Theorem allbutfifvre 45582

Description: Given a sequence of real-valued functions, and 𝑋 that belongs to all but finitely many domains, then its function value is ultimately a real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
allbutfifvre.1	⊢ Ⅎ𝑚𝜑
allbutfifvre.2	⊢ 𝑍 = (ℤ≥‘𝑀)
allbutfifvre.3	⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
allbutfifvre.4	⊢ 𝐷 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
allbutfifvre.5	⊢ (𝜑 → 𝑋 ∈ 𝐷)

Assertion

Ref	Expression
allbutfifvre	⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ)

Distinct variable groups:   𝑚,𝑋,𝑛   𝑚,𝑍   𝜑,𝑛

Allowed substitution hints:   𝜑(𝑚)   𝐷(𝑚,𝑛)   𝐹(𝑚,𝑛)   𝑀(𝑚,𝑛)   𝑍(𝑛)

Proof of Theorem allbutfifvre

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		allbutfifvre.5	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
2		allbutfifvre.4	. . . 4 ⊢ 𝐷 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
3	1, 2	eleqtrdi 2848	. . 3 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚))
4		allbutfifvre.2	. . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀)
5		eqid 2734	. . . 4 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
6	4, 5	allbutfi 45294	. . 3 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚))
7	3, 6	sylib 218	. 2 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚))
8		allbutfifvre.1	. . . . 5 ⊢ Ⅎ𝑚𝜑
9		nfv 1913	. . . . 5 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
10	8, 9	nfan 1898	. . . 4 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
11		simpll 766	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑)
12	4	uztrn2 12927	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍)
13	12	ssd 44968	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍)
14	13	sselda 4008	. . . . . 6 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
15	14	adantll 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
16		allbutfifvre.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
17	16	ffvelcdmda 7124	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑋 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
18	17	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑋 ∈ dom (𝐹‘𝑚) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ))
19	11, 15, 18	syl2anc 583	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝑋 ∈ dom (𝐹‘𝑚) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ))
20	10, 19	ralimdaa 3262	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚) → ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ))
21	20	reximdva 3170	. 2 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚) → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ))
22	7, 21	mpd 15	1 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2103  ∀wral 3063  ∃wrex 3072  ∪ ciun 5025  ∩ ciin 5026  dom cdm 5706  ⟶wf 6575  ‘cfv 6579  ℝcr 11188  ℤ_≥cuz 12908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5327  ax-nul 5334  ax-pow 5393  ax-pr 5457  ax-un 7775  ax-cnex 11245  ax-resscn 11246  ax-pre-lttri 11263  ax-pre-lttrn 11264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rab 3440  df-v 3486  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4354  df-if 4555  df-pw 4630  df-sn 4655  df-pr 4657  df-op 4661  df-uni 4938  df-iun 5027  df-iin 5028  df-br 5177  df-opab 5239  df-mpt 5260  df-id 5604  df-xp 5712  df-rel 5713  df-cnv 5714  df-co 5715  df-dm 5716  df-rn 5717  df-res 5718  df-ima 5719  df-iota 6531  df-fun 6581  df-fn 6582  df-f 6583  df-f1 6584  df-fo 6585  df-f1o 6586  df-fv 6587  df-ov 7457  df-er 8768  df-en 9009  df-dom 9010  df-sdom 9011  df-pnf 11331  df-mnf 11332  df-xr 11333  df-ltxr 11334  df-le 11335  df-neg 11528  df-z 12645  df-uz 12909

This theorem is referenced by:  fnlimabslt  45586