Theorem utopsnneip 24362

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 6871	. . . . . 6 ⊢ (𝑟 = 𝑝 → ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) = ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝))
3	2	eleq2d 2851	. . . . 5 ⊢ (𝑟 = 𝑝 → (𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ 𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
4	3	cbvralvw 3243	. . . 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 2848	. . . . 5 ⊢ (𝑏 = 𝑎 → (𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝) ↔ 𝑎 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
6	5	raleqbi1dv 3333	. . . 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 3427	. 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 487	. . . . . . 7 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → 𝑞 = 𝑝)
10	9	sneqd 4597	. . . . . 6 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → {𝑞} = {𝑝})
11	10	imaeq2d 6052	. . . . 5 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑝}))
12	11	mpteq2dva 5197	. . . 4 ⊢ (𝑞 = 𝑝 → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
13	12	rneqd 5918	. . 3 ⊢ (𝑞 = 𝑝 → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
14	13	cbvmptv 5208	. 2 ⊢ (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞}))) = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
15	1, 8, 14	utopsnneiplem 24361	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  {crab 3417  𝒫 cpw 4558  {csn 4585   ↦ cmpt 5185  ran crn 5652   “ cima 5654  ‘cfv 6525  neicnei 23211  UnifOncust 24314  unifTopcutop 24344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-om 7851  df-1o 8441  df-2o 8442  df-en 8932  df-fin 8935  df-fi 9359  df-top 23008  df-nei 23212  df-ust 24315  df-utop 24345

This theorem is referenced by:  utopsnnei  24363  utopreg  24366  neipcfilu  24409