Theorem utopsnneip 21862

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 6103	. . . . . 6 ⊢ (𝑟 = 𝑝 → ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) = ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝))
3	2	eleq2d 2673	. . . . 5 ⊢ (𝑟 = 𝑝 → (𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ 𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
4	3	cbvralv 3147	. . . 4 ⊢ (∀𝑟 ∈ 𝑏 𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑟) ↔ ∀𝑝 ∈ 𝑏 𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝))
5		eleq1 2676	. . . . 5 ⊢ (𝑏 = 𝑎 → (𝑏 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝) ↔ 𝑎 ∈ ((𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))‘𝑝)))
6	5	raleqbi1dv 3123	. . . 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 271	. . 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 3172	. 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 472	. . . . . . 7 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → 𝑞 = 𝑝)
10	9	sneqd 4137	. . . . . 6 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → {𝑞} = {𝑝})
11	10	imaeq2d 5385	. . . . 5 ⊢ ((𝑞 = 𝑝 ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑝}))
12	11	mpteq2dva 4672	. . . 4 ⊢ (𝑞 = 𝑝 → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
13	12	rneqd 5274	. . 3 ⊢ (𝑞 = 𝑝 → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
14	13	cbvmptv 4678	. 2 ⊢ (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞}))) = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
15	1, 8, 14	utopsnneiplem 21861	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  𝒫 cpw 4108  {csn 4125   ↦ cmpt 4643  ran crn 5039   “ cima 5041  ‘cfv 5804  neicnei 20711  UnifOncust 21813  unifTopcutop 21844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-top 20521  df-nei 20712  df-ust 21814  df-utop 21845

This theorem is referenced by:  utopsnnei  21863  utopreg  21866  neipcfilu  21910