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Theorem ustssco 21937
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 3759 . . . 4 𝑉 ⊆ (𝑉 ∪ (𝑉𝑉))
2 coires1 5617 . . . . . 6 (𝑉 ∘ ( I ↾ 𝑋)) = (𝑉𝑋)
3 ustrel 21934 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)
4 ustssxp 21927 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
5 dmss 5288 . . . . . . . . 9 (𝑉 ⊆ (𝑋 × 𝑋) → dom 𝑉 ⊆ dom (𝑋 × 𝑋))
64, 5syl 17 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → dom 𝑉 ⊆ dom (𝑋 × 𝑋))
7 dmxpid 5310 . . . . . . . 8 dom (𝑋 × 𝑋) = 𝑋
86, 7syl6sseq 3635 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → dom 𝑉𝑋)
9 relssres 5401 . . . . . . 7 ((Rel 𝑉 ∧ dom 𝑉𝑋) → (𝑉𝑋) = 𝑉)
103, 8, 9syl2anc 692 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝑉𝑋) = 𝑉)
112, 10syl5eq 2667 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝑉 ∘ ( I ↾ 𝑋)) = 𝑉)
1211uneq1d 3749 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉𝑉)) = (𝑉 ∪ (𝑉𝑉)))
131, 12syl5sseqr 3638 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉𝑉)))
14 coundi 5600 . . 3 (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = ((𝑉 ∘ ( I ↾ 𝑋)) ∪ (𝑉𝑉))
1513, 14syl6sseqr 3636 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)))
16 ustdiag 21931 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
17 ssequn1 3766 . . . 4 (( I ↾ 𝑋) ⊆ 𝑉 ↔ (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
1816, 17sylib 208 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (( I ↾ 𝑋) ∪ 𝑉) = 𝑉)
1918coeq2d 5249 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (𝑉 ∘ (( I ↾ 𝑋) ∪ 𝑉)) = (𝑉𝑉))
2015, 19sseqtrd 3625 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑉𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cun 3557  wss 3559   I cid 4989   × cxp 5077  dom cdm 5079  cres 5081  ccom 5083  Rel wrel 5084  cfv 5852  UnifOncust 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5815  df-fun 5854  df-fv 5860  df-ust 21923
This theorem is referenced by:  ustexsym  21938  ustex3sym  21940
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