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Theorem nllyi 23301
Description: The property of an n-locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllyi ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ π‘ˆ ∈ 𝐽 ∧ 𝑃 ∈ π‘ˆ) β†’ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑒 βŠ† π‘ˆ ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))
Distinct variable groups:   𝑒,𝐴   𝑒,𝑃   𝑒,π‘ˆ   𝑒,𝐽
Proof of Theorem nllyi
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnlly 23295 . . . 4 (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))
21simprbi 496 . . 3 (𝐽 ∈ 𝑛-Locally 𝐴 β†’ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴)
3 pweq 4608 . . . . . . 7 (π‘₯ = π‘ˆ β†’ 𝒫 π‘₯ = 𝒫 π‘ˆ)
43ineq2d 4204 . . . . . 6 (π‘₯ = π‘ˆ β†’ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯) = (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘ˆ))
54rexeqdv 3318 . . . . 5 (π‘₯ = π‘ˆ β†’ (βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴 ↔ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘ˆ)(𝐽 β†Ύt 𝑒) ∈ 𝐴))
65raleqbi1dv 3325 . . . 4 (π‘₯ = π‘ˆ β†’ (βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴 ↔ βˆ€π‘¦ ∈ π‘ˆ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘ˆ)(𝐽 β†Ύt 𝑒) ∈ 𝐴))
76rspccva 3603 . . 3 ((βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴 ∧ π‘ˆ ∈ 𝐽) β†’ βˆ€π‘¦ ∈ π‘ˆ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘ˆ)(𝐽 β†Ύt 𝑒) ∈ 𝐴)
82, 7sylan 579 . 2 ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ π‘ˆ ∈ 𝐽) β†’ βˆ€π‘¦ ∈ π‘ˆ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘ˆ)(𝐽 β†Ύt 𝑒) ∈ 𝐴)
9 elin 3956 . . . . . . 7 (𝑒 ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘ˆ) ↔ (𝑒 ∈ ((neiβ€˜π½)β€˜{𝑦}) ∧ 𝑒 ∈ 𝒫 π‘ˆ))
10 sneq 4630 . . . . . . . . . 10 (𝑦 = 𝑃 β†’ {𝑦} = {𝑃})
1110fveq2d 6885 . . . . . . . . 9 (𝑦 = 𝑃 β†’ ((neiβ€˜π½)β€˜{𝑦}) = ((neiβ€˜π½)β€˜{𝑃}))
1211eleq2d 2811 . . . . . . . 8 (𝑦 = 𝑃 β†’ (𝑒 ∈ ((neiβ€˜π½)β€˜{𝑦}) ↔ 𝑒 ∈ ((neiβ€˜π½)β€˜{𝑃})))
13 velpw 4599 . . . . . . . . 9 (𝑒 ∈ 𝒫 π‘ˆ ↔ 𝑒 βŠ† π‘ˆ)
1413a1i 11 . . . . . . . 8 (𝑦 = 𝑃 β†’ (𝑒 ∈ 𝒫 π‘ˆ ↔ 𝑒 βŠ† π‘ˆ))
1512, 14anbi12d 630 . . . . . . 7 (𝑦 = 𝑃 β†’ ((𝑒 ∈ ((neiβ€˜π½)β€˜{𝑦}) ∧ 𝑒 ∈ 𝒫 π‘ˆ) ↔ (𝑒 ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ 𝑒 βŠ† π‘ˆ)))
169, 15bitrid 283 . . . . . 6 (𝑦 = 𝑃 β†’ (𝑒 ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘ˆ) ↔ (𝑒 ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ 𝑒 βŠ† π‘ˆ)))
1716anbi1d 629 . . . . 5 (𝑦 = 𝑃 β†’ ((𝑒 ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘ˆ) ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴) ↔ ((𝑒 ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ 𝑒 βŠ† π‘ˆ) ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴)))
18 anass 468 . . . . 5 (((𝑒 ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ 𝑒 βŠ† π‘ˆ) ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴) ↔ (𝑒 ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝑒 βŠ† π‘ˆ ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴)))
1917, 18bitrdi 287 . . . 4 (𝑦 = 𝑃 β†’ ((𝑒 ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘ˆ) ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴) ↔ (𝑒 ∈ ((neiβ€˜π½)β€˜{𝑃}) ∧ (𝑒 βŠ† π‘ˆ ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))))
2019rexbidv2 3166 . . 3 (𝑦 = 𝑃 β†’ (βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘ˆ)(𝐽 β†Ύt 𝑒) ∈ 𝐴 ↔ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑒 βŠ† π‘ˆ ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴)))
2120rspccva 3603 . 2 ((βˆ€π‘¦ ∈ π‘ˆ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘ˆ)(𝐽 β†Ύt 𝑒) ∈ 𝐴 ∧ 𝑃 ∈ π‘ˆ) β†’ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑒 βŠ† π‘ˆ ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))
228, 21stoic3 1770 1 ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ π‘ˆ ∈ 𝐽 ∧ 𝑃 ∈ π‘ˆ) β†’ βˆƒπ‘’ ∈ ((neiβ€˜π½)β€˜{𝑃})(𝑒 βŠ† π‘ˆ ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βˆƒwrex 3062   ∩ cin 3939   βŠ† wss 3940  π’« cpw 4594  {csn 4620  β€˜cfv 6533  (class class class)co 7401   β†Ύt crest 17365  Topctop 22717  neicnei 22923  π‘›-Locally cnlly 23291
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 2695
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 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-ov 7404  df-nlly 23293
This theorem is referenced by:  nlly2i  23302  llycmpkgen  23378
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