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Theorem ustex3sym 22743
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 22742 . . . 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 22740 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑤𝑤))
74, 5, 6syl2anc 584 . . . . . . . 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 5725 . . . . . . . . . 10 ((𝑤𝑤) ⊆ 𝑣 → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑤𝑣))
109adantl 482 . . . . . . . . 9 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑤𝑣))
11 sstr 3978 . . . . . . . . . 10 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → 𝑤𝑣)
12 coss1 5724 . . . . . . . . . 10 (𝑤𝑣 → (𝑤𝑣) ⊆ (𝑣𝑣))
1311, 12syl 17 . . . . . . . . 9 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤𝑣) ⊆ (𝑣𝑣))
1410, 13sstrd 3980 . . . . . . . 8 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑣𝑣))
157, 8, 14syl2anc 584 . . . . . . 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 3980 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉)
183, 17jca 512 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉))
1918ex 413 . . . 4 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) → ((𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉)))
2019reximdva 3278 . . 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 22736 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑣𝑈 (𝑣𝑣) ⊆ 𝑉)
2321, 22r19.29a 3293 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wrex 3143  wss 3939  ccnv 5552  ccom 5557  cfv 6351  UnifOncust 22725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-iota 6311  df-fun 6353  df-fv 6359  df-ust 22726
This theorem is referenced by:  utopreg  22778
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