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Theorem allbutfifvre 44689
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 2841 . . 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 2730 . . . 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 44401 . . 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 1915 . . . . 5 β„²π‘š 𝑛 ∈ 𝑍
108, 9nfan 1900 . . . 4 β„²π‘š(πœ‘ ∧ 𝑛 ∈ 𝑍)
11 simpll 763 . . . . 5 (((πœ‘ ∧ 𝑛 ∈ 𝑍) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ πœ‘)
124uztrn2 12845 . . . . . . . 8 ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘›)) β†’ 𝑗 ∈ 𝑍)
1312ssd 44070 . . . . . . 7 (𝑛 ∈ 𝑍 β†’ (β„€β‰₯β€˜π‘›) βŠ† 𝑍)
1413sselda 3981 . . . . . 6 ((𝑛 ∈ 𝑍 ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘š ∈ 𝑍)
1514adantll 710 . . . . 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 7085 . . . . . 6 (((πœ‘ ∧ π‘š ∈ 𝑍) ∧ 𝑋 ∈ dom (πΉβ€˜π‘š)) β†’ ((πΉβ€˜π‘š)β€˜π‘‹) ∈ ℝ)
1817ex 411 . . . . 5 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (𝑋 ∈ dom (πΉβ€˜π‘š) β†’ ((πΉβ€˜π‘š)β€˜π‘‹) ∈ ℝ))
1911, 15, 18syl2anc 582 . . . 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 3255 . . 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 3166 . 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 394   = wceq 1539  β„²wnf 1783   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  βˆͺ ciun 4996  βˆ© ciin 4997  dom cdm 5675  βŸΆwf 6538  β€˜cfv 6542  β„cr 11111  β„€β‰₯cuz 12826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-pre-lttri 11186  ax-pre-lttrn 11187
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-neg 11451  df-z 12563  df-uz 12827
This theorem is referenced by:  fnlimabslt  44693
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