Theorem axdc2 10350

Description: An apparent strengthening of ax-dc 10347 (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‘(𝑔‘𝑘))))

Distinct variable groups:   𝐴,𝑔,𝑘   𝑔,𝐹,𝑘

Proof of Theorem axdc2

Dummy variables ℎ 𝑠 𝑡 𝑥 𝑦 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axdc2.1	. 2 ⊢ 𝐴 ∈ V
2		eleq1w 2816	. . . . 5 ⊢ (𝑠 = 𝑥 → (𝑠 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
3	2	adantr 480	. . . 4 ⊢ ((𝑠 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑠 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
4		fveq2 6831	. . . . . 6 ⊢ (𝑠 = 𝑥 → (𝐹‘𝑠) = (𝐹‘𝑥))
5	4	eleq2d 2819	. . . . 5 ⊢ (𝑠 = 𝑥 → (𝑡 ∈ (𝐹‘𝑠) ↔ 𝑡 ∈ (𝐹‘𝑥)))
6		eleq1w 2816	. . . . 5 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ (𝐹‘𝑥) ↔ 𝑦 ∈ (𝐹‘𝑥)))
7	5, 6	sylan9bb 509	. . . 4 ⊢ ((𝑠 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑡 ∈ (𝐹‘𝑠) ↔ 𝑦 ∈ (𝐹‘𝑥)))
8	3, 7	anbi12d 632	. . 3 ⊢ ((𝑠 = 𝑥 ∧ 𝑡 = 𝑦) → ((𝑠 ∈ 𝐴 ∧ 𝑡 ∈ (𝐹‘𝑠)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))))
9	8	cbvopabv 5168	. 2 ⊢ {⟨𝑠, 𝑡⟩ ∣ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ (𝐹‘𝑠))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
10		fveq2 6831	. . 3 ⊢ (𝑛 = 𝑥 → (ℎ‘𝑛) = (ℎ‘𝑥))
11	10	cbvmptv 5199	. 2 ⊢ (𝑛 ∈ ω ↦ (ℎ‘𝑛)) = (𝑥 ∈ ω ↦ (ℎ‘𝑥))
12	1, 9, 11	axdc2lem 10349	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1780   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  Vcvv 3438   ∖ cdif 3896  ∅c0 4284  𝒫 cpw 4551  {csn 4577  {copab 5157   ↦ cmpt 5176  suc csuc 6316  ⟶wf 6485  ‘cfv 6489  ωcom 7805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-dc 10347

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-om 7806  df-1o 8394

This theorem is referenced by:  axdc3lem4  10354