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Theorem eltg4i 12151
Description: An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
eltg4i (𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))

Proof of Theorem eltg4i
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-topgen 12068 . . . . . . 7 topGen = (𝑥 ∈ V ↦ {𝑦𝑦 (𝑥 ∩ 𝒫 𝑦)})
21funmpt2 5132 . . . . . 6 Fun topGen
3 funrel 5110 . . . . . 6 (Fun topGen → Rel topGen)
42, 3ax-mp 5 . . . . 5 Rel topGen
5 relelfvdm 5421 . . . . 5 ((Rel topGen ∧ 𝐴 ∈ (topGen‘𝐵)) → 𝐵 ∈ dom topGen)
64, 5mpan 420 . . . 4 (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
7 eltg 12148 . . . 4 (𝐵 ∈ dom topGen → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
86, 7syl 14 . . 3 (𝐴 ∈ (topGen‘𝐵) → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))
98ibi 175 . 2 (𝐴 ∈ (topGen‘𝐵) → 𝐴 (𝐵 ∩ 𝒫 𝐴))
10 inss2 3267 . . . . 5 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
1110unissi 3729 . . . 4 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
12 unipw 4109 . . . 4 𝒫 𝐴 = 𝐴
1311, 12sseqtri 3101 . . 3 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴
1413a1i 9 . 2 (𝐴 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴)
159, 14eqssd 3084 1 (𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  wcel 1465  {cab 2103  Vcvv 2660  cin 3040  wss 3041  𝒫 cpw 3480   cuni 3706  dom cdm 4509  Rel wrel 4514  Fun wfun 5087  cfv 5093  topGenctg 12062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-topgen 12068
This theorem is referenced by:  eltg3  12153  tgdom  12168  tgidm  12170
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