Theorem utopsnnei 23309

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 2738	. . . 4 ⊢ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})
2		imaeq1 5953	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑣 “ {𝑃}) = (𝑉 “ {𝑃}))
3	2	rspceeqv 3567	. . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})) → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
4	1, 3	mpan2 687	. . 3 ⊢ (𝑉 ∈ 𝑈 → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
5	4	3ad2ant2 1132	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
6		utoptop.1	. . . . . 6 ⊢ 𝐽 = (unifTop‘𝑈)
7	6	utopsnneip 23308	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
8	7	3adant2 1129	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
9	8	eleq2d 2824	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))))
10		imaexg 7736	. . . . 5 ⊢ (𝑉 ∈ 𝑈 → (𝑉 “ {𝑃}) ∈ V)
11		eqid 2738	. . . . . 6 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))
12	11	elrnmpt 5854	. . . . 5 ⊢ ((𝑉 “ {𝑃}) ∈ V → ((𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
13	10, 12	syl 17	. . . 4 ⊢ (𝑉 ∈ 𝑈 → ((𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
14	13	3ad2ant2 1132	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
15	9, 14	bitrd 278	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
16	5, 15	mpbird 256	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422  {csn 4558   ↦ cmpt 5153  ran crn 5581   “ cima 5583  ‘cfv 6418  neicnei 22156  UnifOncust 23259  unifTopcutop 23290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-er 8456  df-en 8692  df-fin 8695  df-fi 9100  df-top 21951  df-nei 22157  df-ust 23260  df-utop 23291

This theorem is referenced by:  utop2nei  23310  utop3cls  23311  utopreg  23312