Theorem liminfval 46453

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	⊢ (𝐹 ∈ 𝑉 → (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.

Step	Hyp	Ref	Expression
1		df-liminf 46446	. 2 ⊢ lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
2		imaeq1 6059	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → (𝑥 “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞)))
3	2	ineq1d 4173	. . . . . . 7 ⊢ (𝑥 = 𝐹 → ((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
4	3	infeq1d 9439	. . . . . 6 ⊢ (𝑥 = 𝐹 → inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
5	4	mpteq2dv 5206	. . . . 5 ⊢ (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )))
6		liminfval.1	. . . . . 6 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
7	6	a1i 11	. . . . 5 ⊢ (𝑥 = 𝐹 → 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )))
8	5, 7	eqtr4d 2801	. . . 4 ⊢ (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺)
9	8	rneqd 5930	. . 3 ⊢ (𝑥 = 𝐹 → ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺)
10	9	supeq1d 9407	. 2 ⊢ (𝑥 = 𝐹 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = sup(ran 𝐺, ℝ*, < ))
11		elex 3476	. 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
12		xrltso 13167	. . . 4 ⊢ < Or ℝ*
13	12	supex 9425	. . 3 ⊢ sup(ran 𝐺, ℝ*, < ) ∈ V
14	13	a1i 11	. 2 ⊢ (𝐹 ∈ 𝑉 → sup(ran 𝐺, ℝ*, < ) ∈ V)
15	1, 10, 11, 14	fvmptd3 7015	1 ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∩ cin 3905   ↦ cmpt 5193  ran crn 5664   “ cima 5666  ‘cfv 6538  (class class class)co 7412  supcsup 9401  infcinf 9402  ℝcr 11100  +∞cpnf 11241  ℝ^*cxr 11243   < clt 11244  [,)cico 13375  lim infclsi 46445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-pre-lttri 11175  ax-pre-lttrn 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-po 5571  df-so 5572  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-sup 9403  df-inf 9404  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-liminf 46446

This theorem is referenced by:  liminfcl  46457  liminfvald  46458  liminfval5  46459  liminfresxr  46461  liminfval2  46462  liminfvalxr  46477