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Theorem neival 21745
 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 = 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 21742 . . . 4 (𝐽 ∈ Top → (nei‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}))
32fveq1d 6654 . . 3 (𝐽 ∈ Top → ((nei‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)})‘𝑆))
43adantr 484 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)})‘𝑆))
5 eqid 2798 . . 3 (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)})
6 cleq1lem 14350 . . . . 5 (𝑥 = 𝑆 → ((𝑥𝑔𝑔𝑣) ↔ (𝑆𝑔𝑔𝑣)))
76rexbidv 3256 . . . 4 (𝑥 = 𝑆 → (∃𝑔𝐽 (𝑥𝑔𝑔𝑣) ↔ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)))
87rabbidv 3427 . . 3 (𝑥 = 𝑆 → {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)} = {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)})
91topopn 21549 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
10 elpw2g 5214 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
119, 10syl 17 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1211biimpar 481 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
13 pwexg 5247 . . . . 5 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
14 rabexg 5201 . . . . 5 (𝒫 𝑋 ∈ V → {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)} ∈ V)
159, 13, 143syl 18 . . . 4 (𝐽 ∈ Top → {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)} ∈ V)
1615adantr 484 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)} ∈ V)
175, 8, 12, 16fvmptd3 6775 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)})‘𝑆) = {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)})
184, 17eqtrd 2833 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((nei‘𝐽)‘𝑆) = {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑆𝑔𝑔𝑣)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  {crab 3110  Vcvv 3441   ⊆ wss 3882  𝒫 cpw 4499  ∪ cuni 4803   ↦ cmpt 5113  ‘cfv 6329  Topctop 21536  neicnei 21740 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-top 21537  df-nei 21741 This theorem is referenced by:  isnei  21746
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