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Theorem sdc 37751
Description: Strong dependent choice. Suppose we may choose an element of 𝐴 such that property 𝜓 holds, and suppose that if we have already chosen the first 𝑘 elements (represented here by a function from 1...𝑘 to 𝐴), we may choose another element so that all 𝑘 + 1 elements taken together have property 𝜓. Then there exists an infinite sequence of elements of 𝐴 such that the first 𝑛 terms of this sequence satisfy 𝜓 for all 𝑛. This theorem allows to construct infinite sequences where each term depends on all the previous terms in the sequence. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 3-Jun-2014.)
Hypotheses
Ref Expression
sdc.1 𝑍 = (ℤ𝑀)
sdc.2 (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓𝜒))
sdc.3 (𝑛 = 𝑀 → (𝜓𝜏))
sdc.4 (𝑛 = 𝑘 → (𝜓𝜃))
sdc.5 ((𝑔 = 𝑛 = (𝑘 + 1)) → (𝜓𝜎))
sdc.6 (𝜑𝐴𝑉)
sdc.7 (𝜑𝑀 ∈ ℤ)
sdc.8 (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴𝜏))
sdc.9 ((𝜑𝑘𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴𝜃) → ∃(:(𝑀...(𝑘 + 1))⟶𝐴𝑔 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))
Assertion
Ref Expression
sdc (𝜑 → ∃𝑓(𝑓:𝑍𝐴 ∧ ∀𝑛𝑍 𝜒))
Distinct variable groups:   𝑓,𝑔,,𝑘,𝑛,𝐴   𝑓,𝑀,𝑔,,𝑘,𝑛   𝜒,𝑔   𝜓,𝑓,,𝑘   𝜎,𝑓,𝑔,𝑛   𝜑,𝑛   𝜃,𝑛   ,𝑉   𝜏,,𝑘,𝑛   𝑓,𝑍,𝑔,,𝑘,𝑛   𝜑,𝑔,,𝑘
Allowed substitution hints:   𝜑(𝑓)   𝜓(𝑔,𝑛)   𝜒(𝑓,,𝑘,𝑛)   𝜃(𝑓,𝑔,,𝑘)   𝜏(𝑓,𝑔)   𝜎(,𝑘)   𝑉(𝑓,𝑔,𝑘,𝑛)

Proof of Theorem sdc
Dummy variables 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sdc.1 . 2 𝑍 = (ℤ𝑀)
2 sdc.2 . 2 (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓𝜒))
3 sdc.3 . 2 (𝑛 = 𝑀 → (𝜓𝜏))
4 sdc.4 . 2 (𝑛 = 𝑘 → (𝜓𝜃))
5 sdc.5 . 2 ((𝑔 = 𝑛 = (𝑘 + 1)) → (𝜓𝜎))
6 sdc.6 . 2 (𝜑𝐴𝑉)
7 sdc.7 . 2 (𝜑𝑀 ∈ ℤ)
8 sdc.8 . 2 (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴𝜏))
9 sdc.9 . 2 ((𝜑𝑘𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴𝜃) → ∃(:(𝑀...(𝑘 + 1))⟶𝐴𝑔 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))
10 eqid 2737 . 2 {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} = {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)}
11 eqid 2737 . . . 4 𝑍 = 𝑍
12 oveq2 7439 . . . . . . . 8 (𝑛 = 𝑘 → (𝑀...𝑛) = (𝑀...𝑘))
1312feq2d 6722 . . . . . . 7 (𝑛 = 𝑘 → (𝑔:(𝑀...𝑛)⟶𝐴𝑔:(𝑀...𝑘)⟶𝐴))
1413, 4anbi12d 632 . . . . . 6 (𝑛 = 𝑘 → ((𝑔:(𝑀...𝑛)⟶𝐴𝜓) ↔ (𝑔:(𝑀...𝑘)⟶𝐴𝜃)))
1514cbvrexvw 3238 . . . . 5 (∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓) ↔ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃))
1615abbii 2809 . . . 4 {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} = {𝑔 ∣ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃)}
17 eqid 2737 . . . 4 { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)}
1811, 16, 17mpoeq123i 7509 . . 3 (𝑗𝑍, 𝑓 ∈ {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)}) = (𝑗𝑍, 𝑓 ∈ {𝑔 ∣ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃)} ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
19 eqidd 2738 . . . 4 (𝑗 = 𝑦 → { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
20 eqeq1 2741 . . . . . . 7 (𝑓 = 𝑥 → (𝑓 = ( ↾ (𝑀...𝑘)) ↔ 𝑥 = ( ↾ (𝑀...𝑘))))
21203anbi2d 1443 . . . . . 6 (𝑓 = 𝑥 → ((:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎) ↔ (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))
2221rexbidv 3179 . . . . 5 (𝑓 = 𝑥 → (∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎) ↔ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))
2322abbidv 2808 . . . 4 (𝑓 = 𝑥 → { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
2419, 23cbvmpov 7528 . . 3 (𝑗𝑍, 𝑓 ∈ {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)}) = (𝑦𝑍, 𝑥 ∈ {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
2518, 24eqtr3i 2767 . 2 (𝑗𝑍, 𝑓 ∈ {𝑔 ∣ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃)} ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)}) = (𝑦𝑍, 𝑥 ∈ {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
261, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25sdclem1 37750 1 (𝜑 → ∃𝑓(𝑓:𝑍𝐴 ∧ ∀𝑛𝑍 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  {csn 4626  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  1c1 11156   + caddc 11158  cz 12613  cuz 12878  ...cfz 13547
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-dc 10486  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548
This theorem is referenced by: (None)
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