Theorem axdc3 10342

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 6629	. . . . 5 ⊢ (𝑡 = 𝑠 → (𝑡:suc 𝑛⟶𝐴 ↔ 𝑠:suc 𝑛⟶𝐴))
3		fveq1 6821	. . . . . 6 ⊢ (𝑡 = 𝑠 → (𝑡‘∅) = (𝑠‘∅))
4	3	eqeq1d 2733	. . . . 5 ⊢ (𝑡 = 𝑠 → ((𝑡‘∅) = 𝐶 ↔ (𝑠‘∅) = 𝐶))
5		fveq1 6821	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (𝑡‘suc 𝑗) = (𝑠‘suc 𝑗))
6		fveq1 6821	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → (𝑡‘𝑗) = (𝑠‘𝑗))
7	6	fveq2d 6826	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (𝐹‘(𝑡‘𝑗)) = (𝐹‘(𝑠‘𝑗)))
8	5, 7	eleq12d 2825	. . . . . . 7 ⊢ (𝑡 = 𝑠 → ((𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)) ↔ (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗))))
9	8	ralbidv 3155	. . . . . 6 ⊢ (𝑡 = 𝑠 → (∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)) ↔ ∀𝑗 ∈ 𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗))))
10		suceq 6374	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
11	10	fveq2d 6826	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑠‘suc 𝑗) = (𝑠‘suc 𝑘))
12		2fveq3 6827	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐹‘(𝑠‘𝑗)) = (𝐹‘(𝑠‘𝑘)))
13	11, 12	eleq12d 2825	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗)) ↔ (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))))
14	13	cbvralvw 3210	. . . . . 6 ⊢ (∀𝑗 ∈ 𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗)) ↔ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))
15	9, 14	bitrdi 287	. . . . 5 ⊢ (𝑡 = 𝑠 → (∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)) ↔ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))))
16	2, 4, 15	3anbi123d 1438	. . . 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 3156	. . 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 2801	. 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 2731	. 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 10341	1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436   ∖ cdif 3899  ∅c0 4283  𝒫 cpw 4550  {csn 4576   ↦ cmpt 5172  dom cdm 5616   ↾ cres 5618  suc csuc 6308  ⟶wf 6477  ‘cfv 6481  ωcom 7796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-dc 10334

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-om 7797  df-1o 8385

This theorem is referenced by:  axdc4lem  10343