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Theorem utopsnnei 22831
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 2820 . . . 4 (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})
2 imaeq1 5898 . . . . 5 (𝑣 = 𝑉 → (𝑣 “ {𝑃}) = (𝑉 “ {𝑃}))
32rspceeqv 3617 . . . 4 ((𝑉𝑈 ∧ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})) → ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
41, 3mpan2 689 . . 3 (𝑉𝑈 → ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
543ad2ant2 1130 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
6 utoptop.1 . . . . . 6 𝐽 = (unifTop‘𝑈)
76utopsnneip 22830 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
873adant2 1127 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
98eleq2d 2896 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑉 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))))
10 imaexg 7596 . . . . 5 (𝑉𝑈 → (𝑉 “ {𝑃}) ∈ V)
11 eqid 2820 . . . . . 6 (𝑣𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃}))
1211elrnmpt 5802 . . . . 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 1130 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ((𝑉 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
159, 14bitrd 281 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
165, 15mpbird 259 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  w3a 1083   = wceq 1537  wcel 2114  wrex 3126  Vcvv 3473  {csn 4541  cmpt 5120  ran crn 5530  cima 5532  cfv 6329  neicnei 21678  UnifOncust 22781  unifTopcutop 22812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-pss 3930  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4813  df-int 4851  df-iun 4895  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5434  df-eprel 5439  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6122  df-ord 6168  df-on 6169  df-lim 6170  df-suc 6171  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-ov 7134  df-oprab 7135  df-mpo 7136  df-om 7557  df-wrecs 7923  df-recs 7984  df-rdg 8022  df-1o 8078  df-oadd 8082  df-er 8265  df-en 8486  df-fin 8489  df-fi 8851  df-top 21475  df-nei 21679  df-ust 22782  df-utop 22813
This theorem is referenced by:  utop2nei  22832  utop3cls  22833  utopreg  22834
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