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Mirrors > Home > MPE Home > Th. List > Mathboxes > sdc | Structured version Visualization version GIF version |
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.) |
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))) |
Ref | Expression |
---|---|
sdc | ⊢ (𝜑 → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒)) |
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 2738 | . 2 ⊢ {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} = {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} | |
11 | eqid 2738 | . . . 4 ⊢ 𝑍 = 𝑍 | |
12 | oveq2 7239 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑀...𝑛) = (𝑀...𝑘)) | |
13 | 12 | feq2d 6549 | . . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑔:(𝑀...𝑛)⟶𝐴 ↔ 𝑔:(𝑀...𝑘)⟶𝐴)) |
14 | 13, 4 | anbi12d 634 | . . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓) ↔ (𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃))) |
15 | 14 | cbvrexvw 3371 | . . . . 5 ⊢ (∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓) ↔ ∃𝑘 ∈ 𝑍 (𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃)) |
16 | 15 | abbii 2809 | . . . 4 ⊢ {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)} = {𝑔 ∣ ∃𝑘 ∈ 𝑍 (𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃)} |
17 | eqid 2738 | . . . 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 7305 | . . 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 2739 | . . . 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 2742 | . . . . . . 7 ⊢ (𝑓 = 𝑥 → (𝑓 = (ℎ ↾ (𝑀...𝑘)) ↔ 𝑥 = (ℎ ↾ (𝑀...𝑘)))) | |
21 | 20 | 3anbi2d 1443 | . . . . . 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 3224 | . . . . 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 2808 | . . . 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 7324 | . . 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 2768 | . 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 35664 | 1 ⊢ (𝜑 → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∃wex 1787 ∈ wcel 2111 {cab 2715 ∀wral 3062 ∃wrex 3063 {csn 4555 ↾ cres 5567 ⟶wf 6393 ‘cfv 6397 (class class class)co 7231 ∈ cmpo 7233 1c1 10754 + caddc 10756 ℤcz 12200 ℤ≥cuz 12462 ...cfz 13119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-inf2 9280 ax-dc 10084 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-1st 7779 df-2nd 7780 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-er 8411 df-map 8530 df-en 8647 df-dom 8648 df-sdom 8649 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-nn 11855 df-n0 12115 df-z 12201 df-uz 12463 df-fz 13120 |
This theorem is referenced by: (None) |
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