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Theorem axdc2 10472
Description: An apparent strengthening of ax-dc 10469 (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 2812 . . . . 5 (𝑠 = 𝑥 → (𝑠𝐴𝑥𝐴))
32adantr 480 . . . 4 ((𝑠 = 𝑥𝑡 = 𝑦) → (𝑠𝐴𝑥𝐴))
4 fveq2 6897 . . . . . 6 (𝑠 = 𝑥 → (𝐹𝑠) = (𝐹𝑥))
54eleq2d 2815 . . . . 5 (𝑠 = 𝑥 → (𝑡 ∈ (𝐹𝑠) ↔ 𝑡 ∈ (𝐹𝑥)))
6 eleq1w 2812 . . . . 5 (𝑡 = 𝑦 → (𝑡 ∈ (𝐹𝑥) ↔ 𝑦 ∈ (𝐹𝑥)))
75, 6sylan9bb 509 . . . 4 ((𝑠 = 𝑥𝑡 = 𝑦) → (𝑡 ∈ (𝐹𝑠) ↔ 𝑦 ∈ (𝐹𝑥)))
83, 7anbi12d 631 . . 3 ((𝑠 = 𝑥𝑡 = 𝑦) → ((𝑠𝐴𝑡 ∈ (𝐹𝑠)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
98cbvopabv 5221 . 2 {⟨𝑠, 𝑡⟩ ∣ (𝑠𝐴𝑡 ∈ (𝐹𝑠))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
10 fveq2 6897 . . 3 (𝑛 = 𝑥 → (𝑛) = (𝑥))
1110cbvmptv 5261 . 2 (𝑛 ∈ ω ↦ (𝑛)) = (𝑥 ∈ ω ↦ (𝑥))
121, 9, 11axdc2lem 10471 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 395  wex 1774  wcel 2099  wne 2937  wral 3058  Vcvv 3471  cdif 3944  c0 4323  𝒫 cpw 4603  {csn 4629  {copab 5210  cmpt 5231  suc csuc 6371  wf 6544  cfv 6548  ωcom 7870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-dc 10469
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-om 7871  df-1o 8486
This theorem is referenced by:  axdc3lem4  10476
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