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Theorem neival 23024
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 23021 . . . 4 (𝐽 ∈ Top β†’ (neiβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)}))
32fveq1d 6902 . . 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 479 . 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 2727 . . 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 14967 . . . . 5 (π‘₯ = 𝑆 β†’ ((π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣) ↔ (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)))
76rexbidv 3174 . . . 4 (π‘₯ = 𝑆 β†’ (βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣) ↔ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)))
87rabbidv 3436 . . 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 22826 . . . . 5 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
10 elpw2g 5348 . . . . 5 (𝑋 ∈ 𝐽 β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
119, 10syl 17 . . . 4 (𝐽 ∈ Top β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
1211biimpar 476 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 ∈ 𝒫 𝑋)
13 pwexg 5380 . . . . 5 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
14 rabexg 5335 . . . . 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 479 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ∈ V)
175, 8, 12, 16fvmptd3 7031 . 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 2767 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 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3066  {crab 3428  Vcvv 3471   βŠ† wss 3947  π’« cpw 4604  βˆͺ cuni 4910   ↦ cmpt 5233  β€˜cfv 6551  Topctop 22813  neicnei 23019
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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-top 22814  df-nei 23020
This theorem is referenced by:  isnei  23025
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