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Theorem neival 13728
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 13725 . . . 4 (𝐽 ∈ Top β†’ (neiβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)}))
32fveq1d 5519 . . 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 276 . 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‘†))
51topopn 13593 . . . . 5 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
6 elpw2g 4158 . . . . 5 (𝑋 ∈ 𝐽 β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
75, 6syl 14 . . . 4 (𝐽 ∈ Top β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
87biimpar 297 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 ∈ 𝒫 𝑋)
9 pwexg 4182 . . . . 5 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
10 rabexg 4148 . . . . 5 (𝒫 𝑋 ∈ V β†’ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ∈ V)
115, 9, 103syl 17 . . . 4 (𝐽 ∈ Top β†’ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ∈ V)
1211adantr 276 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ∈ V)
13 sseq1 3180 . . . . . . 7 (π‘₯ = 𝑆 β†’ (π‘₯ βŠ† 𝑔 ↔ 𝑆 βŠ† 𝑔))
1413anbi1d 465 . . . . . 6 (π‘₯ = 𝑆 β†’ ((π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣) ↔ (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)))
1514rexbidv 2478 . . . . 5 (π‘₯ = 𝑆 β†’ (βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣) ↔ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)))
1615rabbidv 2728 . . . 4 (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)})
17 eqid 2177 . . . 4 (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)})
1816, 17fvmptg 5594 . . 3 ((𝑆 ∈ 𝒫 𝑋 ∧ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (𝑆 βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ∈ V) 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)})
198, 12, 18syl2anc 411 . 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)})
204, 19eqtrd 2210 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 set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  βˆƒwrex 2456  {crab 2459  Vcvv 2739   βŠ† wss 3131  π’« cpw 3577  βˆͺ cuni 3811   ↦ cmpt 4066  β€˜cfv 5218  Topctop 13582  neicnei 13723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-top 13583  df-nei 13724
This theorem is referenced by:  isnei  13729
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