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Theorem utopbas 22260
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 22257 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
2 ssrab2 3828 . . . 4 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ⊆ 𝒫 𝑋
31, 2syl6eqss 3796 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
4 ssid 3765 . . . . . 6 𝑋𝑋
54a1i 11 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋𝑋)
6 ustssxp 22229 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → 𝑣 ⊆ (𝑋 × 𝑋))
7 imassrn 5635 . . . . . . . . . 10 (𝑣 “ {𝑥}) ⊆ ran 𝑣
8 rnss 5509 . . . . . . . . . . 11 (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ ran (𝑋 × 𝑋))
9 rnxpid 5725 . . . . . . . . . . 11 ran (𝑋 × 𝑋) = 𝑋
108, 9syl6sseq 3792 . . . . . . . . . 10 (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣𝑋)
117, 10syl5ss 3755 . . . . . . . . 9 (𝑣 ⊆ (𝑋 × 𝑋) → (𝑣 “ {𝑥}) ⊆ 𝑋)
126, 11syl 17 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑥}) ⊆ 𝑋)
1312ralrimiva 3104 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
14 ustne0 22238 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
15 r19.2zb 4205 . . . . . . . 8 (𝑈 ≠ ∅ ↔ (∀𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋 → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋))
1614, 15sylib 208 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → (∀𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋 → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋))
1713, 16mpd 15 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
1817ralrimivw 3105 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥𝑋𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
19 elutop 22258 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 ∈ (unifTop‘𝑈) ↔ (𝑋𝑋 ∧ ∀𝑥𝑋𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)))
205, 18, 19mpbir2and 995 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ (unifTop‘𝑈))
21 elpwuni 4768 . . . 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 222 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = 𝑋)
2423eqcomd 2766 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  {crab 3054  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321   cuni 4588   × cxp 5264  ran crn 5267  cima 5269  cfv 6049  UnifOncust 22224  unifTopcutop 22255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-ust 22225  df-utop 22256
This theorem is referenced by:  utoptopon  22261  utop2nei  22275  utopreg  22277  tuslem  22292
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