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Theorem utopbas 24245
Description: The base of the topology induced by a uniform structure 𝑈. (Contributed by Thierry Arnoux, 5-Dec-2017.)
Assertion
Ref Expression
utopbas (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))

Proof of Theorem utopbas
Dummy variables 𝑎 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 utopval 24242 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
2 ssrab2 4079 . . . 4 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ⊆ 𝒫 𝑋
31, 2eqsstrdi 4027 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
4 ssidd 4006 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋𝑋)
5 ustssxp 24214 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → 𝑣 ⊆ (𝑋 × 𝑋))
6 imassrn 6088 . . . . . . . . . 10 (𝑣 “ {𝑥}) ⊆ ran 𝑣
7 rnss 5949 . . . . . . . . . . 11 (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ ran (𝑋 × 𝑋))
8 rnxpid 6192 . . . . . . . . . . 11 ran (𝑋 × 𝑋) = 𝑋
97, 8sseqtrdi 4023 . . . . . . . . . 10 (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣𝑋)
106, 9sstrid 3994 . . . . . . . . 9 (𝑣 ⊆ (𝑋 × 𝑋) → (𝑣 “ {𝑥}) ⊆ 𝑋)
115, 10syl 17 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑥}) ⊆ 𝑋)
1211ralrimiva 3145 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
13 ustne0 24223 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
14 r19.2zb 4495 . . . . . . . 8 (𝑈 ≠ ∅ ↔ (∀𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋 → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋))
1513, 14sylib 218 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → (∀𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋 → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋))
1612, 15mpd 15 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
1716ralrimivw 3149 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥𝑋𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
18 elutop 24243 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 ∈ (unifTop‘𝑈) ↔ (𝑋𝑋 ∧ ∀𝑥𝑋𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)))
194, 17, 18mpbir2and 713 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ (unifTop‘𝑈))
20 elpwuni 5104 . . . 4 (𝑋 ∈ (unifTop‘𝑈) → ((unifTop‘𝑈) ⊆ 𝒫 𝑋 (unifTop‘𝑈) = 𝑋))
2119, 20syl 17 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ((unifTop‘𝑈) ⊆ 𝒫 𝑋 (unifTop‘𝑈) = 𝑋))
223, 21mpbid 232 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = 𝑋)
2322eqcomd 2742 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  {crab 3435  wss 3950  c0 4332  𝒫 cpw 4599  {csn 4625   cuni 4906   × cxp 5682  ran crn 5685  cima 5687  cfv 6560  UnifOncust 24209  unifTopcutop 24240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ust 24210  df-utop 24241
This theorem is referenced by:  utoptopon  24246  utop2nei  24260  utopreg  24262  tuslem  24276  tuslemOLD  24277
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