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Theorem sdc 33207
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 2621 . 2 {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} = {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)}
11 eqid 2621 . . . 4 𝑍 = 𝑍
12 oveq2 6618 . . . . . . . 8 (𝑛 = 𝑘 → (𝑀...𝑛) = (𝑀...𝑘))
1312feq2d 5993 . . . . . . 7 (𝑛 = 𝑘 → (𝑔:(𝑀...𝑛)⟶𝐴𝑔:(𝑀...𝑘)⟶𝐴))
1413, 4anbi12d 746 . . . . . 6 (𝑛 = 𝑘 → ((𝑔:(𝑀...𝑛)⟶𝐴𝜓) ↔ (𝑔:(𝑀...𝑘)⟶𝐴𝜃)))
1514cbvrexv 3163 . . . . 5 (∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓) ↔ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃))
1615abbii 2736 . . . 4 {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)} = {𝑔 ∣ ∃𝑘𝑍 (𝑔:(𝑀...𝑘)⟶𝐴𝜃)}
17 eqid 2621 . . . 4 { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)}
1811, 16, 17mpt2eq123i 6678 . . 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 2622 . . . 4 (𝑗 = 𝑦 → { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
20 eqeq1 2625 . . . . . . 7 (𝑓 = 𝑥 → (𝑓 = ( ↾ (𝑀...𝑘)) ↔ 𝑥 = ( ↾ (𝑀...𝑘))))
21203anbi2d 1401 . . . . . 6 (𝑓 = 𝑥 → ((:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎) ↔ (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))
2221rexbidv 3046 . . . . 5 (𝑓 = 𝑥 → (∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎) ↔ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))
2322abbidv 2738 . . . 4 (𝑓 = 𝑥 → { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑓 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)} = { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})
2419, 23cbvmpt2v 6695 . . 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 2645 . 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 33206 1 (𝜑 → ∃𝑓(𝑓:𝑍𝐴 ∧ ∀𝑛𝑍 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  {csn 4153  cres 5081  wf 5848  cfv 5852  (class class class)co 6610  cmpt2 6612  1c1 9889   + caddc 9891  cz 11329  cuz 11639  ...cfz 12276
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-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-dc 9220  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
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-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277
This theorem is referenced by: (None)
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