Theorem allbutfifvre 46280

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 2879	. . 3 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚))
4		allbutfifvre.2	. . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀)
5		eqid 2769	. . . 4 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
6	4, 5	allbutfi 45999	. . 3 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚))
7	3, 6	sylib 221	. 2 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚))
8		allbutfifvre.1	. . . . 5 ⊢ Ⅎ𝑚𝜑
9		nfv 1941	. . . . 5 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
10	8, 9	nfan 1926	. . . 4 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
11		simpll 778	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑)
12	4	uztrn2 12880	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍)
13	12	ssd 45691	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍)
14	13	sselda 3945	. . . . . 6 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
15	14	adantll 726	. . . . 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 7080	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑋 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
18	17	ex 417	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑋 ∈ dom (𝐹‘𝑚) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ))
19	11, 15, 18	syl2anc 595	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝑋 ∈ dom (𝐹‘𝑚) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ))
20	10, 19	ralimdaa 3272	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚) → ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ))
21	20	reximdva 3184	. 2 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚) → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ))
22	7, 21	mpd 16	1 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567  Ⅎwnf 1810   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  ∪ ciun 4960  ∩ ciin 4961  dom cdm 5662  ⟶wf 6533  ‘cfv 6537  ℝcr 11098  ℤ_≥cuz 12861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-pre-lttri 11173  ax-pre-lttrn 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-neg 11443  df-z 12591  df-uz 12862

This theorem is referenced by:  fnlimabslt  46284