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Theorem ustssco 24224
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 4177 . . . 4 𝑉 ⊆ (𝑉 ∪ (𝑉𝑉))
2 coires1 6283 . . . . . 6 (𝑉 ∘ ( I ↾ 𝑋)) = (𝑉𝑋)
3 ustrel 24221 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)
4 ustssxp 24214 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
5 dmss 5912 . . . . . . . . 9 (𝑉 ⊆ (𝑋 × 𝑋) → dom 𝑉 ⊆ dom (𝑋 × 𝑋))
64, 5syl 17 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → dom 𝑉 ⊆ dom (𝑋 × 𝑋))
7 dmxpid 5940 . . . . . . . 8 dom (𝑋 × 𝑋) = 𝑋
86, 7sseqtrdi 4023 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → dom 𝑉𝑋)
9 relssres 6039 . . . . . . 7 ((Rel 𝑉 ∧ dom 𝑉𝑋) → (𝑉𝑋) = 𝑉)
103, 8, 9syl2anc 584 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝑉𝑋) = 𝑉)
112, 10eqtrid 2788 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝑉 ∘ ( I ↾ 𝑋)) = 𝑉)
1211uneq1d 4166 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉𝑉)) = (𝑉 ∪ (𝑉𝑉)))
131, 12sseqtrrid 4026 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉𝑉)))
14 coundi 6266 . . 3 (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉𝑉))
1513, 14sseqtrrdi 4024 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)))
16 ustdiag 24218 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
17 ssequn1 4185 . . . 4 (( I ↾ 𝑋) ⊆ 𝑉 ↔ (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
1816, 17sylib 218 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
1918coeq2d 5872 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = (𝑉𝑉))
2015, 19sseqtrd 4019 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑉𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  cun 3948  wss 3950   I cid 5576   × cxp 5682  dom cdm 5684  cres 5686  ccom 5688  Rel wrel 5689  cfv 6560  UnifOncust 24209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-iota 6513  df-fun 6562  df-fv 6568  df-ust 24210
This theorem is referenced by:  ustexsym  24225  ustex3sym  24227
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