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Theorem utopsnneip 22272
Description: The neighborhoods of a point 𝑃 for the topology induced by an uniform space 𝑈. (Contributed by Thierry Arnoux, 13-Jan-2018.)
Hypothesis
Ref Expression
utoptop.1 𝐽 = (unifTop‘𝑈)
Assertion
Ref Expression
utopsnneip ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
Distinct variable groups:   𝑣,𝑃   𝑣,𝑈   𝑣,𝑋
Allowed substitution hint:   𝐽(𝑣)

Proof of Theorem utopsnneip
Dummy variables 𝑝 𝑎 𝑏 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 utoptop.1 . 2 𝐽 = (unifTop‘𝑈)
2 fveq2 6332 . . . . . 6 (𝑟 = 𝑝 → ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) = ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝))
32eleq2d 2836 . . . . 5 (𝑟 = 𝑝 → (𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ 𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
43cbvralv 3320 . . . 4 (∀𝑟𝑏 𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ ∀𝑝𝑏 𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝))
5 eleq1w 2833 . . . . 5 (𝑏 = 𝑎 → (𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝) ↔ 𝑎 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
65raleqbi1dv 3295 . . . 4 (𝑏 = 𝑎 → (∀𝑝𝑏 𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝) ↔ ∀𝑝𝑎 𝑎 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
74, 6syl5bb 272 . . 3 (𝑏 = 𝑎 → (∀𝑟𝑏 𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ ∀𝑝𝑎 𝑎 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
87cbvrabv 3349 . 2 {𝑏 ∈ 𝒫 𝑋 ∣ ∀𝑟𝑏 𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)}
9 simpl 468 . . . . . . 7 ((𝑞 = 𝑝𝑣𝑈) → 𝑞 = 𝑝)
109sneqd 4328 . . . . . 6 ((𝑞 = 𝑝𝑣𝑈) → {𝑞} = {𝑝})
1110imaeq2d 5607 . . . . 5 ((𝑞 = 𝑝𝑣𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑝}))
1211mpteq2dva 4878 . . . 4 (𝑞 = 𝑝 → (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1312rneqd 5491 . . 3 (𝑞 = 𝑝 → ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1413cbvmptv 4884 . 2 (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞}))) = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
151, 8, 14utopsnneiplem 22271 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  𝒫 cpw 4297  {csn 4316  cmpt 4863  ran crn 5250  cima 5252  cfv 6031  neicnei 21122  UnifOncust 22223  unifTopcutop 22254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-fin 8113  df-fi 8473  df-top 20919  df-nei 21123  df-ust 22224  df-utop 22255
This theorem is referenced by:  utopsnnei  22273  utopreg  22276  neipcfilu  22320
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