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Theorem lmff 21015
Description: If 𝐹 converges, there is some upper integer set on which 𝐹 is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)
Hypotheses
Ref Expression
lmff.1 𝑍 = (ℤ𝑀)
lmff.3 (𝜑𝐽 ∈ (TopOn‘𝑋))
lmff.4 (𝜑𝑀 ∈ ℤ)
lmff.5 (𝜑𝐹 ∈ dom (⇝𝑡𝐽))
Assertion
Ref Expression
lmff (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋)
Distinct variable groups:   𝑗,𝐹   𝑗,𝐽   𝑗,𝑀   𝜑,𝑗   𝑗,𝑋   𝑗,𝑍

Proof of Theorem lmff
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmff.5 . . . . . 6 (𝜑𝐹 ∈ dom (⇝𝑡𝐽))
2 eldm2g 5280 . . . . . . 7 (𝐹 ∈ dom (⇝𝑡𝐽) → (𝐹 ∈ dom (⇝𝑡𝐽) ↔ ∃𝑦𝐹, 𝑦⟩ ∈ (⇝𝑡𝐽)))
32ibi 256 . . . . . 6 (𝐹 ∈ dom (⇝𝑡𝐽) → ∃𝑦𝐹, 𝑦⟩ ∈ (⇝𝑡𝐽))
41, 3syl 17 . . . . 5 (𝜑 → ∃𝑦𝐹, 𝑦⟩ ∈ (⇝𝑡𝐽))
5 df-br 4614 . . . . . 6 (𝐹(⇝𝑡𝐽)𝑦 ↔ ⟨𝐹, 𝑦⟩ ∈ (⇝𝑡𝐽))
65exbii 1771 . . . . 5 (∃𝑦 𝐹(⇝𝑡𝐽)𝑦 ↔ ∃𝑦𝐹, 𝑦⟩ ∈ (⇝𝑡𝐽))
74, 6sylibr 224 . . . 4 (𝜑 → ∃𝑦 𝐹(⇝𝑡𝐽)𝑦)
8 lmff.3 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
9 toponmax 20643 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
108, 9syl 17 . . . . . 6 (𝜑𝑋𝐽)
1110adantr 481 . . . . 5 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → 𝑋𝐽)
128lmbr 20972 . . . . . . 7 (𝜑 → (𝐹(⇝𝑡𝐽)𝑦 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑦𝑋 ∧ ∀𝑗𝐽 (𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗))))
1312biimpa 501 . . . . . 6 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑦𝑋 ∧ ∀𝑗𝐽 (𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗)))
1413simp3d 1073 . . . . 5 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → ∀𝑗𝐽 (𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗))
15 lmcl 21011 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑦) → 𝑦𝑋)
168, 15sylan 488 . . . . 5 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → 𝑦𝑋)
17 eleq2 2687 . . . . . . 7 (𝑗 = 𝑋 → (𝑦𝑗𝑦𝑋))
18 feq3 5985 . . . . . . . 8 (𝑗 = 𝑋 → ((𝐹𝑥):𝑥𝑗 ↔ (𝐹𝑥):𝑥𝑋))
1918rexbidv 3045 . . . . . . 7 (𝑗 = 𝑋 → (∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗 ↔ ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋))
2017, 19imbi12d 334 . . . . . 6 (𝑗 = 𝑋 → ((𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗) ↔ (𝑦𝑋 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋)))
2120rspcv 3291 . . . . 5 (𝑋𝐽 → (∀𝑗𝐽 (𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗) → (𝑦𝑋 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋)))
2211, 14, 16, 21syl3c 66 . . . 4 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋)
237, 22exlimddv 1860 . . 3 (𝜑 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋)
24 uzf 11634 . . . 4 :ℤ⟶𝒫 ℤ
25 ffn 6002 . . . 4 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
26 reseq2 5351 . . . . . 6 (𝑥 = (ℤ𝑗) → (𝐹𝑥) = (𝐹 ↾ (ℤ𝑗)))
27 id 22 . . . . . 6 (𝑥 = (ℤ𝑗) → 𝑥 = (ℤ𝑗))
2826, 27feq12d 5990 . . . . 5 (𝑥 = (ℤ𝑗) → ((𝐹𝑥):𝑥𝑋 ↔ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋))
2928rexrn 6317 . . . 4 (ℤ Fn ℤ → (∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋))
3024, 25, 29mp2b 10 . . 3 (∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋)
3123, 30sylib 208 . 2 (𝜑 → ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋)
32 lmff.4 . . . 4 (𝜑𝑀 ∈ ℤ)
33 lmff.1 . . . . 5 𝑍 = (ℤ𝑀)
3433rexuz3 14022 . . . 4 (𝑀 ∈ ℤ → (∃𝑗𝑍𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋) ↔ ∃𝑗 ∈ ℤ ∀𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋)))
3532, 34syl 17 . . 3 (𝜑 → (∃𝑗𝑍𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋) ↔ ∃𝑗 ∈ ℤ ∀𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋)))
3613simp1d 1071 . . . . . . 7 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → 𝐹 ∈ (𝑋pm ℂ))
377, 36exlimddv 1860 . . . . . 6 (𝜑𝐹 ∈ (𝑋pm ℂ))
38 pmfun 7821 . . . . . 6 (𝐹 ∈ (𝑋pm ℂ) → Fun 𝐹)
3937, 38syl 17 . . . . 5 (𝜑 → Fun 𝐹)
40 ffvresb 6349 . . . . 5 (Fun 𝐹 → ((𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋 ↔ ∀𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋)))
4139, 40syl 17 . . . 4 (𝜑 → ((𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋 ↔ ∀𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋)))
4241rexbidv 3045 . . 3 (𝜑 → (∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋 ↔ ∃𝑗𝑍𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋)))
4341rexbidv 3045 . . 3 (𝜑 → (∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋 ↔ ∃𝑗 ∈ ℤ ∀𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋)))
4435, 42, 433bitr4d 300 . 2 (𝜑 → (∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋))
4531, 44mpbird 247 1 (𝜑 → ∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  𝒫 cpw 4130  cop 4154   class class class wbr 4613  dom cdm 5074  ran crn 5075  cres 5076  Fun wfun 5841   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  pm cpm 7803  cc 9878  cz 11321  cuz 11631  TopOnctopon 20618  𝑡clm 20940
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 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-pre-lttri 9954  ax-pre-lttrn 9955
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 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-er 7687  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-neg 10213  df-z 11322  df-uz 11632  df-top 20621  df-topon 20623  df-lm 20943
This theorem is referenced by:  lmle  23007
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