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Theorem allbutfifvre 44391
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 β„²π‘šπœ‘
allbutfifvre.2 𝑍 = (β„€β‰₯β€˜π‘€)
allbutfifvre.3 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (πΉβ€˜π‘š):dom (πΉβ€˜π‘š)βŸΆβ„)
allbutfifvre.4 𝐷 = βˆͺ 𝑛 ∈ 𝑍 ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š)
allbutfifvre.5 (πœ‘ β†’ 𝑋 ∈ 𝐷)
Assertion
Ref Expression
allbutfifvre (πœ‘ β†’ βˆƒπ‘› ∈ 𝑍 βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)((πΉβ€˜π‘š)β€˜π‘‹) ∈ ℝ)
Distinct variable groups:   π‘š,𝑋,𝑛   π‘š,𝑍   πœ‘,𝑛
Allowed substitution hints:   πœ‘(π‘š)   𝐷(π‘š,𝑛)   𝐹(π‘š,𝑛)   𝑀(π‘š,𝑛)   𝑍(𝑛)

Proof of Theorem allbutfifvre
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 allbutfifvre.5 . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝐷)
2 allbutfifvre.4 . . . 4 𝐷 = βˆͺ 𝑛 ∈ 𝑍 ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š)
31, 2eleqtrdi 2844 . . 3 (πœ‘ β†’ 𝑋 ∈ βˆͺ 𝑛 ∈ 𝑍 ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š))
4 allbutfifvre.2 . . . 4 𝑍 = (β„€β‰₯β€˜π‘€)
5 eqid 2733 . . . 4 βˆͺ 𝑛 ∈ 𝑍 ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) = βˆͺ 𝑛 ∈ 𝑍 ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š)
64, 5allbutfi 44103 . . 3 (𝑋 ∈ βˆͺ 𝑛 ∈ 𝑍 ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ↔ βˆƒπ‘› ∈ 𝑍 βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)𝑋 ∈ dom (πΉβ€˜π‘š))
73, 6sylib 217 . 2 (πœ‘ β†’ βˆƒπ‘› ∈ 𝑍 βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)𝑋 ∈ dom (πΉβ€˜π‘š))
8 allbutfifvre.1 . . . . 5 β„²π‘šπœ‘
9 nfv 1918 . . . . 5 β„²π‘š 𝑛 ∈ 𝑍
108, 9nfan 1903 . . . 4 β„²π‘š(πœ‘ ∧ 𝑛 ∈ 𝑍)
11 simpll 766 . . . . 5 (((πœ‘ ∧ 𝑛 ∈ 𝑍) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ πœ‘)
124uztrn2 12841 . . . . . . . 8 ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘›)) β†’ 𝑗 ∈ 𝑍)
1312ssd 43769 . . . . . . 7 (𝑛 ∈ 𝑍 β†’ (β„€β‰₯β€˜π‘›) βŠ† 𝑍)
1413sselda 3983 . . . . . 6 ((𝑛 ∈ 𝑍 ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘š ∈ 𝑍)
1514adantll 713 . . . . 5 (((πœ‘ ∧ 𝑛 ∈ 𝑍) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘š ∈ 𝑍)
16 allbutfifvre.3 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (πΉβ€˜π‘š):dom (πΉβ€˜π‘š)βŸΆβ„)
1716ffvelcdmda 7087 . . . . . 6 (((πœ‘ ∧ π‘š ∈ 𝑍) ∧ 𝑋 ∈ dom (πΉβ€˜π‘š)) β†’ ((πΉβ€˜π‘š)β€˜π‘‹) ∈ ℝ)
1817ex 414 . . . . 5 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (𝑋 ∈ dom (πΉβ€˜π‘š) β†’ ((πΉβ€˜π‘š)β€˜π‘‹) ∈ ℝ))
1911, 15, 18syl2anc 585 . . . 4 (((πœ‘ ∧ 𝑛 ∈ 𝑍) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ (𝑋 ∈ dom (πΉβ€˜π‘š) β†’ ((πΉβ€˜π‘š)β€˜π‘‹) ∈ ℝ))
2010, 19ralimdaa 3258 . . 3 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)𝑋 ∈ dom (πΉβ€˜π‘š) β†’ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)((πΉβ€˜π‘š)β€˜π‘‹) ∈ ℝ))
2120reximdva 3169 . 2 (πœ‘ β†’ (βˆƒπ‘› ∈ 𝑍 βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)𝑋 ∈ dom (πΉβ€˜π‘š) β†’ βˆƒπ‘› ∈ 𝑍 βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)((πΉβ€˜π‘š)β€˜π‘‹) ∈ ℝ))
227, 21mpd 15 1 (πœ‘ β†’ βˆƒπ‘› ∈ 𝑍 βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)((πΉβ€˜π‘š)β€˜π‘‹) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542  β„²wnf 1786   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  βˆͺ ciun 4998  βˆ© ciin 4999  dom cdm 5677  βŸΆwf 6540  β€˜cfv 6544  β„cr 11109  β„€β‰₯cuz 12822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-pre-lttri 11184  ax-pre-lttrn 11185
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-neg 11447  df-z 12559  df-uz 12823
This theorem is referenced by:  fnlimabslt  44395
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