MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  utopbas Structured version   Visualization version   GIF version
Theorem utopbas 24160
Description: The base of the topology induced by a uniform structure π‘ˆ. (Contributed by Thierry Arnoux, 5-Dec-2017.)
Assertion
Ref Expression
utopbas (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ (unifTopβ€˜π‘ˆ))
Proof of Theorem utopbas
Dummy variables π‘Ž 𝑣 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 utopval 24157 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (unifTopβ€˜π‘ˆ) = {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ π‘Ž βˆƒπ‘£ ∈ π‘ˆ (𝑣 β€œ {π‘₯}) βŠ† π‘Ž})
2 ssrab2 4077 . . . 4 {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ π‘Ž βˆƒπ‘£ ∈ π‘ˆ (𝑣 β€œ {π‘₯}) βŠ† π‘Ž} βŠ† 𝒫 𝑋
31, 2eqsstrdi 4036 . . 3 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (unifTopβ€˜π‘ˆ) βŠ† 𝒫 𝑋)
4 ssidd 4005 . . . . 5 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑋 βŠ† 𝑋)
5 ustssxp 24129 . . . . . . . . 9 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑣 ∈ π‘ˆ) β†’ 𝑣 βŠ† (𝑋 Γ— 𝑋))
6 imassrn 6079 . . . . . . . . . 10 (𝑣 β€œ {π‘₯}) βŠ† ran 𝑣
7 rnss 5945 . . . . . . . . . . 11 (𝑣 βŠ† (𝑋 Γ— 𝑋) β†’ ran 𝑣 βŠ† ran (𝑋 Γ— 𝑋))
8 rnxpid 6182 . . . . . . . . . . 11 ran (𝑋 Γ— 𝑋) = 𝑋
97, 8sseqtrdi 4032 . . . . . . . . . 10 (𝑣 βŠ† (𝑋 Γ— 𝑋) β†’ ran 𝑣 βŠ† 𝑋)
106, 9sstrid 3993 . . . . . . . . 9 (𝑣 βŠ† (𝑋 Γ— 𝑋) β†’ (𝑣 β€œ {π‘₯}) βŠ† 𝑋)
115, 10syl 17 . . . . . . . 8 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑣 ∈ π‘ˆ) β†’ (𝑣 β€œ {π‘₯}) βŠ† 𝑋)
1211ralrimiva 3143 . . . . . . 7 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆ€π‘£ ∈ π‘ˆ (𝑣 β€œ {π‘₯}) βŠ† 𝑋)
13 ustne0 24138 . . . . . . . 8 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ π‘ˆ β‰  βˆ…)
14 r19.2zb 4499 . . . . . . . 8 (π‘ˆ β‰  βˆ… ↔ (βˆ€π‘£ ∈ π‘ˆ (𝑣 β€œ {π‘₯}) βŠ† 𝑋 β†’ βˆƒπ‘£ ∈ π‘ˆ (𝑣 β€œ {π‘₯}) βŠ† 𝑋))
1513, 14sylib 217 . . . . . . 7 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (βˆ€π‘£ ∈ π‘ˆ (𝑣 β€œ {π‘₯}) βŠ† 𝑋 β†’ βˆƒπ‘£ ∈ π‘ˆ (𝑣 β€œ {π‘₯}) βŠ† 𝑋))
1612, 15mpd 15 . . . . . 6 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆƒπ‘£ ∈ π‘ˆ (𝑣 β€œ {π‘₯}) βŠ† 𝑋)
1716ralrimivw 3147 . . . . 5 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘£ ∈ π‘ˆ (𝑣 β€œ {π‘₯}) βŠ† 𝑋)
18 elutop 24158 . . . . 5 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (𝑋 ∈ (unifTopβ€˜π‘ˆ) ↔ (𝑋 βŠ† 𝑋 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘£ ∈ π‘ˆ (𝑣 β€œ {π‘₯}) βŠ† 𝑋)))
194, 17, 18mpbir2and 711 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑋 ∈ (unifTopβ€˜π‘ˆ))
20 elpwuni 5112 . . . 4 (𝑋 ∈ (unifTopβ€˜π‘ˆ) β†’ ((unifTopβ€˜π‘ˆ) βŠ† 𝒫 𝑋 ↔ βˆͺ (unifTopβ€˜π‘ˆ) = 𝑋))
2119, 20syl 17 . . 3 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ ((unifTopβ€˜π‘ˆ) βŠ† 𝒫 𝑋 ↔ βˆͺ (unifTopβ€˜π‘ˆ) = 𝑋))
223, 21mpbid 231 . 2 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆͺ (unifTopβ€˜π‘ˆ) = 𝑋)
2322eqcomd 2734 1 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ (unifTopβ€˜π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  βˆ€wral 3058  βˆƒwrex 3067  {crab 3430   βŠ† wss 3949  βˆ…c0 4326  π’« cpw 4606  {csn 4632  βˆͺ cuni 4912   Γ— cxp 5680  ran crn 5683   β€œ cima 5685  β€˜cfv 6553  UnifOncust 24124  unifTopcutop 24155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-ust 24125  df-utop 24156
This theorem is referenced by:  utoptopon  24161  utop2nei  24175  utopreg  24177  tuslem  24191  tuslemOLD  24192
  Copyright terms: Public domain W3C validator