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Theorem ustex3sym 22242
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 815 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
2 simplr 809 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) → 𝑣𝑈)
3 ustex2sym 22241 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣))
41, 2, 3syl2anc 696 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣))
5 simprl 811 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → 𝑤 = 𝑤)
6 simp-5l 829 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → 𝑈 ∈ (UnifOn‘𝑋))
7 simplr 809 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → 𝑤𝑈)
8 ustssco 22239 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑤𝑤))
96, 7, 8syl2anc 696 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → 𝑤 ⊆ (𝑤𝑤))
10 simprr 813 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤𝑤) ⊆ 𝑣)
11 coss2 5434 . . . . . . . . . 10 ((𝑤𝑤) ⊆ 𝑣 → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑤𝑣))
1211adantl 473 . . . . . . . . 9 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑤𝑣))
13 sstr 3752 . . . . . . . . . 10 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → 𝑤𝑣)
14 coss1 5433 . . . . . . . . . 10 (𝑤𝑣 → (𝑤𝑣) ⊆ (𝑣𝑣))
1513, 14syl 17 . . . . . . . . 9 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤𝑣) ⊆ (𝑣𝑣))
1612, 15sstrd 3754 . . . . . . . 8 ((𝑤 ⊆ (𝑤𝑤) ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑣𝑣))
179, 10, 16syl2anc 696 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤 ∘ (𝑤𝑤)) ⊆ (𝑣𝑣))
18 simpllr 817 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑣𝑣) ⊆ 𝑉)
1917, 18sstrd 3754 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉)
205, 19jca 555 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣)) → (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉))
2120ex 449 . . . 4 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑣𝑈) ∧ (𝑣𝑣) ⊆ 𝑉) ∧ 𝑤𝑈) → ((𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑣) → (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉)))
2221reximdva 3155 . . 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 22235 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑣𝑈 (𝑣𝑣) ⊆ 𝑉)
2523, 24r19.29a 3216 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wrex 3051  wss 3715  ccnv 5265  ccom 5270  cfv 6049  UnifOncust 22224
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-iota 6012  df-fun 6051  df-fv 6057  df-ust 22225
This theorem is referenced by:  utopreg  22277
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