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Theorem utopbas 22839
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 22836 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
2 ssrab2 4031 . . . 4 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ⊆ 𝒫 𝑋
31, 2eqsstrdi 3996 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
4 ssidd 3965 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋𝑋)
5 ustssxp 22808 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → 𝑣 ⊆ (𝑋 × 𝑋))
6 imassrn 5918 . . . . . . . . . 10 (𝑣 “ {𝑥}) ⊆ ran 𝑣
7 rnss 5786 . . . . . . . . . . 11 (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ ran (𝑋 × 𝑋))
8 rnxpid 6008 . . . . . . . . . . 11 ran (𝑋 × 𝑋) = 𝑋
97, 8sseqtrdi 3992 . . . . . . . . . 10 (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣𝑋)
106, 9sstrid 3953 . . . . . . . . 9 (𝑣 ⊆ (𝑋 × 𝑋) → (𝑣 “ {𝑥}) ⊆ 𝑋)
115, 10syl 17 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑥}) ⊆ 𝑋)
1211ralrimiva 3174 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
13 ustne0 22817 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
14 r19.2zb 4413 . . . . . . . 8 (𝑈 ≠ ∅ ↔ (∀𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋 → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋))
1513, 14sylib 221 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → (∀𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋 → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋))
1612, 15mpd 15 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
1716ralrimivw 3175 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥𝑋𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
18 elutop 22837 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 ∈ (unifTop‘𝑈) ↔ (𝑋𝑋 ∧ ∀𝑥𝑋𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)))
194, 17, 18mpbir2and 712 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ (unifTop‘𝑈))
20 elpwuni 5002 . . . 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 235 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = 𝑋)
2322eqcomd 2828 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wne 3011  wral 3130  wrex 3131  {crab 3134  wss 3908  c0 4265  𝒫 cpw 4511  {csn 4539   cuni 4813   × cxp 5530  ran crn 5533  cima 5535  cfv 6334  UnifOncust 22803  unifTopcutop 22834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342  df-ust 22804  df-utop 22835
This theorem is referenced by:  utoptopon  22840  utop2nei  22854  utopreg  22856  tuslem  22871
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