MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  utopsnnei Structured version   Visualization version   GIF version

Theorem utopsnnei 22273
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 2771 . . . 4 (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})
2 imaeq1 5602 . . . . . 6 (𝑣 = 𝑉 → (𝑣 “ {𝑃}) = (𝑉 “ {𝑃}))
32eqeq2d 2781 . . . . 5 (𝑣 = 𝑉 → ((𝑉 “ {𝑃}) = (𝑣 “ {𝑃}) ↔ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})))
43rspcev 3460 . . . 4 ((𝑉𝑈 ∧ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})) → ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
51, 4mpan2 671 . . 3 (𝑉𝑈 → ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
653ad2ant2 1128 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
7 utoptop.1 . . . . . 6 𝐽 = (unifTop‘𝑈)
87utopsnneip 22272 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
983adant2 1125 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
109eleq2d 2836 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑉 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))))
11 imaexg 7250 . . . . 5 (𝑉𝑈 → (𝑉 “ {𝑃}) ∈ V)
12 eqid 2771 . . . . . 6 (𝑣𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃}))
1312elrnmpt 5510 . . . . 5 ((𝑉 “ {𝑃}) ∈ V → ((𝑉 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
1411, 13syl 17 . . . 4 (𝑉𝑈 → ((𝑉 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
15143ad2ant2 1128 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ((𝑉 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
1610, 15bitrd 268 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ ∃𝑣𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
176, 16mpbird 247 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1071   = wceq 1631  wcel 2145  wrex 3062  Vcvv 3351  {csn 4316  cmpt 4863  ran crn 5250  cima 5252  cfv 6031  neicnei 21122  UnifOncust 22223  unifTopcutop 22254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-fin 8113  df-fi 8473  df-top 20919  df-nei 21123  df-ust 22224  df-utop 22255
This theorem is referenced by:  utop2nei  22274  utop3cls  22275  utopreg  22276
  Copyright terms: Public domain W3C validator