Theorem axdc3lem 9272

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).) (Unnecessary distinct variable restrictions were removed by David Abernethy, 18-Mar-2014.) (Contributed by Mario Carneiro, 27-Jan-2013.) (Revised by Mario Carneiro, 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 9269	. . . 4 ⊢ ω ∈ V
2		axdc3lem.1	. . . 4 ⊢ 𝐴 ∈ V
3	1, 2	xpex 6962	. . 3 ⊢ (ω × 𝐴) ∈ V
4	3	pwex 4848	. 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 6060	. . . . . . . . 9 ⊢ (𝑠:suc 𝑛⟶𝐴 → 𝑠 ⊆ (suc 𝑛 × 𝐴))
7		peano2 7086	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω)
8		omelon2 7077	. . . . . . . . . . . 12 ⊢ (ω ∈ V → ω ∈ On)
9	1, 8	ax-mp 5	. . . . . . . . . . 11 ⊢ ω ∈ On
10	9	onelssi 5836	. . . . . . . . . 10 ⊢ (suc 𝑛 ∈ ω → suc 𝑛 ⊆ ω)
11		xpss1 5228	. . . . . . . . . 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 3616	. . . . . . . 8 ⊢ ((𝑠:suc 𝑛⟶𝐴 ∧ 𝑛 ∈ ω) → 𝑠 ⊆ (ω × 𝐴))
14		selpw 4165	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 (ω × 𝐴) ↔ 𝑠 ⊆ (ω × 𝐴))
15	13, 14	sylibr 224	. . . . . . 7 ⊢ ((𝑠:suc 𝑛⟶𝐴 ∧ 𝑛 ∈ ω) → 𝑠 ∈ 𝒫 (ω × 𝐴))
16	15	ancoms 469	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑠:suc 𝑛⟶𝐴) → 𝑠 ∈ 𝒫 (ω × 𝐴))
17	16	3ad2antr1 1226	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))) → 𝑠 ∈ 𝒫 (ω × 𝐴))
18	17	rexlimiva 3028	. . . 4 ⊢ (∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))) → 𝑠 ∈ 𝒫 (ω × 𝐴))
19	18	abssi 3677	. . 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 3635	. 2 ⊢ 𝑆 ⊆ 𝒫 (ω × 𝐴)
21	4, 20	ssexi 4803	1 ⊢ 𝑆 ∈ V

Syntax hints:   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158   × cxp 5112  Oncon0 5723  suc csuc 5725  ⟶wf 5884  ‘cfv 5888  ωcom 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-dc 9268

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-om 7066  df-1o 7560

This theorem is referenced by:  axdc3lem2  9273  axdc3lem4  9275