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Theorem tg2 22468
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 6929 . . 3 (𝐴 ∈ (topGenβ€˜π΅) β†’ 𝐡 ∈ dom topGen)
2 eltg2b 22462 . . . 4 (𝐡 ∈ dom topGen β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ βˆ€π‘¦ ∈ 𝐴 βˆƒπ‘₯ ∈ 𝐡 (𝑦 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴)))
3 eleq1 2822 . . . . . . 7 (𝑦 = 𝐢 β†’ (𝑦 ∈ π‘₯ ↔ 𝐢 ∈ π‘₯))
43anbi1d 631 . . . . . 6 (𝑦 = 𝐢 β†’ ((𝑦 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) ↔ (𝐢 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴)))
54rexbidv 3179 . . . . 5 (𝑦 = 𝐢 β†’ (βˆƒπ‘₯ ∈ 𝐡 (𝑦 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐡 (𝐢 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴)))
65rspccv 3610 . . . 4 (βˆ€π‘¦ ∈ 𝐴 βˆƒπ‘₯ ∈ 𝐡 (𝑦 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) β†’ (𝐢 ∈ 𝐴 β†’ βˆƒπ‘₯ ∈ 𝐡 (𝐢 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴)))
72, 6syl6bi 253 . . 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 408 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   βŠ† wss 3949  dom cdm 5677  β€˜cfv 6544  topGenctg 17383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-topgen 17389
This theorem is referenced by:  tgclb  22473  elcls3  22587  pnfnei  22724  mnfnei  22725  tgcnp  22757  tgcmp  22905  2ndcctbss  22959  2ndcdisj  22960  2ndcomap  22962  dis2ndc  22964  ptpjopn  23116  txlm  23152  flftg  23500  alexsublem  23548  alexsubALT  23555  tmdgsum2  23600  xrge0tsms  24350  xrge0tsmsd  32209  iccllysconn  34241  rellysconn  34242  fnessex  35231  ptrecube  36488  islptre  44335
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