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Mirrors > Home > MPE Home > Th. List > axdc3 | Structured version Visualization version GIF version |
Description: Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value 𝐶. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. (Contributed by Mario Carneiro, 27-Jan-2013.) |
Ref | Expression |
---|---|
axdc3.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
axdc3 | ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axdc3.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | feq1 6646 | . . . . 5 ⊢ (𝑡 = 𝑠 → (𝑡:suc 𝑛⟶𝐴 ↔ 𝑠:suc 𝑛⟶𝐴)) | |
3 | fveq1 6838 | . . . . . 6 ⊢ (𝑡 = 𝑠 → (𝑡‘∅) = (𝑠‘∅)) | |
4 | 3 | eqeq1d 2738 | . . . . 5 ⊢ (𝑡 = 𝑠 → ((𝑡‘∅) = 𝐶 ↔ (𝑠‘∅) = 𝐶)) |
5 | fveq1 6838 | . . . . . . . 8 ⊢ (𝑡 = 𝑠 → (𝑡‘suc 𝑗) = (𝑠‘suc 𝑗)) | |
6 | fveq1 6838 | . . . . . . . . 9 ⊢ (𝑡 = 𝑠 → (𝑡‘𝑗) = (𝑠‘𝑗)) | |
7 | 6 | fveq2d 6843 | . . . . . . . 8 ⊢ (𝑡 = 𝑠 → (𝐹‘(𝑡‘𝑗)) = (𝐹‘(𝑠‘𝑗))) |
8 | 5, 7 | eleq12d 2832 | . . . . . . 7 ⊢ (𝑡 = 𝑠 → ((𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)) ↔ (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗)))) |
9 | 8 | ralbidv 3172 | . . . . . 6 ⊢ (𝑡 = 𝑠 → (∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)) ↔ ∀𝑗 ∈ 𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗)))) |
10 | suceq 6381 | . . . . . . . . 9 ⊢ (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘) | |
11 | 10 | fveq2d 6843 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑠‘suc 𝑗) = (𝑠‘suc 𝑘)) |
12 | 2fveq3 6844 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐹‘(𝑠‘𝑗)) = (𝐹‘(𝑠‘𝑘))) | |
13 | 11, 12 | eleq12d 2832 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → ((𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗)) ↔ (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))) |
14 | 13 | cbvralvw 3223 | . . . . . 6 ⊢ (∀𝑗 ∈ 𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗)) ↔ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))) |
15 | 9, 14 | bitrdi 286 | . . . . 5 ⊢ (𝑡 = 𝑠 → (∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)) ↔ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))) |
16 | 2, 4, 15 | 3anbi123d 1436 | . . . 4 ⊢ (𝑡 = 𝑠 → ((𝑡:suc 𝑛⟶𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗))) ↔ (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))))) |
17 | 16 | rexbidv 3173 | . . 3 ⊢ (𝑡 = 𝑠 → (∃𝑛 ∈ ω (𝑡:suc 𝑛⟶𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗))) ↔ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))))) |
18 | 17 | cbvabv 2809 | . 2 ⊢ {𝑡 ∣ ∃𝑛 ∈ ω (𝑡:suc 𝑛⟶𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)))} = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))} |
19 | eqid 2736 | . 2 ⊢ (𝑥 ∈ {𝑡 ∣ ∃𝑛 ∈ ω (𝑡:suc 𝑛⟶𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)))} ↦ {𝑦 ∈ {𝑡 ∣ ∃𝑛 ∈ ω (𝑡:suc 𝑛⟶𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)))} ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)}) = (𝑥 ∈ {𝑡 ∣ ∃𝑛 ∈ ω (𝑡:suc 𝑛⟶𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)))} ↦ {𝑦 ∈ {𝑡 ∣ ∃𝑛 ∈ ω (𝑡:suc 𝑛⟶𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)))} ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)}) | |
20 | 1, 18, 19 | axdc3lem4 10385 | 1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2713 ∀wral 3062 ∃wrex 3071 {crab 3405 Vcvv 3443 ∖ cdif 3905 ∅c0 4280 𝒫 cpw 4558 {csn 4584 ↦ cmpt 5186 dom cdm 5631 ↾ cres 5633 suc csuc 6317 ⟶wf 6489 ‘cfv 6493 ωcom 7798 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-dc 10378 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-om 7799 df-1o 8408 |
This theorem is referenced by: axdc4lem 10387 |
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