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Theorem ustex2sym 23584
Description: In an uniform structure, for any entourage 𝑉, there exists a symmetrical entourage smaller than half 𝑉. (Contributed by Thierry Arnoux, 16-Jan-2018.)
Assertion
Ref Expression
ustex2sym ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑉))
Distinct variable groups:   𝑀,π‘ˆ   𝑀,𝑉   𝑀,𝑋

Proof of Theorem ustex2sym
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 ustexsym 23583 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑣 ∈ π‘ˆ) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑣))
21ad4ant13 750 . . 3 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑣))
3 simprl 770 . . . . . 6 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑣)) β†’ ◑𝑀 = 𝑀)
4 coss1 5816 . . . . . . . . 9 (𝑀 βŠ† 𝑣 β†’ (𝑀 ∘ 𝑀) βŠ† (𝑣 ∘ 𝑀))
5 coss2 5817 . . . . . . . . 9 (𝑀 βŠ† 𝑣 β†’ (𝑣 ∘ 𝑀) βŠ† (𝑣 ∘ 𝑣))
64, 5sstrd 3959 . . . . . . . 8 (𝑀 βŠ† 𝑣 β†’ (𝑀 ∘ 𝑀) βŠ† (𝑣 ∘ 𝑣))
76ad2antll 728 . . . . . . 7 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑣)) β†’ (𝑀 ∘ 𝑀) βŠ† (𝑣 ∘ 𝑣))
8 simpllr 775 . . . . . . 7 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑣)) β†’ (𝑣 ∘ 𝑣) βŠ† 𝑉)
97, 8sstrd 3959 . . . . . 6 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑣)) β†’ (𝑀 ∘ 𝑀) βŠ† 𝑉)
103, 9jca 513 . . . . 5 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑣)) β†’ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑉))
1110ex 414 . . . 4 (((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) β†’ ((◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑣) β†’ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑉)))
1211reximdva 3166 . . 3 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) β†’ (βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑣) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑉)))
132, 12mpd 15 . 2 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑉))
14 ustexhalf 23578 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘£ ∈ π‘ˆ (𝑣 ∘ 𝑣) βŠ† 𝑉)
1513, 14r19.29a 3160 1 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑉))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074   βŠ† wss 3915  β—‘ccnv 5637   ∘ ccom 5642  β€˜cfv 6501  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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6453  df-fun 6503  df-fv 6509  df-ust 23568
This theorem is referenced by:  ustex3sym  23585
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