Theorem axdc2 10362

Description: An apparent strengthening of ax-dc 10359 (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 2820	. . . . 5 ⊢ (𝑠 = 𝑥 → (𝑠 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
3	2	adantr 480	. . . 4 ⊢ ((𝑠 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑠 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
4		fveq2 6834	. . . . . 6 ⊢ (𝑠 = 𝑥 → (𝐹‘𝑠) = (𝐹‘𝑥))
5	4	eleq2d 2823	. . . . 5 ⊢ (𝑠 = 𝑥 → (𝑡 ∈ (𝐹‘𝑠) ↔ 𝑡 ∈ (𝐹‘𝑥)))
6		eleq1w 2820	. . . . 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 5159	. 2 ⊢ {⟨𝑠, 𝑡⟩ ∣ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ (𝐹‘𝑠))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
10		fveq2 6834	. . 3 ⊢ (𝑛 = 𝑥 → (ℎ‘𝑛) = (ℎ‘𝑥))
11	10	cbvmptv 5190	. 2 ⊢ (𝑛 ∈ ω ↦ (ℎ‘𝑛)) = (𝑥 ∈ ω ↦ (ℎ‘𝑥))
12	1, 9, 11	axdc2lem 10361	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 2933  ∀wral 3052  Vcvv 3430   ∖ cdif 3887  ∅c0 4274  𝒫 cpw 4542  {csn 4568  {copab 5148   ↦ cmpt 5167  suc csuc 6319  ⟶wf 6488  ‘cfv 6492  ωcom 7810

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 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-dc 10359

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-om 7811  df-1o 8398

This theorem is referenced by:  axdc3lem4  10366