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Theorem isnlly 22836
Description: The property of being an n-locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
isnlly (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴))
Distinct variable groups:   π‘₯,𝑒,𝑦,𝐴   𝑒,𝐽,π‘₯,𝑦

Proof of Theorem isnlly
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6843 . . . . . . 7 (𝑗 = 𝐽 β†’ (neiβ€˜π‘—) = (neiβ€˜π½))
21fveq1d 6845 . . . . . 6 (𝑗 = 𝐽 β†’ ((neiβ€˜π‘—)β€˜{𝑦}) = ((neiβ€˜π½)β€˜{𝑦}))
32ineq1d 4172 . . . . 5 (𝑗 = 𝐽 β†’ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯) = (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯))
4 oveq1 7365 . . . . . 6 (𝑗 = 𝐽 β†’ (𝑗 β†Ύt 𝑒) = (𝐽 β†Ύt 𝑒))
54eleq1d 2819 . . . . 5 (𝑗 = 𝐽 β†’ ((𝑗 β†Ύt 𝑒) ∈ 𝐴 ↔ (𝐽 β†Ύt 𝑒) ∈ 𝐴))
63, 5rexeqbidv 3319 . . . 4 (𝑗 = 𝐽 β†’ (βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐴 ↔ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴))
76ralbidv 3171 . . 3 (𝑗 = 𝐽 β†’ (βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐴 ↔ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴))
87raleqbi1dv 3306 . 2 (𝑗 = 𝐽 β†’ (βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐴 ↔ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴))
9 df-nlly 22834 . 2 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐴}
108, 9elrab2 3649 1 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3910  π’« cpw 4561  {csn 4587  β€˜cfv 6497  (class class class)co 7358   β†Ύt crest 17307  Topctop 22258  neicnei 22464  π‘›-Locally cnlly 22832
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-ext 2704
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  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-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-nlly 22834
This theorem is referenced by:  nllytop  22840  nllyi  22842  llynlly  22844  nllyss  22847  nllyrest  22853  nllyidm  22856  hausllycmp  22861  cldllycmp  22862  txnlly  23004  cnllycmp  24335
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