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Theorem utopsnnei 24274
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) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))

Proof of Theorem utopsnnei
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . . 4 (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})
2 imaeq1 6075 . . . . 5 (𝑣 = 𝑉 → (𝑣 “ {𝑃}) = (𝑉 “ {𝑃}))
32rspceeqv 3645 . . . 4 ((𝑉𝑈 ∧ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})) → ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
41, 3mpan2 691 . . 3 (𝑉𝑈 → ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
543ad2ant2 1133 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
6 utoptop.1 . . . . . 6 𝐽 = (unifTop‘𝑈)
76utopsnneip 24273 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
873adant2 1130 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
98eleq2d 2825 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑉 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))))
10 imaexg 7936 . . . . 5 (𝑉𝑈 → (𝑉 “ {𝑃}) ∈ V)
11 eqid 2735 . . . . . 6 (𝑣𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃}))
1211elrnmpt 5972 . . . . 5 ((𝑉 “ {𝑃}) ∈ V → ((𝑉 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
1310, 12syl 17 . . . 4 (𝑉𝑈 → ((𝑉 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
14133ad2ant2 1133 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ((𝑉 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
159, 14bitrd 279 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
165, 15mpbird 257 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  {csn 4631  cmpt 5231  ran crn 5690  cima 5692  cfv 6563  neicnei 23121  UnifOncust 24224  unifTopcutop 24255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-2o 8506  df-en 8985  df-fin 8988  df-fi 9449  df-top 22916  df-nei 23122  df-ust 24225  df-utop 24256
This theorem is referenced by:  utop2nei  24275  utop3cls  24276  utopreg  24277
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