Theorem utopsnneip 22253

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 6352	. . . . . 6 ⊢ (𝑟 = 𝑝 → ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) = ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝))
3	2	eleq2d 2825	. . . . 5 ⊢ (𝑟 = 𝑝 → (𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ 𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
4	3	cbvralv 3310	. . . 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 2822	. . . . 5 ⊢ (𝑏 = 𝑎 → (𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝) ↔ 𝑎 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
6	5	raleqbi1dv 3285	. . . 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	syl5bb 272	. . 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 3339	. 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 474	. . . . . . 7 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → 𝑞 = 𝑝)
10	9	sneqd 4333	. . . . . 6 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → {𝑞} = {𝑝})
11	10	imaeq2d 5624	. . . . 5 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑝}))
12	11	mpteq2dva 4896	. . . 4 ⊢ (𝑞 = 𝑝 → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
13	12	rneqd 5508	. . 3 ⊢ (𝑞 = 𝑝 → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
14	13	cbvmptv 4902	. 2 ⊢ (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞}))) = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
15	1, 8, 14	utopsnneiplem 22252	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  {crab 3054  𝒫 cpw 4302  {csn 4321   ↦ cmpt 4881  ran crn 5267   “ cima 5269  ‘cfv 6049  neicnei 21103  UnifOncust 22204  unifTopcutop 22235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-fin 8125  df-fi 8482  df-top 20901  df-nei 21104  df-ust 22205  df-utop 22236

This theorem is referenced by:  utopsnnei  22254  utopreg  22257  neipcfilu  22301