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Theorem axdc3 10141
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 6565 . . . . 5 (𝑡 = 𝑠 → (𝑡:suc 𝑛𝐴𝑠:suc 𝑛𝐴))
3 fveq1 6755 . . . . . 6 (𝑡 = 𝑠 → (𝑡‘∅) = (𝑠‘∅))
43eqeq1d 2740 . . . . 5 (𝑡 = 𝑠 → ((𝑡‘∅) = 𝐶 ↔ (𝑠‘∅) = 𝐶))
5 fveq1 6755 . . . . . . . 8 (𝑡 = 𝑠 → (𝑡‘suc 𝑗) = (𝑠‘suc 𝑗))
6 fveq1 6755 . . . . . . . . 9 (𝑡 = 𝑠 → (𝑡𝑗) = (𝑠𝑗))
76fveq2d 6760 . . . . . . . 8 (𝑡 = 𝑠 → (𝐹‘(𝑡𝑗)) = (𝐹‘(𝑠𝑗)))
85, 7eleq12d 2833 . . . . . . 7 (𝑡 = 𝑠 → ((𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗))))
98ralbidv 3120 . . . . . 6 (𝑡 = 𝑠 → (∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ ∀𝑗𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗))))
10 suceq 6316 . . . . . . . . 9 (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
1110fveq2d 6760 . . . . . . . 8 (𝑗 = 𝑘 → (𝑠‘suc 𝑗) = (𝑠‘suc 𝑘))
12 2fveq3 6761 . . . . . . . 8 (𝑗 = 𝑘 → (𝐹‘(𝑠𝑗)) = (𝐹‘(𝑠𝑘)))
1311, 12eleq12d 2833 . . . . . . 7 (𝑗 = 𝑘 → ((𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗)) ↔ (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
1413cbvralvw 3372 . . . . . 6 (∀𝑗𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗)) ↔ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))
159, 14bitrdi 286 . . . . 5 (𝑡 = 𝑠 → (∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
162, 4, 153anbi123d 1434 . . . 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 3225 . . 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 2812 . 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 2738 . 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 10140 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 395  w3a 1085   = wceq 1539  wex 1783  wcel 2108  {cab 2715  wral 3063  wrex 3064  {crab 3067  Vcvv 3422  cdif 3880  c0 4253  𝒫 cpw 4530  {csn 4558  cmpt 5153  dom cdm 5580  cres 5582  suc csuc 6253  wf 6414  cfv 6418  ωcom 7687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-dc 10133
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267
This theorem is referenced by:  axdc4lem  10142
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