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Theorem allbutfifvre 41963
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, 2eleqtrdi 2925 . . 3 (𝜑𝑋 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
4 allbutfifvre.2 . . . 4 𝑍 = (ℤ𝑀)
5 eqid 2823 . . . 4 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
64, 5allbutfi 41672 . . 3 (𝑋 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚))
73, 6sylib 220 . 2 (𝜑 → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚))
8 allbutfifvre.1 . . . . 5 𝑚𝜑
9 nfv 1915 . . . . 5 𝑚 𝑛𝑍
108, 9nfan 1900 . . . 4 𝑚(𝜑𝑛𝑍)
11 simpll 765 . . . . 5 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
124uztrn2 12265 . . . . . . . 8 ((𝑛𝑍𝑗 ∈ (ℤ𝑛)) → 𝑗𝑍)
1312ssd 41351 . . . . . . 7 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
1413sselda 3969 . . . . . 6 ((𝑛𝑍𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
1514adantll 712 . . . . 5 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
16 allbutfifvre.3 . . . . . . 7 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
1716ffvelrnda 6853 . . . . . 6 (((𝜑𝑚𝑍) ∧ 𝑋 ∈ dom (𝐹𝑚)) → ((𝐹𝑚)‘𝑋) ∈ ℝ)
1817ex 415 . . . . 5 ((𝜑𝑚𝑍) → (𝑋 ∈ dom (𝐹𝑚) → ((𝐹𝑚)‘𝑋) ∈ ℝ))
1911, 15, 18syl2anc 586 . . . 4 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝑋 ∈ dom (𝐹𝑚) → ((𝐹𝑚)‘𝑋) ∈ ℝ))
2010, 19ralimdaa 3219 . . 3 ((𝜑𝑛𝑍) → (∀𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚) → ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ∈ ℝ))
2120reximdva 3276 . 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 398   = wceq 1537  wnf 1784  wcel 2114  wral 3140  wrex 3141   ciun 4921   ciin 4922  dom cdm 5557  wf 6353  cfv 6357  cr 10538  cuz 12246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-pre-lttri 10613  ax-pre-lttrn 10614
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-neg 10875  df-z 11985  df-uz 12247
This theorem is referenced by:  fnlimabslt  41967
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