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Theorem ustssco 24069
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 4167 . . . 4 𝑉 βŠ† (𝑉 βˆͺ (𝑉 ∘ 𝑉))
2 coires1 6256 . . . . . 6 (𝑉 ∘ ( I β†Ύ 𝑋)) = (𝑉 β†Ύ 𝑋)
3 ustrel 24066 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ Rel 𝑉)
4 ustssxp 24059 . . . . . . . . 9 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† (𝑋 Γ— 𝑋))
5 dmss 5895 . . . . . . . . 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 5922 . . . . . . . 8 dom (𝑋 Γ— 𝑋) = 𝑋
86, 7sseqtrdi 4027 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ dom 𝑉 βŠ† 𝑋)
9 relssres 6015 . . . . . . 7 ((Rel 𝑉 ∧ dom 𝑉 βŠ† 𝑋) β†’ (𝑉 β†Ύ 𝑋) = 𝑉)
103, 8, 9syl2anc 583 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝑉 β†Ύ 𝑋) = 𝑉)
112, 10eqtrid 2778 . . . . 5 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝑉 ∘ ( I β†Ύ 𝑋)) = 𝑉)
1211uneq1d 4157 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ ((𝑉 ∘ ( I β†Ύ 𝑋)) βˆͺ (𝑉 ∘ 𝑉)) = (𝑉 βˆͺ (𝑉 ∘ 𝑉)))
131, 12sseqtrrid 4030 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† ((𝑉 ∘ ( I β†Ύ 𝑋)) βˆͺ (𝑉 ∘ 𝑉)))
14 coundi 6239 . . 3 (𝑉 ∘ (( I β†Ύ 𝑋) βˆͺ 𝑉)) = ((𝑉 ∘ ( I β†Ύ 𝑋)) βˆͺ (𝑉 ∘ 𝑉))
1513, 14sseqtrrdi 4028 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† (𝑉 ∘ (( I β†Ύ 𝑋) βˆͺ 𝑉)))
16 ustdiag 24063 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ ( I β†Ύ 𝑋) βŠ† 𝑉)
17 ssequn1 4175 . . . 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 5855 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝑉 ∘ (( I β†Ύ 𝑋) βˆͺ 𝑉)) = (𝑉 ∘ 𝑉))
2015, 19sseqtrd 4017 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 395   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3941   βŠ† wss 3943   I cid 5566   Γ— cxp 5667  dom cdm 5669   β†Ύ cres 5671   ∘ ccom 5673  Rel wrel 5674  β€˜cfv 6536  UnifOncust 24054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6488  df-fun 6538  df-fv 6544  df-ust 24055
This theorem is referenced by:  ustexsym  24070  ustex3sym  24072
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