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Theorem axdc2 9865
Description: An apparent strengthening of ax-dc 9862 (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 2898 . . . . 5 (𝑠 = 𝑥 → (𝑠𝐴𝑥𝐴))
32adantr 484 . . . 4 ((𝑠 = 𝑥𝑡 = 𝑦) → (𝑠𝐴𝑥𝐴))
4 fveq2 6659 . . . . . 6 (𝑠 = 𝑥 → (𝐹𝑠) = (𝐹𝑥))
54eleq2d 2901 . . . . 5 (𝑠 = 𝑥 → (𝑡 ∈ (𝐹𝑠) ↔ 𝑡 ∈ (𝐹𝑥)))
6 eleq1w 2898 . . . . 5 (𝑡 = 𝑦 → (𝑡 ∈ (𝐹𝑥) ↔ 𝑦 ∈ (𝐹𝑥)))
75, 6sylan9bb 513 . . . 4 ((𝑠 = 𝑥𝑡 = 𝑦) → (𝑡 ∈ (𝐹𝑠) ↔ 𝑦 ∈ (𝐹𝑥)))
83, 7anbi12d 633 . . 3 ((𝑠 = 𝑥𝑡 = 𝑦) → ((𝑠𝐴𝑡 ∈ (𝐹𝑠)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
98cbvopabv 5125 . 2 {⟨𝑠, 𝑡⟩ ∣ (𝑠𝐴𝑡 ∈ (𝐹𝑠))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
10 fveq2 6659 . . 3 (𝑛 = 𝑥 → (𝑛) = (𝑥))
1110cbvmptv 5156 . 2 (𝑛 ∈ ω ↦ (𝑛)) = (𝑥 ∈ ω ↦ (𝑥))
121, 9, 11axdc2lem 9864 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 209  wa 399  wex 1781  wcel 2115  wne 3014  wral 3133  Vcvv 3480  cdif 3916  c0 4276  𝒫 cpw 4522  {csn 4550  {copab 5115  cmpt 5133  suc csuc 6181  wf 6340  cfv 6344  ωcom 7571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-dc 9862
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352  df-om 7572  df-1o 8094
This theorem is referenced by:  axdc3lem4  9869
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