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

Theorem ustssco 23582
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 4133 . . . 4 𝑉 βŠ† (𝑉 βˆͺ (𝑉 ∘ 𝑉))
2 coires1 6217 . . . . . 6 (𝑉 ∘ ( I β†Ύ 𝑋)) = (𝑉 β†Ύ 𝑋)
3 ustrel 23579 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ Rel 𝑉)
4 ustssxp 23572 . . . . . . . . 9 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† (𝑋 Γ— 𝑋))
5 dmss 5859 . . . . . . . . 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 5886 . . . . . . . 8 dom (𝑋 Γ— 𝑋) = 𝑋
86, 7sseqtrdi 3995 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ dom 𝑉 βŠ† 𝑋)
9 relssres 5979 . . . . . . 7 ((Rel 𝑉 ∧ dom 𝑉 βŠ† 𝑋) β†’ (𝑉 β†Ύ 𝑋) = 𝑉)
103, 8, 9syl2anc 585 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝑉 β†Ύ 𝑋) = 𝑉)
112, 10eqtrid 2785 . . . . 5 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝑉 ∘ ( I β†Ύ 𝑋)) = 𝑉)
1211uneq1d 4123 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ ((𝑉 ∘ ( I β†Ύ 𝑋)) βˆͺ (𝑉 ∘ 𝑉)) = (𝑉 βˆͺ (𝑉 ∘ 𝑉)))
131, 12sseqtrrid 3998 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† ((𝑉 ∘ ( I β†Ύ 𝑋)) βˆͺ (𝑉 ∘ 𝑉)))
14 coundi 6200 . . 3 (𝑉 ∘ (( I β†Ύ 𝑋) βˆͺ 𝑉)) = ((𝑉 ∘ ( I β†Ύ 𝑋)) βˆͺ (𝑉 ∘ 𝑉))
1513, 14sseqtrrdi 3996 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ 𝑉 βŠ† (𝑉 ∘ (( I β†Ύ 𝑋) βˆͺ 𝑉)))
16 ustdiag 23576 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ ( I β†Ύ 𝑋) βŠ† 𝑉)
17 ssequn1 4141 . . . 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 5819 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ (𝑉 ∘ (( I β†Ύ 𝑋) βˆͺ 𝑉)) = (𝑉 ∘ 𝑉))
2015, 19sseqtrd 3985 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 397   = wceq 1542   ∈ wcel 2107   βˆͺ cun 3909   βŠ† wss 3911   I cid 5531   Γ— cxp 5632  dom cdm 5634   β†Ύ cres 5636   ∘ ccom 5638  Rel wrel 5639  β€˜cfv 6497  UnifOncust 23567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-iota 6449  df-fun 6499  df-fv 6505  df-ust 23568
This theorem is referenced by:  ustexsym  23583  ustex3sym  23585
  Copyright terms: Public domain W3C validator