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Theorem axdc3 9864
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 6488 . . . . 5 (𝑡 = 𝑠 → (𝑡:suc 𝑛𝐴𝑠:suc 𝑛𝐴))
3 fveq1 6662 . . . . . 6 (𝑡 = 𝑠 → (𝑡‘∅) = (𝑠‘∅))
43eqeq1d 2820 . . . . 5 (𝑡 = 𝑠 → ((𝑡‘∅) = 𝐶 ↔ (𝑠‘∅) = 𝐶))
5 fveq1 6662 . . . . . . . 8 (𝑡 = 𝑠 → (𝑡‘suc 𝑗) = (𝑠‘suc 𝑗))
6 fveq1 6662 . . . . . . . . 9 (𝑡 = 𝑠 → (𝑡𝑗) = (𝑠𝑗))
76fveq2d 6667 . . . . . . . 8 (𝑡 = 𝑠 → (𝐹‘(𝑡𝑗)) = (𝐹‘(𝑠𝑗)))
85, 7eleq12d 2904 . . . . . . 7 (𝑡 = 𝑠 → ((𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗))))
98ralbidv 3194 . . . . . 6 (𝑡 = 𝑠 → (∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ ∀𝑗𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗))))
10 suceq 6249 . . . . . . . . 9 (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
1110fveq2d 6667 . . . . . . . 8 (𝑗 = 𝑘 → (𝑠‘suc 𝑗) = (𝑠‘suc 𝑘))
12 2fveq3 6668 . . . . . . . 8 (𝑗 = 𝑘 → (𝐹‘(𝑠𝑗)) = (𝐹‘(𝑠𝑘)))
1311, 12eleq12d 2904 . . . . . . 7 (𝑗 = 𝑘 → ((𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗)) ↔ (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
1413cbvralvw 3447 . . . . . 6 (∀𝑗𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗)) ↔ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))
159, 14syl6bb 288 . . . . 5 (𝑡 = 𝑠 → (∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
162, 4, 153anbi123d 1427 . . . 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 3294 . . 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 2886 . 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 2818 . 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 9863 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 1079   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  cdif 3930  c0 4288  𝒫 cpw 4535  {csn 4557  cmpt 5137  dom cdm 5548  cres 5550  suc csuc 6186  wf 6344  cfv 6348  ωcom 7569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-dc 9856
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-1o 8091
This theorem is referenced by:  axdc4lem  9865
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