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Theorem utopbas 21849
 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 21846 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
2 ssrab2 3650 . . . 4 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ⊆ 𝒫 𝑋
31, 2syl6eqss 3618 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
4 ssid 3587 . . . . . 6 𝑋𝑋
54a1i 11 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋𝑋)
6 ustssxp 21818 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → 𝑣 ⊆ (𝑋 × 𝑋))
7 imassrn 5396 . . . . . . . . . 10 (𝑣 “ {𝑥}) ⊆ ran 𝑣
8 rnss 5275 . . . . . . . . . . 11 (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ ran (𝑋 × 𝑋))
9 rnxpid 5486 . . . . . . . . . . 11 ran (𝑋 × 𝑋) = 𝑋
108, 9syl6sseq 3614 . . . . . . . . . 10 (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣𝑋)
117, 10syl5ss 3579 . . . . . . . . 9 (𝑣 ⊆ (𝑋 × 𝑋) → (𝑣 “ {𝑥}) ⊆ 𝑋)
126, 11syl 17 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑥}) ⊆ 𝑋)
1312ralrimiva 2949 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
14 ustne0 21827 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
15 r19.2zb 4013 . . . . . . . 8 (𝑈 ≠ ∅ ↔ (∀𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋 → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋))
1614, 15sylib 207 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → (∀𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋 → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋))
1713, 16mpd 15 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
1817ralrimivw 2950 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥𝑋𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
19 elutop 21847 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 ∈ (unifTop‘𝑈) ↔ (𝑋𝑋 ∧ ∀𝑥𝑋𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)))
205, 18, 19mpbir2and 959 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ (unifTop‘𝑈))
21 elpwuni 4549 . . . 4 (𝑋 ∈ (unifTop‘𝑈) → ((unifTop‘𝑈) ⊆ 𝒫 𝑋 (unifTop‘𝑈) = 𝑋))
2220, 21syl 17 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ((unifTop‘𝑈) ⊆ 𝒫 𝑋 (unifTop‘𝑈) = 𝑋))
233, 22mpbid 221 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = 𝑋)
2423eqcomd 2616 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   × cxp 5036  ran crn 5039   “ cima 5041  ‘cfv 5804  UnifOncust 21813  unifTopcutop 21844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ust 21814  df-utop 21845 This theorem is referenced by:  utoptopon  21850  utop2nei  21864  utopreg  21866  tuslem  21881
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