MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustexsym Structured version   Visualization version   GIF version
Theorem ustexsym 24148
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 773 . . . 4 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
2 ustinvel 24142 . . . . 5 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘₯ ∈ π‘ˆ) β†’ β—‘π‘₯ ∈ π‘ˆ)
32ad4ant13 749 . . . 4 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ β—‘π‘₯ ∈ π‘ˆ)
4 simplr 767 . . . 4 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ π‘₯ ∈ π‘ˆ)
5 ustincl 24140 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ β—‘π‘₯ ∈ π‘ˆ ∧ π‘₯ ∈ π‘ˆ) β†’ (β—‘π‘₯ ∩ π‘₯) ∈ π‘ˆ)
61, 3, 4, 5syl3anc 1368 . . 3 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ (β—‘π‘₯ ∩ π‘₯) ∈ π‘ˆ)
7 ustrel 24144 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘₯ ∈ π‘ˆ) β†’ Rel π‘₯)
8 dfrel2 6198 . . . . . . 7 (Rel π‘₯ ↔ β—‘β—‘π‘₯ = π‘₯)
97, 8sylib 217 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘₯ ∈ π‘ˆ) β†’ β—‘β—‘π‘₯ = π‘₯)
109ineq1d 4213 . . . . 5 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘₯ ∈ π‘ˆ) β†’ (β—‘β—‘π‘₯ ∩ β—‘π‘₯) = (π‘₯ ∩ β—‘π‘₯))
11 cnvin 6154 . . . . 5 β—‘(β—‘π‘₯ ∩ π‘₯) = (β—‘β—‘π‘₯ ∩ β—‘π‘₯)
12 incom 4203 . . . . 5 (β—‘π‘₯ ∩ π‘₯) = (π‘₯ ∩ β—‘π‘₯)
1310, 11, 123eqtr4g 2793 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘₯ ∈ π‘ˆ) β†’ β—‘(β—‘π‘₯ ∩ π‘₯) = (β—‘π‘₯ ∩ π‘₯))
1413ad4ant13 749 . . 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 4232 . . . 4 (β—‘π‘₯ ∩ π‘₯) βŠ† π‘₯
16 ustssco 24147 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘₯ ∈ π‘ˆ) β†’ π‘₯ βŠ† (π‘₯ ∘ π‘₯))
1716ad4ant13 749 . . . . 5 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ π‘₯ ∈ π‘ˆ) ∧ (π‘₯ ∘ π‘₯) βŠ† 𝑉) β†’ π‘₯ βŠ† (π‘₯ ∘ π‘₯))
18 simpr 483 . . . . 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 5880 . . . . . 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 2744 . . . . 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 630 . . . 4 (𝑀 = (β—‘π‘₯ ∩ π‘₯) β†’ ((◑𝑀 = 𝑀 ∧ 𝑀 βŠ† 𝑉) ↔ (β—‘(β—‘π‘₯ ∩ π‘₯) = (β—‘π‘₯ ∩ π‘₯) ∧ (β—‘π‘₯ ∩ π‘₯) βŠ† 𝑉)))
2625rspcev 3611 . . 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 835 . 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 24143 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘₯ ∈ π‘ˆ (π‘₯ ∘ π‘₯) βŠ† 𝑉)
2927, 28r19.29a 3159 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 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3067   ∩ cin 3948   βŠ† wss 3949  β—‘ccnv 5681   ∘ ccom 5686  Rel wrel 5687  β€˜cfv 6553  UnifOncust 24132
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 7748
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 24133
This theorem is referenced by:  ustex2sym  24149  neipcfilu  24229
  Copyright terms: Public domain W3C validator