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Theorem axdc2 10386
Description: An apparent strengthening of ax-dc 10383 (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 2821 . . . . 5 (𝑠 = 𝑥 → (𝑠𝐴𝑥𝐴))
32adantr 482 . . . 4 ((𝑠 = 𝑥𝑡 = 𝑦) → (𝑠𝐴𝑥𝐴))
4 fveq2 6843 . . . . . 6 (𝑠 = 𝑥 → (𝐹𝑠) = (𝐹𝑥))
54eleq2d 2824 . . . . 5 (𝑠 = 𝑥 → (𝑡 ∈ (𝐹𝑠) ↔ 𝑡 ∈ (𝐹𝑥)))
6 eleq1w 2821 . . . . 5 (𝑡 = 𝑦 → (𝑡 ∈ (𝐹𝑥) ↔ 𝑦 ∈ (𝐹𝑥)))
75, 6sylan9bb 511 . . . 4 ((𝑠 = 𝑥𝑡 = 𝑦) → (𝑡 ∈ (𝐹𝑠) ↔ 𝑦 ∈ (𝐹𝑥)))
83, 7anbi12d 632 . . 3 ((𝑠 = 𝑥𝑡 = 𝑦) → ((𝑠𝐴𝑡 ∈ (𝐹𝑠)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
98cbvopabv 5179 . 2 {⟨𝑠, 𝑡⟩ ∣ (𝑠𝐴𝑡 ∈ (𝐹𝑠))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
10 fveq2 6843 . . 3 (𝑛 = 𝑥 → (𝑛) = (𝑥))
1110cbvmptv 5219 . 2 (𝑛 ∈ ω ↦ (𝑛)) = (𝑥 ∈ ω ↦ (𝑥))
121, 9, 11axdc2lem 10385 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 205  wa 397  wex 1782  wcel 2107  wne 2944  wral 3065  Vcvv 3446  cdif 3908  c0 4283  𝒫 cpw 4561  {csn 4587  {copab 5168  cmpt 5189  suc csuc 6320  wf 6493  cfv 6497  ωcom 7803
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-dc 10383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-om 7804  df-1o 8413
This theorem is referenced by:  axdc3lem4  10390
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