ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  bastop1 GIF version

Theorem bastop1 12089
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 12069 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (topGen‘𝐵) ⊆ (topGen‘𝐽))
2 tgtop 12074 . . . . . 6 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
32adantr 272 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (topGen‘𝐽) = 𝐽)
41, 3sseqtrd 3099 . . . 4 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (topGen‘𝐵) ⊆ 𝐽)
5 eqss 3076 . . . . 5 ((topGen‘𝐵) = 𝐽 ↔ ((topGen‘𝐵) ⊆ 𝐽𝐽 ⊆ (topGen‘𝐵)))
65baib 885 . . . 4 ((topGen‘𝐵) ⊆ 𝐽 → ((topGen‘𝐵) = 𝐽𝐽 ⊆ (topGen‘𝐵)))
74, 6syl 14 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → ((topGen‘𝐵) = 𝐽𝐽 ⊆ (topGen‘𝐵)))
8 dfss3 3051 . . 3 (𝐽 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐽 𝑥 ∈ (topGen‘𝐵))
97, 8syl6bb 195 . 2 ((𝐽 ∈ Top ∧ 𝐵𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥𝐽 𝑥 ∈ (topGen‘𝐵)))
10 ssexg 4025 . . . . 5 ((𝐵𝐽𝐽 ∈ Top) → 𝐵 ∈ V)
1110ancoms 266 . . . 4 ((𝐽 ∈ Top ∧ 𝐵𝐽) → 𝐵 ∈ V)
12 eltg3 12063 . . . 4 (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
1311, 12syl 14 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
1413ralbidv 2409 . 2 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (∀𝑥𝐽 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦)))
159, 14bitrd 187 1 ((𝐽 ∈ Top ∧ 𝐵𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1312  wex 1449  wcel 1461  wral 2388  Vcvv 2655  wss 3035   cuni 3700  cfv 5079  topGenctg 11972  Topctop 12001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-topgen 11978  df-top 12002
This theorem is referenced by:  bastop2  12090
  Copyright terms: Public domain W3C validator