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Theorem ustssco 22929
 Description: In an uniform structure, any entourage 𝑉 is a subset of its composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.)
Assertion
Ref Expression
ustssco ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑉𝑉))

Proof of Theorem ustssco
StepHypRef Expression
1 ssun1 4079 . . . 4 𝑉 ⊆ (𝑉 ∪ (𝑉𝑉))
2 coires1 6099 . . . . . 6 (𝑉 ∘ ( I ↾ 𝑋)) = (𝑉𝑋)
3 ustrel 22926 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)
4 ustssxp 22919 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
5 dmss 5748 . . . . . . . . 9 (𝑉 ⊆ (𝑋 × 𝑋) → dom 𝑉 ⊆ dom (𝑋 × 𝑋))
64, 5syl 17 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → dom 𝑉 ⊆ dom (𝑋 × 𝑋))
7 dmxpid 5776 . . . . . . . 8 dom (𝑋 × 𝑋) = 𝑋
86, 7sseqtrdi 3944 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → dom 𝑉𝑋)
9 relssres 5869 . . . . . . 7 ((Rel 𝑉 ∧ dom 𝑉𝑋) → (𝑉𝑋) = 𝑉)
103, 8, 9syl2anc 587 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝑉𝑋) = 𝑉)
112, 10syl5eq 2805 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝑉 ∘ ( I ↾ 𝑋)) = 𝑉)
1211uneq1d 4069 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉𝑉)) = (𝑉 ∪ (𝑉𝑉)))
131, 12sseqtrrid 3947 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉𝑉)))
14 coundi 6082 . . 3 (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉𝑉))
1513, 14sseqtrrdi 3945 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)))
16 ustdiag 22923 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
17 ssequn1 4087 . . . 4 (( I ↾ 𝑋) ⊆ 𝑉 ↔ (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
1816, 17sylib 221 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
1918coeq2d 5708 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = (𝑉𝑉))
2015, 19sseqtrd 3934 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑉𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∪ cun 3858   ⊆ wss 3860   I cid 5433   × cxp 5526  dom cdm 5528   ↾ cres 5530   ∘ ccom 5532  Rel wrel 5533  ‘cfv 6340  UnifOncust 22914 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 22915 This theorem is referenced by:  ustexsym  22930  ustex3sym  22932
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