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Theorem ustexsym 22301
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 791 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
2 simplr 785 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑈)
3 ustinvel 22295 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥𝑈)
41, 2, 3syl2anc 579 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑈)
5 ustincl 22293 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈𝑥𝑈) → (𝑥𝑥) ∈ 𝑈)
61, 4, 2, 5syl3anc 1490 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ∈ 𝑈)
7 ustrel 22297 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → Rel 𝑥)
8 dfrel2 5768 . . . . . . 7 (Rel 𝑥𝑥 = 𝑥)
97, 8sylib 209 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥 = 𝑥)
109ineq1d 3977 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → (𝑥𝑥) = (𝑥𝑥))
11 cnvin 5725 . . . . 5 (𝑥𝑥) = (𝑥𝑥)
12 incom 3969 . . . . 5 (𝑥𝑥) = (𝑥𝑥)
1310, 11, 123eqtr4g 2824 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → (𝑥𝑥) = (𝑥𝑥))
141, 2, 13syl2anc 579 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) = (𝑥𝑥))
15 inss2 3995 . . . 4 (𝑥𝑥) ⊆ 𝑥
16 ustssco 22300 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑈) → 𝑥 ⊆ (𝑥𝑥))
171, 2, 16syl2anc 579 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥 ⊆ (𝑥𝑥))
18 simpr 477 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ⊆ 𝑉)
1917, 18sstrd 3773 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → 𝑥𝑉)
2015, 19syl5ss 3774 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → (𝑥𝑥) ⊆ 𝑉)
21 cnveq 5466 . . . . . 6 (𝑤 = (𝑥𝑥) → 𝑤 = (𝑥𝑥))
22 id 22 . . . . . 6 (𝑤 = (𝑥𝑥) → 𝑤 = (𝑥𝑥))
2321, 22eqeq12d 2780 . . . . 5 (𝑤 = (𝑥𝑥) → (𝑤 = 𝑤(𝑥𝑥) = (𝑥𝑥)))
24 sseq1 3788 . . . . 5 (𝑤 = (𝑥𝑥) → (𝑤𝑉 ↔ (𝑥𝑥) ⊆ 𝑉))
2523, 24anbi12d 624 . . . 4 (𝑤 = (𝑥𝑥) → ((𝑤 = 𝑤𝑤𝑉) ↔ ((𝑥𝑥) = (𝑥𝑥) ∧ (𝑥𝑥) ⊆ 𝑉)))
2625rspcev 3462 . . 3 (((𝑥𝑥) ∈ 𝑈 ∧ ((𝑥𝑥) = (𝑥𝑥) ∧ (𝑥𝑥) ⊆ 𝑉)) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
276, 14, 20, 26syl12anc 865 . 2 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑉) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
28 ustexhalf 22296 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑥𝑈 (𝑥𝑥) ⊆ 𝑉)
2927, 28r19.29a 3225 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wrex 3056  cin 3733  wss 3734  ccnv 5278  ccom 5283  Rel wrel 5284  cfv 6070  UnifOncust 22285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-iota 6033  df-fun 6072  df-fv 6078  df-ust 22286
This theorem is referenced by:  ustex2sym  22302  neipcfilu  22382
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