Theorem utopsnneip 24474

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) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (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.

Step	Hyp	Ref	Expression
1		utoptop.1	. 2 ⊢ 𝐽 = (unifTop‘𝑈)
2		fveq2 6878	. . . . . 6 ⊢ (𝑟 = 𝑝 → ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) = ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝))
3	2	eleq2d 2846	. . . . 5 ⊢ (𝑟 = 𝑝 → (𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ 𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
4	3	cbvralvw 3240	. . . 4 ⊢ (∀𝑟 ∈ 𝑏 𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ ∀𝑝 ∈ 𝑏 𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝))
5		eleq1w 2843	. . . . 5 ⊢ (𝑏 = 𝑎 → (𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝) ↔ 𝑎 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
6	5	raleqbi1dv 3329	. . . 4 ⊢ (𝑏 = 𝑎 → (∀𝑝 ∈ 𝑏 𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝) ↔ ∀𝑝 ∈ 𝑎 𝑎 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
7	4, 6	bitrid 286	. . 3 ⊢ (𝑏 = 𝑎 → (∀𝑟 ∈ 𝑏 𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ ∀𝑝 ∈ 𝑎 𝑎 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
8	7	cbvrabv 3422	. 2 ⊢ {𝑏 ∈ 𝒫 𝑋 ∣ ∀𝑟 ∈ 𝑏 𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)}
9		simpl 488	. . . . . . 7 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → 𝑞 = 𝑝)
10	9	sneqd 4596	. . . . . 6 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → {𝑞} = {𝑝})
11	10	imaeq2d 6056	. . . . 5 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑝}))
12	11	mpteq2dva 5198	. . . 4 ⊢ (𝑞 = 𝑝 → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
13	12	rneqd 5922	. . 3 ⊢ (𝑞 = 𝑝 → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
14	13	cbvmptv 5209	. 2 ⊢ (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞}))) = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
15	1, 8, 14	utopsnneiplem 24473	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  𝒫 cpw 4557  {csn 4584   ↦ cmpt 5186  ran crn 5656   “ cima 5658  ‘cfv 6533  neicnei 23322  UnifOncust 24426  unifTopcutop 24456

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-om 7863  df-1o 8455  df-2o 8456  df-en 8953  df-fin 8956  df-fi 9381  df-top 23119  df-nei 23323  df-ust 24427  df-utop 24457

This theorem is used by:  utopsnnei  24475  utopreg  24478  neipcfilu  24521