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Theorem utopbas 24091
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 24088 . . . 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 4072 . . . 4 {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ π‘Ž βˆƒπ‘£ ∈ π‘ˆ (𝑣 β€œ {π‘₯}) βŠ† π‘Ž} βŠ† 𝒫 𝑋
31, 2eqsstrdi 4031 . . 3 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (unifTopβ€˜π‘ˆ) βŠ† 𝒫 𝑋)
4 ssidd 4000 . . . . 5 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑋 βŠ† 𝑋)
5 ustssxp 24060 . . . . . . . . 9 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑣 ∈ π‘ˆ) β†’ 𝑣 βŠ† (𝑋 Γ— 𝑋))
6 imassrn 6063 . . . . . . . . . 10 (𝑣 β€œ {π‘₯}) βŠ† ran 𝑣
7 rnss 5931 . . . . . . . . . . 11 (𝑣 βŠ† (𝑋 Γ— 𝑋) β†’ ran 𝑣 βŠ† ran (𝑋 Γ— 𝑋))
8 rnxpid 6165 . . . . . . . . . . 11 ran (𝑋 Γ— 𝑋) = 𝑋
97, 8sseqtrdi 4027 . . . . . . . . . 10 (𝑣 βŠ† (𝑋 Γ— 𝑋) β†’ ran 𝑣 βŠ† 𝑋)
106, 9sstrid 3988 . . . . . . . . 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 3140 . . . . . . 7 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆ€π‘£ ∈ π‘ˆ (𝑣 β€œ {π‘₯}) βŠ† 𝑋)
13 ustne0 24069 . . . . . . . 8 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ π‘ˆ β‰  βˆ…)
14 r19.2zb 4490 . . . . . . . 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 3144 . . . . 5 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘£ ∈ π‘ˆ (𝑣 β€œ {π‘₯}) βŠ† 𝑋)
18 elutop 24089 . . . . 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 710 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑋 ∈ (unifTopβ€˜π‘ˆ))
20 elpwuni 5101 . . . 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 2732 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 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  βˆƒwrex 3064  {crab 3426   βŠ† wss 3943  βˆ…c0 4317  π’« cpw 4597  {csn 4623  βˆͺ cuni 4902   Γ— cxp 5667  ran crn 5670   β€œ cima 5672  β€˜cfv 6536  UnifOncust 24055  unifTopcutop 24086
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fv 6544  df-ust 24056  df-utop 24087
This theorem is referenced by:  utoptopon  24092  utop2nei  24106  utopreg  24108  tuslem  24122  tuslemOLD  24123
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