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Theorem axdc2 9309
Description: An apparent strengthening of ax-dc 9306 (but derived from it) which shows that there is a denumerable sequence 𝑔 for any function that maps elements of a set 𝐴 to nonempty subsets of 𝐴 such that 𝑔(𝑥 + 1) ∈ 𝐹(𝑔(𝑥)) for all 𝑥 ∈ ω. The finitistic version of this can be proven by induction, but the infinite version requires this new axiom. (Contributed by Mario Carneiro, 25-Jan-2013.)
Hypothesis
Ref Expression
axdc2.1 𝐴 ∈ V
Assertion
Ref Expression
axdc2 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
Distinct variable groups:   𝐴,𝑔,𝑘   𝑔,𝐹,𝑘

Proof of Theorem axdc2
Dummy variables 𝑠 𝑡 𝑥 𝑦 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axdc2.1 . 2 𝐴 ∈ V
2 eleq1 2718 . . . . 5 (𝑠 = 𝑥 → (𝑠𝐴𝑥𝐴))
32adantr 480 . . . 4 ((𝑠 = 𝑥𝑡 = 𝑦) → (𝑠𝐴𝑥𝐴))
4 fveq2 6229 . . . . . 6 (𝑠 = 𝑥 → (𝐹𝑠) = (𝐹𝑥))
54eleq2d 2716 . . . . 5 (𝑠 = 𝑥 → (𝑡 ∈ (𝐹𝑠) ↔ 𝑡 ∈ (𝐹𝑥)))
6 eleq1 2718 . . . . 5 (𝑡 = 𝑦 → (𝑡 ∈ (𝐹𝑥) ↔ 𝑦 ∈ (𝐹𝑥)))
75, 6sylan9bb 736 . . . 4 ((𝑠 = 𝑥𝑡 = 𝑦) → (𝑡 ∈ (𝐹𝑠) ↔ 𝑦 ∈ (𝐹𝑥)))
83, 7anbi12d 747 . . 3 ((𝑠 = 𝑥𝑡 = 𝑦) → ((𝑠𝐴𝑡 ∈ (𝐹𝑠)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
98cbvopabv 4755 . 2 {⟨𝑠, 𝑡⟩ ∣ (𝑠𝐴𝑡 ∈ (𝐹𝑠))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
10 fveq2 6229 . . 3 (𝑛 = 𝑥 → (𝑛) = (𝑥))
1110cbvmptv 4783 . 2 (𝑛 ∈ ω ↦ (𝑛)) = (𝑥 ∈ ω ↦ (𝑥))
121, 9, 11axdc2lem 9308 1 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wex 1744  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cdif 3604  c0 3948  𝒫 cpw 4191  {csn 4210  {copab 4745  cmpt 4762  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-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-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-fv 5934  df-om 7108  df-1o 7605
This theorem is referenced by:  axdc3lem4  9313
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