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Theorem axdc3 9867
 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 ((𝐶𝐴𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
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.
StepHypRef Expression
1 axdc3.1 . 2 𝐴 ∈ V
2 feq1 6468 . . . . 5 (𝑡 = 𝑠 → (𝑡:suc 𝑛𝐴𝑠:suc 𝑛𝐴))
3 fveq1 6644 . . . . . 6 (𝑡 = 𝑠 → (𝑡‘∅) = (𝑠‘∅))
43eqeq1d 2800 . . . . 5 (𝑡 = 𝑠 → ((𝑡‘∅) = 𝐶 ↔ (𝑠‘∅) = 𝐶))
5 fveq1 6644 . . . . . . . 8 (𝑡 = 𝑠 → (𝑡‘suc 𝑗) = (𝑠‘suc 𝑗))
6 fveq1 6644 . . . . . . . . 9 (𝑡 = 𝑠 → (𝑡𝑗) = (𝑠𝑗))
76fveq2d 6649 . . . . . . . 8 (𝑡 = 𝑠 → (𝐹‘(𝑡𝑗)) = (𝐹‘(𝑠𝑗)))
85, 7eleq12d 2884 . . . . . . 7 (𝑡 = 𝑠 → ((𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗))))
98ralbidv 3162 . . . . . 6 (𝑡 = 𝑠 → (∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ ∀𝑗𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗))))
10 suceq 6224 . . . . . . . . 9 (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
1110fveq2d 6649 . . . . . . . 8 (𝑗 = 𝑘 → (𝑠‘suc 𝑗) = (𝑠‘suc 𝑘))
12 2fveq3 6650 . . . . . . . 8 (𝑗 = 𝑘 → (𝐹‘(𝑠𝑗)) = (𝐹‘(𝑠𝑘)))
1311, 12eleq12d 2884 . . . . . . 7 (𝑗 = 𝑘 → ((𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗)) ↔ (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
1413cbvralvw 3396 . . . . . 6 (∀𝑗𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗)) ↔ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))
159, 14syl6bb 290 . . . . 5 (𝑡 = 𝑠 → (∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
162, 4, 153anbi123d 1433 . . . 4 (𝑡 = 𝑠 → ((𝑡:suc 𝑛𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗))) ↔ (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
1716rexbidv 3256 . . 3 (𝑡 = 𝑠 → (∃𝑛 ∈ ω (𝑡:suc 𝑛𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗))) ↔ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
1817cbvabv 2866 . 2 {𝑡 ∣ ∃𝑛 ∈ ω (𝑡:suc 𝑛𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)))} = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
19 eqid 2798 . 2 (𝑥 ∈ {𝑡 ∣ ∃𝑛 ∈ ω (𝑡:suc 𝑛𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)))} ↦ {𝑦 ∈ {𝑡 ∣ ∃𝑛 ∈ ω (𝑡:suc 𝑛𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)))} ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)}) = (𝑥 ∈ {𝑡 ∣ ∃𝑛 ∈ ω (𝑡:suc 𝑛𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)))} ↦ {𝑦 ∈ {𝑡 ∣ ∃𝑛 ∈ ω (𝑡:suc 𝑛𝐴 ∧ (𝑡‘∅) = 𝐶 ∧ ∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)))} ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)})
201, 18, 19axdc3lem4 9866 1 ((𝐶𝐴𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∖ cdif 3878  ∅c0 4243  𝒫 cpw 4497  {csn 4525   ↦ cmpt 5110  dom cdm 5519   ↾ cres 5521  suc csuc 6161  ⟶wf 6320  ‘cfv 6324  ωcom 7562 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-dc 9859 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7563  df-1o 8087 This theorem is referenced by:  axdc4lem  9868
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