Theorem sdc 37730

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 = (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 2734	. 2 ⊢ {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} = {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)}
11		eqid 2734	. . . 4 ⊢ 𝑍 = 𝑍
12		oveq2 7438	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑀...𝑛) = (𝑀...𝑘))
13	12	feq2d 6722	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑔:(𝑀...𝑛)⟶𝐴 ↔ 𝑔:(𝑀...𝑘)⟶𝐴))
14	13, 4	anbi12d 632	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓) ↔ (𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃)))
15	14	cbvrexvw 3235	. . . . 5 ⊢ (∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓) ↔ ∃𝑘 ∈ 𝑍 (𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃))
16	15	abbii 2806	. . . 4 ⊢ {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} = {𝑔 ∣ ∃𝑘 ∈ 𝑍 (𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃)}
17		eqid 2734	. . . 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 7508	. . 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 2735	. . . 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 2738	. . . . . . 7 ⊢ (𝑓 = 𝑥 → (𝑓 = (ℎ ↾ (𝑀...𝑘)) ↔ 𝑥 = (ℎ ↾ (𝑀...𝑘))))
21	20	3anbi2d 1440	. . . . . 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 3176	. . . . 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 2805	. . . 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 7527	. . 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 2764	. 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 37729	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1536  ∃wex 1775   ∈ wcel 2105  {cab 2711  ∀wral 3058  ∃wrex 3067  {csn 4630   ↾ cres 5690  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430   ∈ cmpo 7432  1c1 11153   + caddc 11155  ℤcz 12610  ℤ_≥cuz 12875  ...cfz 13543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-dc 10483  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544

This theorem is referenced by: (None)