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Theorem sdc 35011
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 2819 . 2 {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} = {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)}
11 eqid 2819 . . . 4 𝑍 = 𝑍
12 oveq2 7156 . . . . . . . 8 (𝑛 = 𝑘 → (𝑀...𝑛) = (𝑀...𝑘))
1312feq2d 6493 . . . . . . 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 3449 . . . . 5 (∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓) ↔ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃))
1615abbii 2884 . . . 4 {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} = {𝑔 ∣ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃)}
17 eqid 2819 . . . 4 { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)}
1811, 16, 17mpoeq123i 7222 . . 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 2820 . . . 4 (𝑗 = 𝑦 → { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
20 eqeq1 2823 . . . . . . 7 (𝑓 = 𝑥 → (𝑓 = ( ↾ (𝑀...𝑘)) ↔ 𝑥 = ( ↾ (𝑀...𝑘))))
21203anbi2d 1435 . . . . . 6 (𝑓 = 𝑥 → ((:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎) ↔ (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))
2221rexbidv 3295 . . . . 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, 23cbvmpov 7241 . . 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 2844 . 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 35010 1 (𝜑 → ∃𝑓(𝑓:𝑍𝐴 ∧ ∀𝑛𝑍 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1082   = wceq 1531  wex 1774  wcel 2108  {cab 2797  wral 3136  wrex 3137  {csn 4559  cres 5550  wf 6344  cfv 6348  (class class class)co 7148  cmpo 7150  1c1 10530   + caddc 10532  cz 11973  cuz 12235  ...cfz 12884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-dc 9860  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885
This theorem is referenced by: (None)
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