MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tg2 Structured version   Visualization version   GIF version

Theorem tg2 22331
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 6880 . . 3 (𝐴 ∈ (topGenβ€˜π΅) β†’ 𝐡 ∈ dom topGen)
2 eltg2b 22325 . . . 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 3172 . . . . 5 (𝑦 = 𝐢 β†’ (βˆƒπ‘₯ ∈ 𝐡 (𝑦 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐡 (𝐢 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴)))
65rspccv 3577 . . . 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 3061  βˆƒwrex 3070   βŠ† wss 3911  dom cdm 5634  β€˜cfv 6497  topGenctg 17324
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-topgen 17330
This theorem is referenced by:  tgclb  22336  elcls3  22450  pnfnei  22587  mnfnei  22588  tgcnp  22620  tgcmp  22768  2ndcctbss  22822  2ndcdisj  22823  2ndcomap  22825  dis2ndc  22827  ptpjopn  22979  txlm  23015  flftg  23363  alexsublem  23411  alexsubALT  23418  tmdgsum2  23463  xrge0tsms  24213  xrge0tsmsd  31948  iccllysconn  33901  rellysconn  33902  fnessex  34864  ptrecube  36124  islptre  43946
  Copyright terms: Public domain W3C validator