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Theorem isnlly 23324
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 6884 . . . . . . 7 (𝑗 = 𝐽 β†’ (neiβ€˜π‘—) = (neiβ€˜π½))
21fveq1d 6886 . . . . . 6 (𝑗 = 𝐽 β†’ ((neiβ€˜π‘—)β€˜{𝑦}) = ((neiβ€˜π½)β€˜{𝑦}))
32ineq1d 4206 . . . . 5 (𝑗 = 𝐽 β†’ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯) = (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯))
4 oveq1 7411 . . . . . 6 (𝑗 = 𝐽 β†’ (𝑗 β†Ύt 𝑒) = (𝐽 β†Ύt 𝑒))
54eleq1d 2812 . . . . 5 (𝑗 = 𝐽 β†’ ((𝑗 β†Ύt 𝑒) ∈ 𝐴 ↔ (𝐽 β†Ύt 𝑒) ∈ 𝐴))
63, 5rexeqbidv 3337 . . . 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 3327 . 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 23322 . 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 3681 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 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064   ∩ cin 3942  π’« cpw 4597  {csn 4623  β€˜cfv 6536  (class class class)co 7404   β†Ύt crest 17373  Topctop 22746  neicnei 22952  π‘›-Locally cnlly 23320
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-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-nlly 23322
This theorem is referenced by:  nllytop  23328  nllyi  23330  llynlly  23332  nllyss  23335  nllyrest  23341  nllyidm  23344  hausllycmp  23349  cldllycmp  23350  txnlly  23492  cnllycmp  24833
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