Theorem utopsnnei 21863

Description: Images of singletons by entourages 𝑉 are neighborhoods of those singletons. (Contributed by Thierry Arnoux, 13-Jan-2018.)

Hypothesis

Ref	Expression
utoptop.1	⊢ 𝐽 = (unifTop‘𝑈)

Assertion

Ref	Expression
utopsnnei	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))

Proof of Theorem utopsnnei

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . . 4 ⊢ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})
2		imaeq1 5380	. . . . . 6 ⊢ (𝑣 = 𝑉 → (𝑣 “ {𝑃}) = (𝑉 “ {𝑃}))
3	2	eqeq2d 2620	. . . . 5 ⊢ (𝑣 = 𝑉 → ((𝑉 “ {𝑃}) = (𝑣 “ {𝑃}) ↔ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})))
4	3	rspcev 3282	. . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})) → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
5	1, 4	mpan2 703	. . 3 ⊢ (𝑉 ∈ 𝑈 → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
6	5	3ad2ant2 1076	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
7		utoptop.1	. . . . . 6 ⊢ 𝐽 = (unifTop‘𝑈)
8	7	utopsnneip 21862	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
9	8	3adant2 1073	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
10	9	eleq2d 2673	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))))
11		imaexg 6995	. . . . 5 ⊢ (𝑉 ∈ 𝑈 → (𝑉 “ {𝑃}) ∈ V)
12		eqid 2610	. . . . . 6 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))
13	12	elrnmpt 5293	. . . . 5 ⊢ ((𝑉 “ {𝑃}) ∈ V → ((𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
14	11, 13	syl 17	. . . 4 ⊢ (𝑉 ∈ 𝑈 → ((𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
15	14	3ad2ant2 1076	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
16	10, 15	bitrd 267	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
17	6, 16	mpbird 246	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))

Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  {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:  utop2nei  21864  utop3cls  21865  utopreg  21866