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Theorem liminfval 40309
Description: The inferior limit of a set 𝐹. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
liminfval.1 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
Assertion
Ref Expression
liminfval (𝐹𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
Distinct variable group:   𝑘,𝐹
Allowed substitution hints:   𝐺(𝑘)   𝑉(𝑘)

Proof of Theorem liminfval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-liminf 40302 . 2 lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
2 imaeq1 5496 . . . . . . . . 9 (𝑥 = 𝐹 → (𝑥 “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞)))
32ineq1d 3846 . . . . . . . 8 (𝑥 = 𝐹 → ((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
43infeq1d 8424 . . . . . . 7 (𝑥 = 𝐹 → inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
54mpteq2dv 4778 . . . . . 6 (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )))
6 liminfval.1 . . . . . . 7 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
76a1i 11 . . . . . 6 (𝑥 = 𝐹𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )))
85, 7eqtr4d 2688 . . . . 5 (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺)
98rneqd 5385 . . . 4 (𝑥 = 𝐹 → ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺)
109supeq1d 8393 . . 3 (𝑥 = 𝐹 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = sup(ran 𝐺, ℝ*, < ))
1110adantl 481 . 2 ((𝐹𝑉𝑥 = 𝐹) → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = sup(ran 𝐺, ℝ*, < ))
12 elex 3243 . 2 (𝐹𝑉𝐹 ∈ V)
13 xrltso 12012 . . . 4 < Or ℝ*
1413supex 8410 . . 3 sup(ran 𝐺, ℝ*, < ) ∈ V
1514a1i 11 . 2 (𝐹𝑉 → sup(ran 𝐺, ℝ*, < ) ∈ V)
161, 11, 12, 15fvmptd2 39759 1 (𝐹𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  cmpt 4762  ran crn 5144  cima 5146  cfv 5926  (class class class)co 6690  supcsup 8387  infcinf 8388  cr 9973  +∞cpnf 10109  *cxr 10111   < clt 10112  [,)cico 12215  lim infclsi 40301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-liminf 40302
This theorem is referenced by:  liminfcl  40313  liminfvald  40314  liminfval5  40315  liminfresxr  40317  liminfval2  40318  liminfvalxr  40333
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