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Theorem axdc2 9859
Description: An apparent strengthening of ax-dc 9856 (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 eleq1w 2892 . . . . 5 (𝑠 = 𝑥 → (𝑠𝐴𝑥𝐴))
32adantr 481 . . . 4 ((𝑠 = 𝑥𝑡 = 𝑦) → (𝑠𝐴𝑥𝐴))
4 fveq2 6663 . . . . . 6 (𝑠 = 𝑥 → (𝐹𝑠) = (𝐹𝑥))
54eleq2d 2895 . . . . 5 (𝑠 = 𝑥 → (𝑡 ∈ (𝐹𝑠) ↔ 𝑡 ∈ (𝐹𝑥)))
6 eleq1w 2892 . . . . 5 (𝑡 = 𝑦 → (𝑡 ∈ (𝐹𝑥) ↔ 𝑦 ∈ (𝐹𝑥)))
75, 6sylan9bb 510 . . . 4 ((𝑠 = 𝑥𝑡 = 𝑦) → (𝑡 ∈ (𝐹𝑠) ↔ 𝑦 ∈ (𝐹𝑥)))
83, 7anbi12d 630 . . 3 ((𝑠 = 𝑥𝑡 = 𝑦) → ((𝑠𝐴𝑡 ∈ (𝐹𝑠)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
98cbvopabv 5129 . 2 {⟨𝑠, 𝑡⟩ ∣ (𝑠𝐴𝑡 ∈ (𝐹𝑠))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
10 fveq2 6663 . . 3 (𝑛 = 𝑥 → (𝑛) = (𝑥))
1110cbvmptv 5160 . 2 (𝑛 ∈ ω ↦ (𝑛)) = (𝑥 ∈ ω ↦ (𝑥))
121, 9, 11axdc2lem 9858 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 207  wa 396  wex 1771  wcel 2105  wne 3013  wral 3135  Vcvv 3492  cdif 3930  c0 4288  𝒫 cpw 4535  {csn 4557  {copab 5119  cmpt 5137  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-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-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-fv 6356  df-om 7570  df-1o 8091
This theorem is referenced by:  axdc3lem4  9863
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