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Theorem lmff 21327
 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 5475 . . . . . . 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 4805 . . . . . 6 (𝐹(⇝𝑡𝐽)𝑦 ↔ ⟨𝐹, 𝑦⟩ ∈ (⇝𝑡𝐽))
65exbii 1923 . . . . 5 (∃𝑦 𝐹(⇝𝑡𝐽)𝑦 ↔ ∃𝑦𝐹, 𝑦⟩ ∈ (⇝𝑡𝐽))
74, 6sylibr 224 . . . 4 (𝜑 → ∃𝑦 𝐹(⇝𝑡𝐽)𝑦)
8 lmff.3 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
9 lmcl 21323 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑦) → 𝑦𝑋)
108, 9sylan 489 . . . . 5 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → 𝑦𝑋)
11 eleq2 2828 . . . . . . 7 (𝑗 = 𝑋 → (𝑦𝑗𝑦𝑋))
12 feq3 6189 . . . . . . . 8 (𝑗 = 𝑋 → ((𝐹𝑥):𝑥𝑗 ↔ (𝐹𝑥):𝑥𝑋))
1312rexbidv 3190 . . . . . . 7 (𝑗 = 𝑋 → (∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗 ↔ ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋))
1411, 13imbi12d 333 . . . . . 6 (𝑗 = 𝑋 → ((𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗) ↔ (𝑦𝑋 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋)))
158lmbr 21284 . . . . . . . 8 (𝜑 → (𝐹(⇝𝑡𝐽)𝑦 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑦𝑋 ∧ ∀𝑗𝐽 (𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗))))
1615biimpa 502 . . . . . . 7 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑦𝑋 ∧ ∀𝑗𝐽 (𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗)))
1716simp3d 1139 . . . . . 6 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → ∀𝑗𝐽 (𝑦𝑗 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑗))
18 toponmax 20952 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
198, 18syl 17 . . . . . . 7 (𝜑𝑋𝐽)
2019adantr 472 . . . . . 6 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → 𝑋𝐽)
2114, 17, 20rspcdva 3455 . . . . 5 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → (𝑦𝑋 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋))
2210, 21mpd 15 . . . 4 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋)
237, 22exlimddv 2012 . . 3 (𝜑 → ∃𝑥 ∈ ran ℤ(𝐹𝑥):𝑥𝑋)
24 uzf 11902 . . . 4 :ℤ⟶𝒫 ℤ
25 ffn 6206 . . . 4 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
26 reseq2 5546 . . . . . 6 (𝑥 = (ℤ𝑗) → (𝐹𝑥) = (𝐹 ↾ (ℤ𝑗)))
27 id 22 . . . . . 6 (𝑥 = (ℤ𝑗) → 𝑥 = (ℤ𝑗))
2826, 27feq12d 6194 . . . . 5 (𝑥 = (ℤ𝑗) → ((𝐹𝑥):𝑥𝑋 ↔ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋))
2928rexrn 6525 . . . 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 14307 . . . 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)))
3616simp1d 1137 . . . . . . 7 ((𝜑𝐹(⇝𝑡𝐽)𝑦) → 𝐹 ∈ (𝑋pm ℂ))
377, 36exlimddv 2012 . . . . . 6 (𝜑𝐹 ∈ (𝑋pm ℂ))
38 pmfun 8045 . . . . . 6 (𝐹 ∈ (𝑋pm ℂ) → Fun 𝐹)
3937, 38syl 17 . . . . 5 (𝜑 → Fun 𝐹)
40 ffvresb 6558 . . . . 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 3190 . . 3 (𝜑 → (∃𝑗𝑍 (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑋 ↔ ∃𝑗𝑍𝑥 ∈ (ℤ𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝑋)))
4341rexbidv 3190 . . 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 383   ∧ w3a 1072   = wceq 1632  ∃wex 1853   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051  𝒫 cpw 4302  ⟨cop 4327   class class class wbr 4804  dom cdm 5266  ran crn 5267   ↾ cres 5268  Fun wfun 6043   Fn wfn 6044  ⟶wf 6045  ‘cfv 6049  (class class class)co 6814   ↑pm cpm 8026  ℂcc 10146  ℤcz 11589  ℤ≥cuz 11899  TopOnctopon 20937  ⇝𝑡clm 21252 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-pre-lttri 10222  ax-pre-lttrn 10223 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-er 7913  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-neg 10481  df-z 11590  df-uz 11900  df-top 20921  df-topon 20938  df-lm 21255 This theorem is referenced by:  lmle  23319
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