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Theorem neival 22957
Description: Value of the set of neighborhoods of a subset of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
neifval.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
neival ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((neiβ€˜π½)β€˜π‘†) = {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)})
Distinct variable groups:   𝑣,𝑔,𝐽   𝑆,𝑔,𝑣   𝑔,𝑋,𝑣

Proof of Theorem neival
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . . 5 𝑋 = βˆͺ 𝐽
21neifval 22954 . . . 4 (𝐽 ∈ Top β†’ (neiβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)}))
32fveq1d 6886 . . 3 (𝐽 ∈ Top β†’ ((neiβ€˜π½)β€˜π‘†) = ((π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)})β€˜π‘†))
43adantr 480 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((neiβ€˜π½)β€˜π‘†) = ((π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)})β€˜π‘†))
5 eqid 2726 . . 3 (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)}) = (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)})
6 cleq1lem 14933 . . . . 5 (π‘₯ = 𝑆 β†’ ((π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣) ↔ (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)))
76rexbidv 3172 . . . 4 (π‘₯ = 𝑆 β†’ (βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣) ↔ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)))
87rabbidv 3434 . . 3 (π‘₯ = 𝑆 β†’ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} = {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)})
91topopn 22759 . . . . 5 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
10 elpw2g 5337 . . . . 5 (𝑋 ∈ 𝐽 β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
119, 10syl 17 . . . 4 (𝐽 ∈ Top β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
1211biimpar 477 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 ∈ 𝒫 𝑋)
13 pwexg 5369 . . . . 5 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
14 rabexg 5324 . . . . 5 (𝒫 𝑋 ∈ V β†’ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ∈ V)
159, 13, 143syl 18 . . . 4 (𝐽 ∈ Top β†’ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ∈ V)
1615adantr 480 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ∈ V)
175, 8, 12, 16fvmptd3 7014 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)})β€˜π‘†) = {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)})
184, 17eqtrd 2766 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((neiβ€˜π½)β€˜π‘†) = {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064  {crab 3426  Vcvv 3468   βŠ† wss 3943  π’« cpw 4597  βˆͺ cuni 4902   ↦ cmpt 5224  β€˜cfv 6536  Topctop 22746  neicnei 22952
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
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-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 22747  df-nei 22953
This theorem is referenced by:  isnei  22958
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