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Theorem axdc3 10210
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 6581 . . . . 5 (𝑡 = 𝑠 → (𝑡:suc 𝑛𝐴𝑠:suc 𝑛𝐴))
3 fveq1 6773 . . . . . 6 (𝑡 = 𝑠 → (𝑡‘∅) = (𝑠‘∅))
43eqeq1d 2740 . . . . 5 (𝑡 = 𝑠 → ((𝑡‘∅) = 𝐶 ↔ (𝑠‘∅) = 𝐶))
5 fveq1 6773 . . . . . . . 8 (𝑡 = 𝑠 → (𝑡‘suc 𝑗) = (𝑠‘suc 𝑗))
6 fveq1 6773 . . . . . . . . 9 (𝑡 = 𝑠 → (𝑡𝑗) = (𝑠𝑗))
76fveq2d 6778 . . . . . . . 8 (𝑡 = 𝑠 → (𝐹‘(𝑡𝑗)) = (𝐹‘(𝑠𝑗)))
85, 7eleq12d 2833 . . . . . . 7 (𝑡 = 𝑠 → ((𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗))))
98ralbidv 3112 . . . . . 6 (𝑡 = 𝑠 → (∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ ∀𝑗𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗))))
10 suceq 6331 . . . . . . . . 9 (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
1110fveq2d 6778 . . . . . . . 8 (𝑗 = 𝑘 → (𝑠‘suc 𝑗) = (𝑠‘suc 𝑘))
12 2fveq3 6779 . . . . . . . 8 (𝑗 = 𝑘 → (𝐹‘(𝑠𝑗)) = (𝐹‘(𝑠𝑘)))
1311, 12eleq12d 2833 . . . . . . 7 (𝑗 = 𝑘 → ((𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗)) ↔ (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
1413cbvralvw 3383 . . . . . 6 (∀𝑗𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗)) ↔ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))
159, 14bitrdi 287 . . . . 5 (𝑡 = 𝑠 → (∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
162, 4, 153anbi123d 1435 . . . 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 3226 . . 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 2811 . 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 10209 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 396  w3a 1086   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wral 3064  wrex 3065  {crab 3068  Vcvv 3432  cdif 3884  c0 4256  𝒫 cpw 4533  {csn 4561  cmpt 5157  dom cdm 5589  cres 5591  suc csuc 6268  wf 6429  cfv 6433  ωcom 7712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-dc 10202
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297
This theorem is referenced by:  axdc4lem  10211
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