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Theorem ustex3sym 23277
Description: In an uniform structure, for any entourage 𝑉, there exists a symmetrical entourage smaller than a third of 𝑉. (Contributed by Thierry Arnoux, 16-Jan-2018.)
Assertion
Ref Expression
ustex3sym ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉))
Distinct variable groups:   𝑤,𝑈   𝑤,𝑉   𝑤,𝑋

Proof of Theorem ustex3sym
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 ustex2sym 23276 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣))
21ad4ant13 747 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣))
3 simprl 767 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → 𝑤 = 𝑤)
4 simp-5l 781 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → 𝑈 ∈ (UnifOn‘𝑋))
5 simplr 765 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → 𝑤𝑈)
6 ustssco 23274 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑤𝑤))
74, 5, 6syl2anc 583 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → 𝑤 ⊆ (𝑤𝑤))
8 simprr 769 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤𝑤) ⊆ 𝑣)
9 coss2 5754 . . . . . . . . . 10 ((𝑤𝑤) ⊆ 𝑣 → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑤𝑣))
109adantl 481 . . . . . . . . 9 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑤𝑣))
11 sstr 3925 . . . . . . . . . 10 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → 𝑤𝑣)
12 coss1 5753 . . . . . . . . . 10 (𝑤𝑣 → (𝑤𝑣) ⊆ (𝑣𝑣))
1311, 12syl 17 . . . . . . . . 9 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤𝑣) ⊆ (𝑣𝑣))
1410, 13sstrd 3927 . . . . . . . 8 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑣𝑣))
157, 8, 14syl2anc 583 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑣𝑣))
16 simpllr 772 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑣𝑣) ⊆ 𝑉)
1715, 16sstrd 3927 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉)
183, 17jca 511 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉))
1918ex 412 . . . 4 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) → ((𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉)))
2019reximdva 3202 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) → (∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉)))
212, 20mpd 15 . 2 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉))
22 ustexhalf 23270 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑣𝑈 (𝑣𝑣) ⊆ 𝑉)
2321, 22r19.29a 3217 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2108  wrex 3064  wss 3883  ccnv 5579  ccom 5584  cfv 6418  UnifOncust 23259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-iota 6376  df-fun 6420  df-fv 6426  df-ust 23260
This theorem is referenced by:  utopreg  23312
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