Theorem axdc3 10460

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.)

Hypothesis

Ref	Expression
axdc3.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
axdc3	⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))

Distinct variable groups:   𝐴,𝑔,𝑘   𝐶,𝑔,𝑘   𝑔,𝐹,𝑘

Proof of Theorem axdc3

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

Step	Hyp	Ref	Expression
1		axdc3.1	. 2 ⊢ 𝐴 ∈ V
2		feq1 6684	. . . . 5 ⊢ (𝑡 = 𝑠 → (𝑡:suc 𝑛⟶𝐴 ↔ 𝑠:suc 𝑛⟶𝐴))
3		fveq1 6881	. . . . . 6 ⊢ (𝑡 = 𝑠 → (𝑡‘∅) = (𝑠‘∅))
4	3	eqeq1d 2764	. . . . 5 ⊢ (𝑡 = 𝑠 → ((𝑡‘∅) = 𝐶 ↔ (𝑠‘∅) = 𝐶))
5		fveq1 6881	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (𝑡‘suc 𝑗) = (𝑠‘suc 𝑗))
6		fveq1 6881	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → (𝑡‘𝑗) = (𝑠‘𝑗))
7	6	fveq2d 6886	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (𝐹‘(𝑡‘𝑗)) = (𝐹‘(𝑠‘𝑗)))
8	5, 7	eleq12d 2856	. . . . . . 7 ⊢ (𝑡 = 𝑠 → ((𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)) ↔ (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗))))
9	8	ralbidv 3187	. . . . . 6 ⊢ (𝑡 = 𝑠 → (∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)) ↔ ∀𝑗 ∈ 𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗))))
10		suceq 6430	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
11	10	fveq2d 6886	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑠‘suc 𝑗) = (𝑠‘suc 𝑘))
12		2fveq3 6887	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐹‘(𝑠‘𝑗)) = (𝐹‘(𝑠‘𝑘)))
13	11, 12	eleq12d 2856	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗)) ↔ (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))))
14	13	cbvralvw 3242	. . . . . 6 ⊢ (∀𝑗 ∈ 𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗)) ↔ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))
15	9, 14	bitrdi 290	. . . . 5 ⊢ (𝑡 = 𝑠 → (∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)) ↔ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))))
16	2, 4, 15	3anbi123d 1464	. . . 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 3188	. . 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 2832	. 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 2762	. 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 10459	1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2740  ∀wral 3078  ∃wrex 3088  {crab 3414  Vcvv 3453   ∖ cdif 3899  ∅c0 4282  𝒫 cpw 4560  {csn 4587   ↦ cmpt 5190  dom cdm 5659   ↾ cres 5661  suc csuc 6363  ⟶wf 6533  ‘cfv 6537  ωcom 7866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-dc 10452

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7867  df-1o 8459

This theorem is used by:  axdc4lem  10461