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Theorem utopsnneip 22857
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 6649 . . . . . 6 (𝑟 = 𝑝 → ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) = ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝))
32eleq2d 2878 . . . . 5 (𝑟 = 𝑝 → (𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ 𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
43cbvralvw 3399 . . . 4 (∀𝑟𝑏 𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ ∀𝑝𝑏 𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝))
5 eleq1w 2875 . . . . 5 (𝑏 = 𝑎 → (𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝) ↔ 𝑎 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
65raleqbi1dv 3359 . . . 4 (𝑏 = 𝑎 → (∀𝑝𝑏 𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝) ↔ ∀𝑝𝑎 𝑎 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
74, 6syl5bb 286 . . 3 (𝑏 = 𝑎 → (∀𝑟𝑏 𝑏 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ ∀𝑝𝑎 𝑎 ∈ ((𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
87cbvrabv 3442 . 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 486 . . . . . . 7 ((𝑞 = 𝑝𝑣𝑈) → 𝑞 = 𝑝)
109sneqd 4540 . . . . . 6 ((𝑞 = 𝑝𝑣𝑈) → {𝑞} = {𝑝})
1110imaeq2d 5900 . . . . 5 ((𝑞 = 𝑝𝑣𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑝}))
1211mpteq2dva 5128 . . . 4 (𝑞 = 𝑝 → (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1312rneqd 5776 . . 3 (𝑞 = 𝑝 → ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1413cbvmptv 5136 . 2 (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞}))) = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
151, 8, 14utopsnneiplem 22856 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  {crab 3113  𝒫 cpw 4500  {csn 4528  cmpt 5113  ran crn 5524  cima 5526  cfv 6328  neicnei 21705  UnifOncust 22808  unifTopcutop 22839
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-fin 8500  df-fi 8863  df-top 21502  df-nei 21706  df-ust 22809  df-utop 22840
This theorem is referenced by:  utopsnnei  22858  utopreg  22861  neipcfilu  22905
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