Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustexsym Structured version   Visualization version   GIF version

Theorem ustexsym 21829
 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 794 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
2 simplr 788 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑈)
3 ustinvel 21823 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥𝑈)
41, 2, 3syl2anc 691 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑈)
5 ustincl 21821 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈𝑥𝑈) → (𝑥𝑥) ∈ 𝑈)
61, 4, 2, 5syl3anc 1318 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ∈ 𝑈)
7 ustrel 21825 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → Rel 𝑥)
8 dfrel2 5502 . . . . . . 7 (Rel 𝑥𝑥 = 𝑥)
97, 8sylib 207 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥 = 𝑥)
109ineq1d 3775 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → (𝑥𝑥) = (𝑥𝑥))
11 cnvin 5459 . . . . 5 (𝑥𝑥) = (𝑥𝑥)
12 incom 3767 . . . . 5 (𝑥𝑥) = (𝑥𝑥)
1310, 11, 123eqtr4g 2669 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → (𝑥𝑥) = (𝑥𝑥))
141, 2, 13syl2anc 691 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) = (𝑥𝑥))
15 inss2 3796 . . . 4 (𝑥𝑥) ⊆ 𝑥
16 ustssco 21828 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥 ⊆ (𝑥𝑥))
171, 2, 16syl2anc 691 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥 ⊆ (𝑥𝑥))
18 simpr 476 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ⊆ 𝑉)
1917, 18sstrd 3578 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑉)
2015, 19syl5ss 3579 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ⊆ 𝑉)
21 cnveq 5218 . . . . . 6 (𝑤 = (𝑥𝑥) → 𝑤 = (𝑥𝑥))
22 id 22 . . . . . 6 (𝑤 = (𝑥𝑥) → 𝑤 = (𝑥𝑥))
2321, 22eqeq12d 2625 . . . . 5 (𝑤 = (𝑥𝑥) → (𝑤 = 𝑤(𝑥𝑥) = (𝑥𝑥)))
24 sseq1 3589 . . . . 5 (𝑤 = (𝑥𝑥) → (𝑤𝑉 ↔ (𝑥𝑥) ⊆ 𝑉))
2523, 24anbi12d 743 . . . 4 (𝑤 = (𝑥𝑥) → ((𝑤 = 𝑤𝑤𝑉) ↔ ((𝑥𝑥) = (𝑥𝑥) ∧ (𝑥𝑥) ⊆ 𝑉)))
2625rspcev 3282 . . 3 (((𝑥𝑥) ∈ 𝑈 ∧ ((𝑥𝑥) = (𝑥𝑥) ∧ (𝑥𝑥) ⊆ 𝑉)) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
276, 14, 20, 26syl12anc 1316 . 2 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
28 ustexhalf 21824 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑥𝑈 (𝑥𝑥) ⊆ 𝑉)
2927, 28r19.29a 3060 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  ◡ccnv 5037   ∘ ccom 5042  Rel wrel 5043  ‘cfv 5804  UnifOncust 21813 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-ust 21814 This theorem is referenced by:  ustex2sym  21830  neipcfilu  21910
 Copyright terms: Public domain W3C validator