Theorem axdc3 10364

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 6640	. . . . 5 ⊢ (𝑡 = 𝑠 → (𝑡:suc 𝑛⟶𝐴 ↔ 𝑠:suc 𝑛⟶𝐴))
3		fveq1 6833	. . . . . 6 ⊢ (𝑡 = 𝑠 → (𝑡‘∅) = (𝑠‘∅))
4	3	eqeq1d 2738	. . . . 5 ⊢ (𝑡 = 𝑠 → ((𝑡‘∅) = 𝐶 ↔ (𝑠‘∅) = 𝐶))
5		fveq1 6833	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (𝑡‘suc 𝑗) = (𝑠‘suc 𝑗))
6		fveq1 6833	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → (𝑡‘𝑗) = (𝑠‘𝑗))
7	6	fveq2d 6838	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (𝐹‘(𝑡‘𝑗)) = (𝐹‘(𝑠‘𝑗)))
8	5, 7	eleq12d 2830	. . . . . . 7 ⊢ (𝑡 = 𝑠 → ((𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)) ↔ (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗))))
9	8	ralbidv 3159	. . . . . 6 ⊢ (𝑡 = 𝑠 → (∀𝑗 ∈ 𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡‘𝑗)) ↔ ∀𝑗 ∈ 𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗))))
10		suceq 6385	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
11	10	fveq2d 6838	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑠‘suc 𝑗) = (𝑠‘suc 𝑘))
12		2fveq3 6839	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐹‘(𝑠‘𝑗)) = (𝐹‘(𝑠‘𝑘)))
13	11, 12	eleq12d 2830	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠‘𝑗)) ↔ (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))))
14	13	cbvralvw 3214	. . . . . 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 3160	. . 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 2806	. 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 10363	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 2113  {cab 2714  ∀wral 3051  ∃wrex 3060  {crab 3399  Vcvv 3440   ∖ cdif 3898  ∅c0 4285  𝒫 cpw 4554  {csn 4580   ↦ cmpt 5179  dom cdm 5624   ↾ cres 5626  suc csuc 6319  ⟶wf 6488  ‘cfv 6492  ωcom 7808

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-dc 10356

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397

This theorem is referenced by:  axdc4lem  10365