 𝑥 ∈ (topGen‘𝐵))
97, 8bitrdi 287 . 2 ((𝐽 ∈ Top ∧ 𝐵𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥𝐽 𝑥 ∈ (topGen‘𝐵)))
10 ssexg 5246 . . . . 5 ((𝐵𝐽𝐽 ∈ Top) → 𝐵 ∈ V)
1110ancoms 459 . . . 4 ((𝐽 ∈ Top ∧ 𝐵𝐽) → 𝐵 ∈ V)
12 eltg3 22100 . . . 4 (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
1311, 12syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
1413ralbidv 3108 . 2 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (∀𝑥𝐽 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦)))
159, 14bitrd 278 1 ((𝐽 ∈ Top ∧ 𝐵𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wex 1782  wcel 2106  wral 3064  Vcvv 3430  wss 3887   cuni 4840  cfv 6427  topGenctg 17136  Topctop 22030
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-iota 6385  df-fun 6429  df-fv 6435  df-topgen 17142  df-top 22031
This theorem is referenced by:  bastop2  22132
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