Theorem axdc3 10376

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 6648	. . . . 5 ⊢ (𝑡 = 𝑠 → (𝑡:suc 𝑛⟶𝐴 ↔ 𝑠:suc 𝑛⟶𝐴))
3		fveq1 6841	. . . . . 6 ⊢ (𝑡 = 𝑠 → (𝑡‘∅) = (𝑠‘∅))
4	3	eqeq1d 2739	. . . . 5 ⊢ (𝑡 = 𝑠 → ((𝑡‘∅) = 𝐶 ↔ (𝑠‘∅) = 𝐶))
5		fveq1 6841	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (𝑡‘suc 𝑗) = (𝑠‘suc 𝑗))
6		fveq1 6841	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → (𝑡‘𝑗) = (𝑠‘𝑗))
7	6	fveq2d 6846	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (𝐹‘(𝑡‘𝑗)) = (𝐹‘(𝑠‘𝑗)))
8	5, 7	eleq12d 2831	. . . . . . 7 ⊢ (𝑡 = 𝑠 → ((𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)) ↔ (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗))))
9	8	ralbidv 3161	. . . . . 6 ⊢ (𝑡 = 𝑠 → (∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)) ↔ ∀𝑗 ∈ 𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗))))
10		suceq 6393	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
11	10	fveq2d 6846	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑠‘suc 𝑗) = (𝑠‘suc 𝑘))
12		2fveq3 6847	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐹‘(𝑠‘𝑗)) = (𝐹‘(𝑠‘𝑘)))
13	11, 12	eleq12d 2831	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗)) ↔ (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))))
14	13	cbvralvw 3216	. . . . . 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 1439	. . . 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 3162	. . 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 2807	. 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 2737	. 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 10375	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 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442   ∖ cdif 3900  ∅c0 4287  𝒫 cpw 4556  {csn 4582   ↦ cmpt 5181  dom cdm 5632   ↾ cres 5634  suc csuc 6327  ⟶wf 6496  ‘cfv 6500  ωcom 7818

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-dc 10368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-1o 8407

This theorem is referenced by:  axdc4lem  10377