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Theorem ustexsym 22929
 Description: In an uniform structure, for any entourage 𝑉, there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)
Assertion
Ref Expression
ustexsym ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
Distinct variable groups:   𝑤,𝑈   𝑤,𝑉
Allowed substitution hint:   𝑋(𝑤)

Proof of Theorem ustexsym
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simplll 774 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
2 ustinvel 22923 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥𝑈)
32ad4ant13 750 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑈)
4 simplr 768 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑈)
5 ustincl 22921 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈𝑥𝑈) → (𝑥𝑥) ∈ 𝑈)
61, 3, 4, 5syl3anc 1368 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ∈ 𝑈)
7 ustrel 22925 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → Rel 𝑥)
8 dfrel2 6023 . . . . . . 7 (Rel 𝑥𝑥 = 𝑥)
97, 8sylib 221 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥 = 𝑥)
109ineq1d 4118 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → (𝑥𝑥) = (𝑥𝑥))
11 cnvin 5980 . . . . 5 (𝑥𝑥) = (𝑥𝑥)
12 incom 4108 . . . . 5 (𝑥𝑥) = (𝑥𝑥)
1310, 11, 123eqtr4g 2818 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → (𝑥𝑥) = (𝑥𝑥))
1413ad4ant13 750 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) = (𝑥𝑥))
15 inss2 4136 . . . 4 (𝑥𝑥) ⊆ 𝑥
16 ustssco 22928 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥 ⊆ (𝑥𝑥))
1716ad4ant13 750 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥 ⊆ (𝑥𝑥))
18 simpr 488 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ⊆ 𝑉)
1917, 18sstrd 3904 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑉)
2015, 19sstrid 3905 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ⊆ 𝑉)
21 cnveq 5719 . . . . . 6 (𝑤 = (𝑥𝑥) → 𝑤 = (𝑥𝑥))
22 id 22 . . . . . 6 (𝑤 = (𝑥𝑥) → 𝑤 = (𝑥𝑥))
2321, 22eqeq12d 2774 . . . . 5 (𝑤 = (𝑥𝑥) → (𝑤 = 𝑤(𝑥𝑥) = (𝑥𝑥)))
24 sseq1 3919 . . . . 5 (𝑤 = (𝑥𝑥) → (𝑤𝑉 ↔ (𝑥𝑥) ⊆ 𝑉))
2523, 24anbi12d 633 . . . 4 (𝑤 = (𝑥𝑥) → ((𝑤 = 𝑤𝑤𝑉) ↔ ((𝑥𝑥) = (𝑥𝑥) ∧ (𝑥𝑥) ⊆ 𝑉)))
2625rspcev 3543 . . 3 (((𝑥𝑥) ∈ 𝑈 ∧ ((𝑥𝑥) = (𝑥𝑥) ∧ (𝑥𝑥) ⊆ 𝑉)) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
276, 14, 20, 26syl12anc 835 . 2 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
28 ustexhalf 22924 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑥𝑈 (𝑥𝑥) ⊆ 𝑉)
2927, 28r19.29a 3213 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3071   ∩ cin 3859   ⊆ wss 3860  ◡ccnv 5527   ∘ ccom 5532  Rel wrel 5533  ‘cfv 6340  UnifOncust 22913 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-iota 6299  df-fun 6342  df-fv 6348  df-ust 22914 This theorem is referenced by:  ustex2sym  22930  neipcfilu  23010
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