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Theorem bastop1 21018
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 20993 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (topGen‘𝐵) ⊆ (topGen‘𝐽))
2 tgtop 20998 . . . . . 6 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
32adantr 466 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (topGen‘𝐽) = 𝐽)
41, 3sseqtrd 3790 . . . 4 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (topGen‘𝐵) ⊆ 𝐽)
5 eqss 3767 . . . . 5 ((topGen‘𝐵) = 𝐽 ↔ ((topGen‘𝐵) ⊆ 𝐽𝐽 ⊆ (topGen‘𝐵)))
65baib 525 . . . 4 ((topGen‘𝐵) ⊆ 𝐽 → ((topGen‘𝐵) = 𝐽𝐽 ⊆ (topGen‘𝐵)))
74, 6syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → ((topGen‘𝐵) = 𝐽𝐽 ⊆ (topGen‘𝐵)))
8 dfss3 3741 . . 3 (𝐽 ⊆ (topGen‘𝐵) ↔ ∀𝑥𝐽 𝑥 ∈ (topGen‘𝐵))
97, 8syl6bb 276 . 2 ((𝐽 ∈ Top ∧ 𝐵𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥𝐽 𝑥 ∈ (topGen‘𝐵)))
10 ssexg 4938 . . . . 5 ((𝐵𝐽𝐽 ∈ Top) → 𝐵 ∈ V)
1110ancoms 455 . . . 4 ((𝐽 ∈ Top ∧ 𝐵𝐽) → 𝐵 ∈ V)
12 eltg3 20987 . . . 4 (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
1311, 12syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
1413ralbidv 3135 . 2 ((𝐽 ∈ Top ∧ 𝐵𝐽) → (∀𝑥𝐽 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦)))
159, 14bitrd 268 1 ((𝐽 ∈ Top ∧ 𝐵𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥𝐽𝑦(𝑦𝐵𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wral 3061  Vcvv 3351  wss 3723   cuni 4574  cfv 6031  topGenctg 16306  Topctop 20918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-topgen 16312  df-top 20919
This theorem is referenced by:  bastop2  21019
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