MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustexsym Structured version   Visualization version   GIF version
Theorem ustexsym 23712
Description: In an uniform structure, for any entourage 𝑉, there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)
Assertion
Ref Expression
ustexsym ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑉))
Distinct variable groups:   𝑀,π‘ˆ   𝑀,𝑉
Allowed substitution hint:   𝑋(𝑀)
Proof of Theorem ustexsym
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 simplll 774 . . . 4 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
2 ustinvel 23706 . . . . 5 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘₯ ∈ π‘ˆ) β†’ β—‘π‘₯ ∈ π‘ˆ)
32ad4ant13 750 . . . 4 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ β—‘π‘₯ ∈ π‘ˆ)
4 simplr 768 . . . 4 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ π‘₯ ∈ π‘ˆ)
5 ustincl 23704 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ β—‘π‘₯ ∈ π‘ˆ ∧ π‘₯ ∈ π‘ˆ) β†’ (β—‘π‘₯ ∩ π‘₯) ∈ π‘ˆ)
61, 3, 4, 5syl3anc 1372 . . 3 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ (β—‘π‘₯ ∩ π‘₯) ∈ π‘ˆ)
7 ustrel 23708 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘₯ ∈ π‘ˆ) β†’ Rel π‘₯)
8 dfrel2 6186 . . . . . . 7 (Rel π‘₯ ↔ β—‘β—‘π‘₯ = π‘₯)
97, 8sylib 217 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘₯ ∈ π‘ˆ) β†’ β—‘β—‘π‘₯ = π‘₯)
109ineq1d 4211 . . . . 5 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘₯ ∈ π‘ˆ) β†’ (β—‘β—‘π‘₯ ∩ β—‘π‘₯) = (π‘₯ ∩ β—‘π‘₯))
11 cnvin 6142 . . . . 5 β—‘(β—‘π‘₯ ∩ π‘₯) = (β—‘β—‘π‘₯ ∩ β—‘π‘₯)
12 incom 4201 . . . . 5 (β—‘π‘₯ ∩ π‘₯) = (π‘₯ ∩ β—‘π‘₯)
1310, 11, 123eqtr4g 2798 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘₯ ∈ π‘ˆ) β†’ β—‘(β—‘π‘₯ ∩ π‘₯) = (β—‘π‘₯ ∩ π‘₯))
1413ad4ant13 750 . . 3 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ β—‘(β—‘π‘₯ ∩ π‘₯) = (β—‘π‘₯ ∩ π‘₯))
15 inss2 4229 . . . 4 (β—‘π‘₯ ∩ π‘₯) βŠ† π‘₯
16 ustssco 23711 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘₯ ∈ π‘ˆ) β†’ π‘₯ βŠ† (π‘₯ ∘ π‘₯))
1716ad4ant13 750 . . . . 5 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ π‘₯ βŠ† (π‘₯ ∘ π‘₯))
18 simpr 486 . . . . 5 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ (π‘₯ ∘ π‘₯) βŠ† 𝑉)
1917, 18sstrd 3992 . . . 4 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ π‘₯ βŠ† 𝑉)
2015, 19sstrid 3993 . . 3 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ (β—‘π‘₯ ∩ π‘₯) βŠ† 𝑉)
21 cnveq 5872 . . . . . 6 (𝑀 = (β—‘π‘₯ ∩ π‘₯) β†’ ◑𝑀 = β—‘(β—‘π‘₯ ∩ π‘₯))
22 id 22 . . . . . 6 (𝑀 = (β—‘π‘₯ ∩ π‘₯) β†’ 𝑀 = (β—‘π‘₯ ∩ π‘₯))
2321, 22eqeq12d 2749 . . . . 5 (𝑀 = (β—‘π‘₯ ∩ π‘₯) β†’ (◑𝑀 = 𝑀 ↔ β—‘(β—‘π‘₯ ∩ π‘₯) = (β—‘π‘₯ ∩ π‘₯)))
24 sseq1 4007 . . . . 5 (𝑀 = (β—‘π‘₯ ∩ π‘₯) β†’ (𝑀 βŠ† 𝑉 ↔ (β—‘π‘₯ ∩ π‘₯) βŠ† 𝑉))
2523, 24anbi12d 632 . . . 4 (𝑀 = (β—‘π‘₯ ∩ π‘₯) β†’ ((◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑉) ↔ (β—‘(β—‘π‘₯ ∩ π‘₯) = (β—‘π‘₯ ∩ π‘₯) ∧ (β—‘π‘₯ ∩ π‘₯) βŠ† 𝑉)))
2625rspcev 3613 . . 3 (((β—‘π‘₯ ∩ π‘₯) ∈ π‘ˆ ∧ (β—‘(β—‘π‘₯ ∩ π‘₯) = (β—‘π‘₯ ∩ π‘₯) ∧ (β—‘π‘₯ ∩ π‘₯) βŠ† 𝑉)) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑉))
276, 14, 20, 26syl12anc 836 . 2 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑉))
28 ustexhalf 23707 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘₯ ∈ π‘ˆ (π‘₯ ∘ π‘₯) βŠ† 𝑉)
2927, 28r19.29a 3163 1 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑉))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071   ∩ cin 3947   βŠ† wss 3948  β—‘ccnv 5675   ∘ ccom 5680  Rel wrel 5681  β€˜cfv 6541  UnifOncust 23696
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6493  df-fun 6543  df-fv 6549  df-ust 23697
This theorem is referenced by:  ustex2sym  23713  neipcfilu  23793
  Copyright terms: Public domain W3C validator