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Theorem utopbas 21949
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 21946 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
2 ssrab2 3666 . . . 4 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ⊆ 𝒫 𝑋
31, 2syl6eqss 3634 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
4 ssid 3603 . . . . . 6 𝑋𝑋
54a1i 11 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋𝑋)
6 ustssxp 21918 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → 𝑣 ⊆ (𝑋 × 𝑋))
7 imassrn 5436 . . . . . . . . . 10 (𝑣 “ {𝑥}) ⊆ ran 𝑣
8 rnss 5314 . . . . . . . . . . 11 (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ ran (𝑋 × 𝑋))
9 rnxpid 5526 . . . . . . . . . . 11 ran (𝑋 × 𝑋) = 𝑋
108, 9syl6sseq 3630 . . . . . . . . . 10 (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣𝑋)
117, 10syl5ss 3594 . . . . . . . . 9 (𝑣 ⊆ (𝑋 × 𝑋) → (𝑣 “ {𝑥}) ⊆ 𝑋)
126, 11syl 17 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑥}) ⊆ 𝑋)
1312ralrimiva 2960 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
14 ustne0 21927 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
15 r19.2zb 4033 . . . . . . . 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 2961 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥𝑋𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
19 elutop 21947 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 ∈ (unifTop‘𝑈) ↔ (𝑋𝑋 ∧ ∀𝑥𝑋𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)))
205, 18, 19mpbir2and 956 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ (unifTop‘𝑈))
21 elpwuni 4579 . . . 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 2627 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148   cuni 4402   × cxp 5072  ran crn 5075  cima 5077  cfv 5847  UnifOncust 21913  unifTopcutop 21944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-ust 21914  df-utop 21945
This theorem is referenced by:  utoptopon  21950  utop2nei  21964  utopreg  21966  tuslem  21981
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