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Theorem sdc 33894
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 us to construct infinite seqeunces 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 2764 . 2 {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} = {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)}
11 eqid 2764 . . . 4 𝑍 = 𝑍
12 oveq2 6849 . . . . . . . 8 (𝑛 = 𝑘 → (𝑀...𝑛) = (𝑀...𝑘))
1312feq2d 6208 . . . . . . 7 (𝑛 = 𝑘 → (𝑔:(𝑀...𝑛)⟶𝐴𝑔:(𝑀...𝑘)⟶𝐴))
1413, 4anbi12d 624 . . . . . 6 (𝑛 = 𝑘 → ((𝑔:(𝑀...𝑛)⟶𝐴𝜓) ↔ (𝑔:(𝑀...𝑘)⟶𝐴𝜃)))
1514cbvrexv 3319 . . . . 5 (∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓) ↔ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃))
1615abbii 2881 . . . 4 {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} = {𝑔 ∣ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃)}
17 eqid 2764 . . . 4 { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)}
1811, 16, 17mpt2eq123i 6915 . . 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 2765 . . . 4 (𝑗 = 𝑦 → { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
20 eqeq1 2768 . . . . . . 7 (𝑓 = 𝑥 → (𝑓 = ( ↾ (𝑀...𝑘)) ↔ 𝑥 = ( ↾ (𝑀...𝑘))))
21203anbi2d 1565 . . . . . 6 (𝑓 = 𝑥 → ((:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎) ↔ (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))
2221rexbidv 3198 . . . . 5 (𝑓 = 𝑥 → (∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎) ↔ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))
2322abbidv 2883 . . . 4 (𝑓 = 𝑥 → { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
2419, 23cbvmpt2v 6932 . . 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 2788 . 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 33893 1 (𝜑 → ∃𝑓(𝑓:𝑍𝐴 ∧ ∀𝑛𝑍 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  {cab 2750  wral 3054  wrex 3055  {csn 4333  cres 5278  wf 6063  cfv 6067  (class class class)co 6841  cmpt2 6843  1c1 10189   + caddc 10191  cz 11623  cuz 11885  ...cfz 12532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-inf2 8752  ax-dc 9520  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-er 7946  df-map 8061  df-en 8160  df-dom 8161  df-sdom 8162  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-nn 11274  df-n0 11538  df-z 11624  df-uz 11886  df-fz 12533
This theorem is referenced by: (None)
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