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Theorem ustssco 24139
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 4174 . . . 4 𝑉 βŠ† (𝑉 βˆͺ (𝑉 ∘ 𝑉))
2 coires1 6273 . . . . . 6 (𝑉 ∘ ( I β†Ύ 𝑋)) = (𝑉 β†Ύ 𝑋)
3 ustrel 24136 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ Rel 𝑉)
4 ustssxp 24129 . . . . . . . . 9 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† (𝑋 Γ— 𝑋))
5 dmss 5909 . . . . . . . . 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 5936 . . . . . . . 8 dom (𝑋 Γ— 𝑋) = 𝑋
86, 7sseqtrdi 4032 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ dom 𝑉 βŠ† 𝑋)
9 relssres 6031 . . . . . . 7 ((Rel 𝑉 ∧ dom 𝑉 βŠ† 𝑋) β†’ (𝑉 β†Ύ 𝑋) = 𝑉)
103, 8, 9syl2anc 582 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝑉 β†Ύ 𝑋) = 𝑉)
112, 10eqtrid 2780 . . . . 5 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝑉 ∘ ( I β†Ύ 𝑋)) = 𝑉)
1211uneq1d 4163 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ ((𝑉 ∘ ( I β†Ύ 𝑋)) βˆͺ (𝑉 ∘ 𝑉)) = (𝑉 βˆͺ (𝑉 ∘ 𝑉)))
131, 12sseqtrrid 4035 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† ((𝑉 ∘ ( I β†Ύ 𝑋)) βˆͺ (𝑉 ∘ 𝑉)))
14 coundi 6256 . . 3 (𝑉 ∘ (( I β†Ύ 𝑋) βˆͺ 𝑉)) = ((𝑉 ∘ ( I β†Ύ 𝑋)) βˆͺ (𝑉 ∘ 𝑉))
1513, 14sseqtrrdi 4033 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† (𝑉 ∘ (( I β†Ύ 𝑋) βˆͺ 𝑉)))
16 ustdiag 24133 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ ( I β†Ύ 𝑋) βŠ† 𝑉)
17 ssequn1 4182 . . . 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 5869 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝑉 ∘ (( I β†Ύ 𝑋) βˆͺ 𝑉)) = (𝑉 ∘ 𝑉))
2015, 19sseqtrd 4022 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 394   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3947   βŠ† wss 3949   I cid 5579   Γ— cxp 5680  dom cdm 5682   β†Ύ cres 5684   ∘ ccom 5686  Rel wrel 5687  β€˜cfv 6553  UnifOncust 24124
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561  df-ust 24125
This theorem is referenced by:  ustexsym  24140  ustex3sym  24142
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