Theorem utopsnneip 24213

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 6840	. . . . . 6 ⊢ (𝑟 = 𝑝 → ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) = ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝))
3	2	eleq2d 2822	. . . . 5 ⊢ (𝑟 = 𝑝 → (𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ 𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
4	3	cbvralvw 3215	. . . 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 2819	. . . . 5 ⊢ (𝑏 = 𝑎 → (𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝) ↔ 𝑎 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
6	5	raleqbi1dv 3305	. . . 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 283	. . 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 3399	. 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 482	. . . . . . 7 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → 𝑞 = 𝑝)
10	9	sneqd 4579	. . . . . 6 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → {𝑞} = {𝑝})
11	10	imaeq2d 6025	. . . . 5 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑝}))
12	11	mpteq2dva 5178	. . . 4 ⊢ (𝑞 = 𝑝 → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
13	12	rneqd 5893	. . 3 ⊢ (𝑞 = 𝑝 → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
14	13	cbvmptv 5189	. 2 ⊢ (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞}))) = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
15	1, 8, 14	utopsnneiplem 24212	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389  𝒫 cpw 4541  {csn 4567   ↦ cmpt 5166  ran crn 5632   “ cima 5634  ‘cfv 6498  neicnei 23062  UnifOncust 24165  unifTopcutop 24195

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-om 7818  df-1o 8405  df-2o 8406  df-en 8894  df-fin 8897  df-fi 9324  df-top 22859  df-nei 23063  df-ust 24166  df-utop 24196

This theorem is referenced by:  utopsnnei  24214  utopreg  24217  neipcfilu  24260