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Theorem ustexsym 22751
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 771 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
2 ustinvel 22745 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥𝑈)
32ad4ant13 747 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑈)
4 simplr 765 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑈)
5 ustincl 22743 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈𝑥𝑈) → (𝑥𝑥) ∈ 𝑈)
61, 3, 4, 5syl3anc 1363 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ∈ 𝑈)
7 ustrel 22747 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → Rel 𝑥)
8 dfrel2 6039 . . . . . . 7 (Rel 𝑥𝑥 = 𝑥)
97, 8sylib 219 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥 = 𝑥)
109ineq1d 4185 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → (𝑥𝑥) = (𝑥𝑥))
11 cnvin 5996 . . . . 5 (𝑥𝑥) = (𝑥𝑥)
12 incom 4175 . . . . 5 (𝑥𝑥) = (𝑥𝑥)
1310, 11, 123eqtr4g 2878 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → (𝑥𝑥) = (𝑥𝑥))
1413ad4ant13 747 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) = (𝑥𝑥))
15 inss2 4203 . . . 4 (𝑥𝑥) ⊆ 𝑥
16 ustssco 22750 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥 ⊆ (𝑥𝑥))
1716ad4ant13 747 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥 ⊆ (𝑥𝑥))
18 simpr 485 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ⊆ 𝑉)
1917, 18sstrd 3974 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑉)
2015, 19sstrid 3975 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ⊆ 𝑉)
21 cnveq 5737 . . . . . 6 (𝑤 = (𝑥𝑥) → 𝑤 = (𝑥𝑥))
22 id 22 . . . . . 6 (𝑤 = (𝑥𝑥) → 𝑤 = (𝑥𝑥))
2321, 22eqeq12d 2834 . . . . 5 (𝑤 = (𝑥𝑥) → (𝑤 = 𝑤(𝑥𝑥) = (𝑥𝑥)))
24 sseq1 3989 . . . . 5 (𝑤 = (𝑥𝑥) → (𝑤𝑉 ↔ (𝑥𝑥) ⊆ 𝑉))
2523, 24anbi12d 630 . . . 4 (𝑤 = (𝑥𝑥) → ((𝑤 = 𝑤𝑤𝑉) ↔ ((𝑥𝑥) = (𝑥𝑥) ∧ (𝑥𝑥) ⊆ 𝑉)))
2625rspcev 3620 . . 3 (((𝑥𝑥) ∈ 𝑈 ∧ ((𝑥𝑥) = (𝑥𝑥) ∧ (𝑥𝑥) ⊆ 𝑉)) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
276, 14, 20, 26syl12anc 832 . 2 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
28 ustexhalf 22746 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑥𝑈 (𝑥𝑥) ⊆ 𝑉)
2927, 28r19.29a 3286 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3136  cin 3932  wss 3933  ccnv 5547  ccom 5552  Rel wrel 5553  cfv 6348  UnifOncust 22735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-ust 22736
This theorem is referenced by:  ustex2sym  22752  neipcfilu  22832
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