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Theorem isnlly 23386
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 6897 . . . . . . 7 (𝑗 = 𝐽 β†’ (neiβ€˜π‘—) = (neiβ€˜π½))
21fveq1d 6899 . . . . . 6 (𝑗 = 𝐽 β†’ ((neiβ€˜π‘—)β€˜{𝑦}) = ((neiβ€˜π½)β€˜{𝑦}))
32ineq1d 4211 . . . . 5 (𝑗 = 𝐽 β†’ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯) = (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯))
4 oveq1 7427 . . . . . 6 (𝑗 = 𝐽 β†’ (𝑗 β†Ύt 𝑒) = (𝐽 β†Ύt 𝑒))
54eleq1d 2814 . . . . 5 (𝑗 = 𝐽 β†’ ((𝑗 β†Ύt 𝑒) ∈ 𝐴 ↔ (𝐽 β†Ύt 𝑒) ∈ 𝐴))
63, 5rexeqbidv 3340 . . . 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 3174 . . 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 3330 . 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 23384 . 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 3685 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 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  βˆƒwrex 3067   ∩ cin 3946  π’« cpw 4603  {csn 4629  β€˜cfv 6548  (class class class)co 7420   β†Ύt crest 17402  Topctop 22808  neicnei 23014  π‘›-Locally cnlly 23382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-nlly 23384
This theorem is referenced by:  nllytop  23390  nllyi  23392  llynlly  23394  nllyss  23397  nllyrest  23403  nllyidm  23406  hausllycmp  23411  cldllycmp  23412  txnlly  23554  cnllycmp  24895
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