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Theorem eltg3i 21499
Description: The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
eltg3i ((𝐵𝑉𝐴𝐵) → 𝐴 ∈ (topGen‘𝐵))

Proof of Theorem eltg3i
StepHypRef Expression
1 simpr 485 . . . 4 ((𝐵𝑉𝐴𝐵) → 𝐴𝐵)
2 pwuni 4868 . . . 4 𝐴 ⊆ 𝒫 𝐴
3 ssin 4206 . . . 4 ((𝐴𝐵𝐴 ⊆ 𝒫 𝐴) ↔ 𝐴 ⊆ (𝐵 ∩ 𝒫 𝐴))
41, 2, 3sylanblc 589 . . 3 ((𝐵𝑉𝐴𝐵) → 𝐴 ⊆ (𝐵 ∩ 𝒫 𝐴))
54unissd 4856 . 2 ((𝐵𝑉𝐴𝐵) → 𝐴 (𝐵 ∩ 𝒫 𝐴))
6 eltg 21495 . . 3 (𝐵𝑉 → ( 𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
76adantr 481 . 2 ((𝐵𝑉𝐴𝐵) → ( 𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
85, 7mpbird 258 1 ((𝐵𝑉𝐴𝐵) → 𝐴 ∈ (topGen‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  cin 3934  wss 3935  𝒫 cpw 4537   cuni 4832  cfv 6349  topGenctg 16701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-topgen 16707
This theorem is referenced by:  eltg3  21500  tgiun  21517  tgidm  21518  tgrest  21697  leordtval2  21750  fnemeet1  33612  fnejoin2  33615  ontgval  33677
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