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Theorem tg2 22459
Description: Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
Assertion
Ref Expression
tg2 ((𝐴 ∈ (topGenβ€˜π΅) ∧ 𝐢 ∈ 𝐴) β†’ βˆƒπ‘₯ ∈ 𝐡 (𝐢 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡   π‘₯,𝐢

Proof of Theorem tg2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6925 . . 3 (𝐴 ∈ (topGenβ€˜π΅) β†’ 𝐡 ∈ dom topGen)
2 eltg2b 22453 . . . 4 (𝐡 ∈ dom topGen β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ βˆ€π‘¦ ∈ 𝐴 βˆƒπ‘₯ ∈ 𝐡 (𝑦 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴)))
3 eleq1 2821 . . . . . . 7 (𝑦 = 𝐢 β†’ (𝑦 ∈ π‘₯ ↔ 𝐢 ∈ π‘₯))
43anbi1d 630 . . . . . 6 (𝑦 = 𝐢 β†’ ((𝑦 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) ↔ (𝐢 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴)))
54rexbidv 3178 . . . . 5 (𝑦 = 𝐢 β†’ (βˆƒπ‘₯ ∈ 𝐡 (𝑦 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐡 (𝐢 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴)))
65rspccv 3609 . . . 4 (βˆ€π‘¦ ∈ 𝐴 βˆƒπ‘₯ ∈ 𝐡 (𝑦 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) β†’ (𝐢 ∈ 𝐴 β†’ βˆƒπ‘₯ ∈ 𝐡 (𝐢 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴)))
72, 6syl6bi 252 . . 3 (𝐡 ∈ dom topGen β†’ (𝐴 ∈ (topGenβ€˜π΅) β†’ (𝐢 ∈ 𝐴 β†’ βˆƒπ‘₯ ∈ 𝐡 (𝐢 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴))))
81, 7mpcom 38 . 2 (𝐴 ∈ (topGenβ€˜π΅) β†’ (𝐢 ∈ 𝐴 β†’ βˆƒπ‘₯ ∈ 𝐡 (𝐢 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴)))
98imp 407 1 ((𝐴 ∈ (topGenβ€˜π΅) ∧ 𝐢 ∈ 𝐴) β†’ βˆƒπ‘₯ ∈ 𝐡 (𝐢 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3947  dom cdm 5675  β€˜cfv 6540  topGenctg 17379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-topgen 17385
This theorem is referenced by:  tgclb  22464  elcls3  22578  pnfnei  22715  mnfnei  22716  tgcnp  22748  tgcmp  22896  2ndcctbss  22950  2ndcdisj  22951  2ndcomap  22953  dis2ndc  22955  ptpjopn  23107  txlm  23143  flftg  23491  alexsublem  23539  alexsubALT  23546  tmdgsum2  23591  xrge0tsms  24341  xrge0tsmsd  32196  iccllysconn  34229  rellysconn  34230  fnessex  35219  ptrecube  36476  islptre  44321
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