Theorem axdc2 10371

Description: An apparent strengthening of ax-dc 10368 (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 2819	. . . . 5 ⊢ (𝑠 = 𝑥 → (𝑠 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
3	2	adantr 480	. . . 4 ⊢ ((𝑠 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑠 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
4		fveq2 6840	. . . . . 6 ⊢ (𝑠 = 𝑥 → (𝐹‘𝑠) = (𝐹‘𝑥))
5	4	eleq2d 2822	. . . . 5 ⊢ (𝑠 = 𝑥 → (𝑡 ∈ (𝐹‘𝑠) ↔ 𝑡 ∈ (𝐹‘𝑥)))
6		eleq1w 2819	. . . . 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 633	. . 3 ⊢ ((𝑠 = 𝑥 ∧ 𝑡 = 𝑦) → ((𝑠 ∈ 𝐴 ∧ 𝑡 ∈ (𝐹‘𝑠)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))))
9	8	cbvopabv 5158	. 2 ⊢ {⟨𝑠, 𝑡⟩ ∣ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ (𝐹‘𝑠))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
10		fveq2 6840	. . 3 ⊢ (𝑛 = 𝑥 → (ℎ‘𝑛) = (ℎ‘𝑥))
11	10	cbvmptv 5189	. 2 ⊢ (𝑛 ∈ ω ↦ (ℎ‘𝑛)) = (𝑥 ∈ ω ↦ (ℎ‘𝑥))
12	1, 9, 11	axdc2lem 10370	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 1781   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  Vcvv 3429   ∖ cdif 3886  ∅c0 4273  𝒫 cpw 4541  {csn 4567  {copab 5147   ↦ cmpt 5166  suc csuc 6325  ⟶wf 6494  ‘cfv 6498  ωcom 7817

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-dc 10368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-om 7818  df-1o 8405

This theorem is referenced by:  axdc3lem4  10375