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

Theorem ustssco 22750
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 4145 . . . 4 𝑉 ⊆ (𝑉 ∪ (𝑉𝑉))
2 coires1 6110 . . . . . 6 (𝑉 ∘ ( I ↾ 𝑋)) = (𝑉𝑋)
3 ustrel 22747 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)
4 ustssxp 22740 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
5 dmss 5764 . . . . . . . . 9 (𝑉 ⊆ (𝑋 × 𝑋) → dom 𝑉 ⊆ dom (𝑋 × 𝑋))
64, 5syl 17 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → dom 𝑉 ⊆ dom (𝑋 × 𝑋))
7 dmxpid 5793 . . . . . . . 8 dom (𝑋 × 𝑋) = 𝑋
86, 7sseqtrdi 4014 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → dom 𝑉𝑋)
9 relssres 5886 . . . . . . 7 ((Rel 𝑉 ∧ dom 𝑉𝑋) → (𝑉𝑋) = 𝑉)
103, 8, 9syl2anc 584 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝑉𝑋) = 𝑉)
112, 10syl5eq 2865 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝑉 ∘ ( I ↾ 𝑋)) = 𝑉)
1211uneq1d 4135 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉𝑉)) = (𝑉 ∪ (𝑉𝑉)))
131, 12sseqtrrid 4017 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉𝑉)))
14 coundi 6093 . . 3 (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉𝑉))
1513, 14sseqtrrdi 4015 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)))
16 ustdiag 22744 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
17 ssequn1 4153 . . . 4 (( I ↾ 𝑋) ⊆ 𝑉 ↔ (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
1816, 17sylib 219 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
1918coeq2d 5726 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = (𝑉𝑉))
2015, 19sseqtrd 4004 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑉𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cun 3931  wss 3933   I cid 5452   × cxp 5546  dom cdm 5548  cres 5550  ccom 5552  Rel wrel 5553  cfv 6348  UnifOncust 22735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-ust 22736
This theorem is referenced by:  ustexsym  22751  ustex3sym  22753
  Copyright terms: Public domain W3C validator