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Theorem ustexsym 22240
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 815 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
2 simplr 809 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑈)
3 ustinvel 22234 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥𝑈)
41, 2, 3syl2anc 696 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑈)
5 ustincl 22232 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈𝑥𝑈) → (𝑥𝑥) ∈ 𝑈)
61, 4, 2, 5syl3anc 1477 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ∈ 𝑈)
7 ustrel 22236 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → Rel 𝑥)
8 dfrel2 5741 . . . . . . 7 (Rel 𝑥𝑥 = 𝑥)
97, 8sylib 208 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥 = 𝑥)
109ineq1d 3956 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → (𝑥𝑥) = (𝑥𝑥))
11 cnvin 5698 . . . . 5 (𝑥𝑥) = (𝑥𝑥)
12 incom 3948 . . . . 5 (𝑥𝑥) = (𝑥𝑥)
1310, 11, 123eqtr4g 2819 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → (𝑥𝑥) = (𝑥𝑥))
141, 2, 13syl2anc 696 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) = (𝑥𝑥))
15 inss2 3977 . . . 4 (𝑥𝑥) ⊆ 𝑥
16 ustssco 22239 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥 ⊆ (𝑥𝑥))
171, 2, 16syl2anc 696 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥 ⊆ (𝑥𝑥))
18 simpr 479 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ⊆ 𝑉)
1917, 18sstrd 3754 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑉)
2015, 19syl5ss 3755 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ⊆ 𝑉)
21 cnveq 5451 . . . . . 6 (𝑤 = (𝑥𝑥) → 𝑤 = (𝑥𝑥))
22 id 22 . . . . . 6 (𝑤 = (𝑥𝑥) → 𝑤 = (𝑥𝑥))
2321, 22eqeq12d 2775 . . . . 5 (𝑤 = (𝑥𝑥) → (𝑤 = 𝑤(𝑥𝑥) = (𝑥𝑥)))
24 sseq1 3767 . . . . 5 (𝑤 = (𝑥𝑥) → (𝑤𝑉 ↔ (𝑥𝑥) ⊆ 𝑉))
2523, 24anbi12d 749 . . . 4 (𝑤 = (𝑥𝑥) → ((𝑤 = 𝑤𝑤𝑉) ↔ ((𝑥𝑥) = (𝑥𝑥) ∧ (𝑥𝑥) ⊆ 𝑉)))
2625rspcev 3449 . . 3 (((𝑥𝑥) ∈ 𝑈 ∧ ((𝑥𝑥) = (𝑥𝑥) ∧ (𝑥𝑥) ⊆ 𝑉)) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
276, 14, 20, 26syl12anc 1475 . 2 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
28 ustexhalf 22235 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑥𝑈 (𝑥𝑥) ⊆ 𝑉)
2927, 28r19.29a 3216 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wrex 3051  cin 3714  wss 3715  ccnv 5265  ccom 5270  Rel wrel 5271  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:  ustex2sym  22241  neipcfilu  22321
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