Theorem axdc2 9860

Description: An apparent strengthening of ax-dc 9857 (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 2872	. . . . 5 ⊢ (𝑠 = 𝑥 → (𝑠 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
3	2	adantr 484	. . . 4 ⊢ ((𝑠 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑠 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
4		fveq2 6645	. . . . . 6 ⊢ (𝑠 = 𝑥 → (𝐹‘𝑠) = (𝐹‘𝑥))
5	4	eleq2d 2875	. . . . 5 ⊢ (𝑠 = 𝑥 → (𝑡 ∈ (𝐹‘𝑠) ↔ 𝑡 ∈ (𝐹‘𝑥)))
6		eleq1w 2872	. . . . 5 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ (𝐹‘𝑥) ↔ 𝑦 ∈ (𝐹‘𝑥)))
7	5, 6	sylan9bb 513	. . . 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 5102	. 2 ⊢ {⟨𝑠, 𝑡⟩ ∣ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ (𝐹‘𝑠))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
10		fveq2 6645	. . 3 ⊢ (𝑛 = 𝑥 → (ℎ‘𝑛) = (ℎ‘𝑥))
11	10	cbvmptv 5133	. 2 ⊢ (𝑛 ∈ ω ↦ (ℎ‘𝑛)) = (𝑥 ∈ ω ↦ (ℎ‘𝑥))
12	1, 9, 11	axdc2lem 9859	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  Vcvv 3441   ∖ cdif 3878  ∅c0 4243  𝒫 cpw 4497  {csn 4525  {copab 5092   ↦ cmpt 5110  suc csuc 6161  ⟶wf 6320  ‘cfv 6324  ωcom 7560

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-dc 9857

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-om 7561  df-1o 8085

This theorem is referenced by:  axdc3lem4  9864