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Theorem ustex3sym 21961
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 simplll 797 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
2 simplr 791 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) → 𝑣𝑈)
3 ustex2sym 21960 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣))
41, 2, 3syl2anc 692 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣))
5 simprl 793 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → 𝑤 = 𝑤)
6 simp-5l 807 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → 𝑈 ∈ (UnifOn‘𝑋))
7 simplr 791 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → 𝑤𝑈)
8 ustssco 21958 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑤𝑤))
96, 7, 8syl2anc 692 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → 𝑤 ⊆ (𝑤𝑤))
10 simprr 795 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤𝑤) ⊆ 𝑣)
11 coss2 5248 . . . . . . . . . 10 ((𝑤𝑤) ⊆ 𝑣 → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑤𝑣))
1211adantl 482 . . . . . . . . 9 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑤𝑣))
13 sstr 3596 . . . . . . . . . 10 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → 𝑤𝑣)
14 coss1 5247 . . . . . . . . . 10 (𝑤𝑣 → (𝑤𝑣) ⊆ (𝑣𝑣))
1513, 14syl 17 . . . . . . . . 9 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤𝑣) ⊆ (𝑣𝑣))
1612, 15sstrd 3598 . . . . . . . 8 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑣𝑣))
179, 10, 16syl2anc 692 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑣𝑣))
18 simpllr 798 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑣𝑣) ⊆ 𝑉)
1917, 18sstrd 3598 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉)
205, 19jca 554 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉))
2120ex 450 . . . 4 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) → ((𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉)))
2221reximdva 3013 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) → (∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉)))
234, 22mpd 15 . 2 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉))
24 ustexhalf 21954 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑣𝑈 (𝑣𝑣) ⊆ 𝑉)
2523, 24r19.29a 3073 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wrex 2909  wss 3560  ccnv 5083  ccom 5088  cfv 5857  UnifOncust 21943
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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-iota 5820  df-fun 5859  df-fv 5865  df-ust 21944
This theorem is referenced by:  utopreg  21996
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