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Theorem allbutfifvre 39343
 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 𝑚𝜑
allbutfifvre.2 𝑍 = (ℤ𝑀)
allbutfifvre.3 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
allbutfifvre.4 𝐷 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
allbutfifvre.5 (𝜑𝑋𝐷)
Assertion
Ref Expression
allbutfifvre (𝜑 → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘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.
StepHypRef Expression
1 allbutfifvre.5 . . . 4 (𝜑𝑋𝐷)
2 allbutfifvre.4 . . . 4 𝐷 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
31, 2syl6eleq 2708 . . 3 (𝜑𝑋 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
4 allbutfifvre.2 . . . 4 𝑍 = (ℤ𝑀)
5 eqid 2621 . . . 4 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
64, 5allbutfi 39115 . . 3 (𝑋 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚))
73, 6sylib 208 . 2 (𝜑 → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚))
8 allbutfifvre.1 . . . . 5 𝑚𝜑
9 nfv 1840 . . . . 5 𝑚 𝑛𝑍
108, 9nfan 1825 . . . 4 𝑚(𝜑𝑛𝑍)
11 simpll 789 . . . . 5 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
124uztrn2 11665 . . . . . . . 8 ((𝑛𝑍𝑗 ∈ (ℤ𝑛)) → 𝑗𝑍)
1312ssd 38774 . . . . . . 7 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
1413sselda 3588 . . . . . 6 ((𝑛𝑍𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
1514adantll 749 . . . . 5 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
16 allbutfifvre.3 . . . . . . 7 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
1716ffvelrnda 6325 . . . . . 6 (((𝜑𝑚𝑍) ∧ 𝑋 ∈ dom (𝐹𝑚)) → ((𝐹𝑚)‘𝑋) ∈ ℝ)
1817ex 450 . . . . 5 ((𝜑𝑚𝑍) → (𝑋 ∈ dom (𝐹𝑚) → ((𝐹𝑚)‘𝑋) ∈ ℝ))
1911, 15, 18syl2anc 692 . . . 4 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝑋 ∈ dom (𝐹𝑚) → ((𝐹𝑚)‘𝑋) ∈ ℝ))
2010, 19ralimdaa 2954 . . 3 ((𝜑𝑛𝑍) → (∀𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚) → ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ∈ ℝ))
2120reximdva 3013 . 2 (𝜑 → (∃𝑛𝑍𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚) → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ∈ ℝ))
227, 21mpd 15 1 (𝜑 → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ∈ ℝ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480  Ⅎwnf 1705   ∈ wcel 1987  ∀wral 2908  ∃wrex 2909  ∪ ciun 4492  ∩ ciin 4493  dom cdm 5084  ⟶wf 5853  ‘cfv 5857  ℝcr 9895  ℤ≥cuz 11647 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-pre-lttri 9970  ax-pre-lttrn 9971 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-neg 10229  df-z 11338  df-uz 11648 This theorem is referenced by:  fnlimabslt  39347
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