MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nllyss Structured version   Visualization version   GIF version
Theorem nllyss 22983
Description: The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllyss (𝐴 βŠ† 𝐡 β†’ 𝑛-Locally 𝐴 βŠ† 𝑛-Locally 𝐡)
Proof of Theorem nllyss
Dummy variables 𝑗 𝑒 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3975 . . . . . . 7 (𝐴 βŠ† 𝐡 β†’ ((𝑗 β†Ύt 𝑒) ∈ 𝐴 β†’ (𝑗 β†Ύt 𝑒) ∈ 𝐡))
21reximdv 3170 . . . . . 6 (𝐴 βŠ† 𝐡 β†’ (βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐴 β†’ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐡))
32ralimdv 3169 . . . . 5 (𝐴 βŠ† 𝐡 β†’ (βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐴 β†’ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐡))
43ralimdv 3169 . . . 4 (𝐴 βŠ† 𝐡 β†’ (βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐴 β†’ βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐡))
54anim2d 612 . . 3 (𝐴 βŠ† 𝐡 β†’ ((𝑗 ∈ Top ∧ βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐴) β†’ (𝑗 ∈ Top ∧ βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐡)))
6 isnlly 22972 . . 3 (𝑗 ∈ 𝑛-Locally 𝐴 ↔ (𝑗 ∈ Top ∧ βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐴))
7 isnlly 22972 . . 3 (𝑗 ∈ 𝑛-Locally 𝐡 ↔ (𝑗 ∈ Top ∧ βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐡))
85, 6, 73imtr4g 295 . 2 (𝐴 βŠ† 𝐡 β†’ (𝑗 ∈ 𝑛-Locally 𝐴 β†’ 𝑗 ∈ 𝑛-Locally 𝐡))
98ssrdv 3988 1 (𝐴 βŠ† 𝐡 β†’ 𝑛-Locally 𝐴 βŠ† 𝑛-Locally 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  {csn 4628  β€˜cfv 6543  (class class class)co 7408   β†Ύt crest 17365  Topctop 22394  neicnei 22600  π‘›-Locally cnlly 22968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-nlly 22970
This theorem is referenced by:  iinllyconn  34240  cvmlift3  34314
  Copyright terms: Public domain W3C validator