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Theorem axdc3 9314
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 6064 . . . . 5 (𝑡 = 𝑠 → (𝑡:suc 𝑛𝐴𝑠:suc 𝑛𝐴))
3 fveq1 6228 . . . . . 6 (𝑡 = 𝑠 → (𝑡‘∅) = (𝑠‘∅))
43eqeq1d 2653 . . . . 5 (𝑡 = 𝑠 → ((𝑡‘∅) = 𝐶 ↔ (𝑠‘∅) = 𝐶))
5 fveq1 6228 . . . . . . . 8 (𝑡 = 𝑠 → (𝑡‘suc 𝑗) = (𝑠‘suc 𝑗))
6 fveq1 6228 . . . . . . . . 9 (𝑡 = 𝑠 → (𝑡𝑗) = (𝑠𝑗))
76fveq2d 6233 . . . . . . . 8 (𝑡 = 𝑠 → (𝐹‘(𝑡𝑗)) = (𝐹‘(𝑠𝑗)))
85, 7eleq12d 2724 . . . . . . 7 (𝑡 = 𝑠 → ((𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗))))
98ralbidv 3015 . . . . . 6 (𝑡 = 𝑠 → (∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ ∀𝑗𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗))))
10 suceq 5828 . . . . . . . . 9 (𝑗 = 𝑘 → suc 𝑗 = suc 𝑘)
1110fveq2d 6233 . . . . . . . 8 (𝑗 = 𝑘 → (𝑠‘suc 𝑗) = (𝑠‘suc 𝑘))
12 fveq2 6229 . . . . . . . . 9 (𝑗 = 𝑘 → (𝑠𝑗) = (𝑠𝑘))
1312fveq2d 6233 . . . . . . . 8 (𝑗 = 𝑘 → (𝐹‘(𝑠𝑗)) = (𝐹‘(𝑠𝑘)))
1411, 13eleq12d 2724 . . . . . . 7 (𝑗 = 𝑘 → ((𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗)) ↔ (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
1514cbvralv 3201 . . . . . 6 (∀𝑗𝑛 (𝑠‘suc 𝑗) ∈ (𝐹‘(𝑠𝑗)) ↔ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))
169, 15syl6bb 276 . . . . 5 (𝑡 = 𝑠 → (∀𝑗𝑛 (𝑡‘suc 𝑗) ∈ (𝐹‘(𝑡𝑗)) ↔ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
172, 4, 163anbi123d 1439 . . . 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)))))
1817rexbidv 3081 . . 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)))))
1918cbvabv 2776 . 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)))}
20 eqid 2651 . 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)})
211, 19, 20axdc3lem4 9313 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 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  c0 3948  𝒫 cpw 4191  {csn 4210  cmpt 4762  dom cdm 5143  cres 5145  suc csuc 5763  wf 5922  cfv 5926  ωcom 7107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-dc 9306
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605
This theorem is referenced by:  axdc4lem  9315
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