Theorem sdc 35182

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 = (ℤ≥‘𝑀)
sdc.2	⊢ (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓 ↔ 𝜒))
sdc.3	⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜏))
sdc.4	⊢ (𝑛 = 𝑘 → (𝜓 ↔ 𝜃))
sdc.5	⊢ ((𝑔 = ℎ ∧ 𝑛 = (𝑘 + 1)) → (𝜓 ↔ 𝜎))
sdc.6	⊢ (𝜑 → 𝐴 ∈ 𝑉)
sdc.7	⊢ (𝜑 → 𝑀 ∈ ℤ)
sdc.8	⊢ (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴 ∧ 𝜏))
sdc.9	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃) → ∃ℎ(ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑔 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)))

Assertion

Ref	Expression
sdc	⊢ (𝜑 → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒))

Distinct variable groups:   𝑓,𝑔,ℎ,𝑘,𝑛,𝐴   𝑓,𝑀,𝑔,ℎ,𝑘,𝑛   𝜒,𝑔   𝜓,𝑓,ℎ,𝑘   𝜎,𝑓,𝑔,𝑛   𝜑,𝑛   𝜃,𝑛   ℎ,𝑉   𝜏,ℎ,𝑘,𝑛   𝑓,𝑍,𝑔,ℎ,𝑘,𝑛   𝜑,𝑔,ℎ,𝑘

Allowed substitution hints:   𝜑(𝑓)   𝜓(𝑔,𝑛)   𝜒(𝑓,ℎ,𝑘,𝑛)   𝜃(𝑓,𝑔,ℎ,𝑘)   𝜏(𝑓,𝑔)   𝜎(ℎ,𝑘)   𝑉(𝑓,𝑔,𝑘,𝑛)

Proof of Theorem sdc

Dummy variables 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sdc.1	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		sdc.2	. 2 ⊢ (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓 ↔ 𝜒))
3		sdc.3	. 2 ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜏))
4		sdc.4	. 2 ⊢ (𝑛 = 𝑘 → (𝜓 ↔ 𝜃))
5		sdc.5	. 2 ⊢ ((𝑔 = ℎ ∧ 𝑛 = (𝑘 + 1)) → (𝜓 ↔ 𝜎))
6		sdc.6	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		sdc.7	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
8		sdc.8	. 2 ⊢ (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴 ∧ 𝜏))
9		sdc.9	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃) → ∃ℎ(ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑔 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)))
10		eqid 2798	. 2 ⊢ {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} = {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)}
11		eqid 2798	. . . 4 ⊢ 𝑍 = 𝑍
12		oveq2 7143	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑀...𝑛) = (𝑀...𝑘))
13	12	feq2d 6473	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑔:(𝑀...𝑛)⟶𝐴 ↔ 𝑔:(𝑀...𝑘)⟶𝐴))
14	13, 4	anbi12d 633	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓) ↔ (𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃)))
15	14	cbvrexvw 3397	. . . . 5 ⊢ (∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓) ↔ ∃𝑘 ∈ 𝑍 (𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃))
16	15	abbii 2863	. . . 4 ⊢ {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} = {𝑔 ∣ ∃𝑘 ∈ 𝑍 (𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃)}
17		eqid 2798	. . . 4 ⊢ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑓 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑓 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)}
18	11, 16, 17	mpoeq123i 7209	. . 3 ⊢ (𝑗 ∈ 𝑍, 𝑓 ∈ {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} ↦ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑓 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)}) = (𝑗 ∈ 𝑍, 𝑓 ∈ {𝑔 ∣ ∃𝑘 ∈ 𝑍 (𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃)} ↦ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑓 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})
19		eqidd 2799	. . . 4 ⊢ (𝑗 = 𝑦 → {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑓 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑓 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})
20		eqeq1 2802	. . . . . . 7 ⊢ (𝑓 = 𝑥 → (𝑓 = (ℎ ↾ (𝑀...𝑘)) ↔ 𝑥 = (ℎ ↾ (𝑀...𝑘))))
21	20	3anbi2d 1438	. . . . . 6 ⊢ (𝑓 = 𝑥 → ((ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑓 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎) ↔ (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)))
22	21	rexbidv 3256	. . . . 5 ⊢ (𝑓 = 𝑥 → (∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑓 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎) ↔ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)))
23	22	abbidv 2862	. . . 4 ⊢ (𝑓 = 𝑥 → {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑓 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)} = {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})
24	19, 23	cbvmpov 7228	. . 3 ⊢ (𝑗 ∈ 𝑍, 𝑓 ∈ {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} ↦ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑓 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)}) = (𝑦 ∈ 𝑍, 𝑥 ∈ {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} ↦ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})
25	18, 24	eqtr3i 2823	. 2 ⊢ (𝑗 ∈ 𝑍, 𝑓 ∈ {𝑔 ∣ ∃𝑘 ∈ 𝑍 (𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃)} ↦ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑓 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)}) = (𝑦 ∈ 𝑍, 𝑥 ∈ {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} ↦ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})
26	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25	sdclem1 35181	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  {csn 4525   ↾ cres 5521  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  1c1 10527   + caddc 10529  ℤcz 11969  ℤ_≥cuz 12231  ...cfz 12885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-dc 9857  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886

This theorem is referenced by: (None)