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Theorem ustssco 23710
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 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† (𝑉 ∘ 𝑉))
Proof of Theorem ustssco
StepHypRef Expression
1 ssun1 4171 . . . 4 𝑉 βŠ† (𝑉 βˆͺ (𝑉 ∘ 𝑉))
2 coires1 6260 . . . . . 6 (𝑉 ∘ ( I β†Ύ 𝑋)) = (𝑉 β†Ύ 𝑋)
3 ustrel 23707 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ Rel 𝑉)
4 ustssxp 23700 . . . . . . . . 9 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† (𝑋 Γ— 𝑋))
5 dmss 5900 . . . . . . . . 9 (𝑉 βŠ† (𝑋 Γ— 𝑋) β†’ dom 𝑉 βŠ† dom (𝑋 Γ— 𝑋))
64, 5syl 17 . . . . . . . 8 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ dom 𝑉 βŠ† dom (𝑋 Γ— 𝑋))
7 dmxpid 5927 . . . . . . . 8 dom (𝑋 Γ— 𝑋) = 𝑋
86, 7sseqtrdi 4031 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ dom 𝑉 βŠ† 𝑋)
9 relssres 6020 . . . . . . 7 ((Rel 𝑉 ∧ dom 𝑉 βŠ† 𝑋) β†’ (𝑉 β†Ύ 𝑋) = 𝑉)
103, 8, 9syl2anc 584 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝑉 β†Ύ 𝑋) = 𝑉)
112, 10eqtrid 2784 . . . . 5 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝑉 ∘ ( I β†Ύ 𝑋)) = 𝑉)
1211uneq1d 4161 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ ((𝑉 ∘ ( I β†Ύ 𝑋)) βˆͺ (𝑉 ∘ 𝑉)) = (𝑉 βˆͺ (𝑉 ∘ 𝑉)))
131, 12sseqtrrid 4034 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† ((𝑉 ∘ ( I β†Ύ 𝑋)) βˆͺ (𝑉 ∘ 𝑉)))
14 coundi 6243 . . 3 (𝑉 ∘ (( I β†Ύ 𝑋) βˆͺ 𝑉)) = ((𝑉 ∘ ( I β†Ύ 𝑋)) βˆͺ (𝑉 ∘ 𝑉))
1513, 14sseqtrrdi 4032 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† (𝑉 ∘ (( I β†Ύ 𝑋) βˆͺ 𝑉)))
16 ustdiag 23704 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ ( I β†Ύ 𝑋) βŠ† 𝑉)
17 ssequn1 4179 . . . 4 (( I β†Ύ 𝑋) βŠ† 𝑉 ↔ (( I β†Ύ 𝑋) βˆͺ 𝑉) = 𝑉)
1816, 17sylib 217 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (( I β†Ύ 𝑋) βˆͺ 𝑉) = 𝑉)
1918coeq2d 5860 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝑉 ∘ (( I β†Ύ 𝑋) βˆͺ 𝑉)) = (𝑉 ∘ 𝑉))
2015, 19sseqtrd 4021 1 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† (𝑉 ∘ 𝑉))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βˆͺ cun 3945   βŠ† wss 3947   I cid 5572   Γ— cxp 5673  dom cdm 5675   β†Ύ cres 5677   ∘ ccom 5679  Rel wrel 5680  β€˜cfv 6540  UnifOncust 23695
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6492  df-fun 6542  df-fv 6548  df-ust 23696
This theorem is referenced by:  ustexsym  23711  ustex3sym  23713
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