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Theorem neival 20816
Description: 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 = 𝐽
Assertion
Ref Expression
neival ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) = {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)})
Distinct variable groups:   𝑣,𝑔,𝐽   𝑆,𝑔,𝑣   𝑔,𝑋,𝑣

Proof of Theorem neival
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . . 5 𝑋 = 𝐽
21neifval 20813 . . . 4 (𝐽 ∈ Top → (nei‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}))
32fveq1d 6150 . . 3 (𝐽 ∈ Top → ((nei‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)})‘𝑆))
43adantr 481 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)})‘𝑆))
51topopn 20636 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
6 elpw2g 4787 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
75, 6syl 17 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
87biimpar 502 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
9 pwexg 4810 . . . . 5 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
10 rabexg 4772 . . . . 5 (𝒫 𝑋 ∈ V → {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)} ∈ V)
115, 9, 103syl 18 . . . 4 (𝐽 ∈ Top → {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)} ∈ V)
1211adantr 481 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)} ∈ V)
13 sseq1 3605 . . . . . . 7 (𝑥 = 𝑆 → (𝑥𝑔𝑆𝑔))
1413anbi1d 740 . . . . . 6 (𝑥 = 𝑆 → ((𝑥𝑔𝑔𝑣) ↔ (𝑆𝑔𝑔𝑣)))
1514rexbidv 3045 . . . . 5 (𝑥 = 𝑆 → (∃𝑔𝐽 (𝑥𝑔𝑔𝑣) ↔ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)))
1615rabbidv 3177 . . . 4 (𝑥 = 𝑆 → {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)} = {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)})
17 eqid 2621 . . . 4 (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)})
1816, 17fvmptg 6237 . . 3 ((𝑆 ∈ 𝒫 𝑋 ∧ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)} ∈ V) → ((𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)})‘𝑆) = {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)})
198, 12, 18syl2anc 692 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)})‘𝑆) = {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)})
204, 19eqtrd 2655 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) = {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130   cuni 4402  cmpt 4673  cfv 5847  Topctop 20617  neicnei 20811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-top 20621  df-nei 20812
This theorem is referenced by:  isnei  20817
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