Theorem axdc3lem 10429

Description: The class 𝑆 of finite approximations to the DC sequence is a set. (We derive here the stronger statement that 𝑆 is a subset of a specific set, namely 𝒫 (ω × 𝐴).) (Contributed by Mario Carneiro, 27-Jan-2013.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 18-Mar-2014.)

Hypotheses

Ref	Expression
axdc3lem.1	⊢ 𝐴 ∈ V
axdc3lem.2	⊢ 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))}

Assertion

Ref	Expression
axdc3lem	⊢ 𝑆 ∈ V

Distinct variable group:   𝐴,𝑛,𝑠

Allowed substitution hints:   𝐴(𝑘)   𝐶(𝑘,𝑛,𝑠)   𝑆(𝑘,𝑛,𝑠)   𝐹(𝑘,𝑛,𝑠)

Proof of Theorem axdc3lem

Step	Hyp	Ref	Expression
1		dcomex 10426	. . . 4 ⊢ ω ∈ V
2		axdc3lem.1	. . . 4 ⊢ 𝐴 ∈ V
3	1, 2	xpex 7724	. . 3 ⊢ (ω × 𝐴) ∈ V
4	3	pwex 5372	. 2 ⊢ 𝒫 (ω × 𝐴) ∈ V
5		axdc3lem.2	. . 3 ⊢ 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))}
6		fssxp 6733	. . . . . . . . 9 ⊢ (𝑠:suc 𝑛⟶𝐴 → 𝑠 ⊆ (suc 𝑛 × 𝐴))
7		peano2 7865	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω)
8		omelon2 7852	. . . . . . . . . . . 12 ⊢ (ω ∈ V → ω ∈ On)
9	1, 8	ax-mp 5	. . . . . . . . . . 11 ⊢ ω ∈ On
10	9	onelssi 6469	. . . . . . . . . 10 ⊢ (suc 𝑛 ∈ ω → suc 𝑛 ⊆ ω)
11		xpss1 5689	. . . . . . . . . 10 ⊢ (suc 𝑛 ⊆ ω → (suc 𝑛 × 𝐴) ⊆ (ω × 𝐴))
12	7, 10, 11	3syl 18	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → (suc 𝑛 × 𝐴) ⊆ (ω × 𝐴))
13	6, 12	sylan9ss 3992	. . . . . . . 8 ⊢ ((𝑠:suc 𝑛⟶𝐴 ∧ 𝑛 ∈ ω) → 𝑠 ⊆ (ω × 𝐴))
14		velpw 4602	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 (ω × 𝐴) ↔ 𝑠 ⊆ (ω × 𝐴))
15	13, 14	sylibr 233	. . . . . . 7 ⊢ ((𝑠:suc 𝑛⟶𝐴 ∧ 𝑛 ∈ ω) → 𝑠 ∈ 𝒫 (ω × 𝐴))
16	15	ancoms 459	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑠:suc 𝑛⟶𝐴) → 𝑠 ∈ 𝒫 (ω × 𝐴))
17	16	3ad2antr1 1188	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))) → 𝑠 ∈ 𝒫 (ω × 𝐴))
18	17	rexlimiva 3147	. . . 4 ⊢ (∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))) → 𝑠 ∈ 𝒫 (ω × 𝐴))
19	18	abssi 4064	. . 3 ⊢ {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))} ⊆ 𝒫 (ω × 𝐴)
20	5, 19	eqsstri 4013	. 2 ⊢ 𝑆 ⊆ 𝒫 (ω × 𝐴)
21	4, 20	ssexi 5316	1 ⊢ 𝑆 ∈ V

Syntax hints:   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ⊆ wss 3945  ∅c0 4319  𝒫 cpw 4597   × cxp 5668  Oncon0 6354  suc csuc 6356  ⟶wf 6529  ‘cfv 6533  ωcom 7839

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-dc 10425

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-om 7840  df-1o 8450

This theorem is referenced by:  axdc3lem2  10430  axdc3lem4  10432