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Theorem neifval 12384
 Description: Value of the neighborhood function on the subsets 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
neifval (𝐽 ∈ Top → (nei‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}))
Distinct variable groups:   𝑣,𝑔,𝑥,𝐽   𝑔,𝑋,𝑣,𝑥

Proof of Theorem neifval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . 4 𝑋 = 𝐽
21topopn 12250 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 4114 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 mptexg 5657 . . 3 (𝒫 𝑋 ∈ V → (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}) ∈ V)
52, 3, 43syl 17 . 2 (𝐽 ∈ Top → (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}) ∈ V)
6 unieq 3755 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1eqtr4di 2192 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 3522 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
9 rexeq 2632 . . . . 5 (𝑗 = 𝐽 → (∃𝑔𝑗 (𝑥𝑔𝑔𝑣) ↔ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)))
108, 9rabeqbidv 2686 . . . 4 (𝑗 = 𝐽 → {𝑣 ∈ 𝒫 𝑗 ∣ ∃𝑔𝑗 (𝑥𝑔𝑔𝑣)} = {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)})
118, 10mpteq12dv 4020 . . 3 (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 𝑗 ↦ {𝑣 ∈ 𝒫 𝑗 ∣ ∃𝑔𝑗 (𝑥𝑔𝑔𝑣)}) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}))
12 df-nei 12383 . . 3 nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 ↦ {𝑣 ∈ 𝒫 𝑗 ∣ ∃𝑔𝑗 (𝑥𝑔𝑔𝑣)}))
1311, 12fvmptg 5509 . 2 ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}) ∈ V) → (nei‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}))
145, 13mpdan 418 1 (𝐽 ∈ Top → (nei‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  ∃wrex 2419  {crab 2422  Vcvv 2691   ⊆ wss 3078  𝒫 cpw 3517  ∪ cuni 3746   ↦ cmpt 3999  ‘cfv 5135  Topctop 12239  neicnei 12382 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123  ax-coll 4053  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1738  df-eu 2004  df-mo 2005  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-reu 2425  df-rab 2427  df-v 2693  df-sbc 2916  df-csb 3010  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-iun 3825  df-br 3940  df-opab 4000  df-mpt 4001  df-id 4226  df-xp 4557  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562  df-res 4563  df-ima 4564  df-iota 5100  df-fun 5137  df-fn 5138  df-f 5139  df-f1 5140  df-fo 5141  df-f1o 5142  df-fv 5143  df-top 12240  df-nei 12383 This theorem is referenced by:  neif  12385  neival  12387
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