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Theorem axdc3lem 9310
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 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
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
StepHypRef Expression
1 dcomex 9307 . . . 4 ω ∈ V
2 axdc3lem.1 . . . 4 𝐴 ∈ V
31, 2xpex 7004 . . 3 (ω × 𝐴) ∈ V
43pwex 4878 . 2 𝒫 (ω × 𝐴) ∈ V
5 axdc3lem.2 . . 3 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
6 fssxp 6098 . . . . . . . . 9 (𝑠:suc 𝑛𝐴𝑠 ⊆ (suc 𝑛 × 𝐴))
7 peano2 7128 . . . . . . . . . 10 (𝑛 ∈ ω → suc 𝑛 ∈ ω)
8 omelon2 7119 . . . . . . . . . . . 12 (ω ∈ V → ω ∈ On)
91, 8ax-mp 5 . . . . . . . . . . 11 ω ∈ On
109onelssi 5874 . . . . . . . . . 10 (suc 𝑛 ∈ ω → suc 𝑛 ⊆ ω)
11 xpss1 5161 . . . . . . . . . 10 (suc 𝑛 ⊆ ω → (suc 𝑛 × 𝐴) ⊆ (ω × 𝐴))
127, 10, 113syl 18 . . . . . . . . 9 (𝑛 ∈ ω → (suc 𝑛 × 𝐴) ⊆ (ω × 𝐴))
136, 12sylan9ss 3649 . . . . . . . 8 ((𝑠:suc 𝑛𝐴𝑛 ∈ ω) → 𝑠 ⊆ (ω × 𝐴))
14 selpw 4198 . . . . . . . 8 (𝑠 ∈ 𝒫 (ω × 𝐴) ↔ 𝑠 ⊆ (ω × 𝐴))
1513, 14sylibr 224 . . . . . . 7 ((𝑠:suc 𝑛𝐴𝑛 ∈ ω) → 𝑠 ∈ 𝒫 (ω × 𝐴))
1615ancoms 468 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑠:suc 𝑛𝐴) → 𝑠 ∈ 𝒫 (ω × 𝐴))
17163ad2antr1 1246 . . . . 5 ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))) → 𝑠 ∈ 𝒫 (ω × 𝐴))
1817rexlimiva 3057 . . . 4 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → 𝑠 ∈ 𝒫 (ω × 𝐴))
1918abssi 3710 . . 3 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ 𝒫 (ω × 𝐴)
205, 19eqsstri 3668 . 2 𝑆 ⊆ 𝒫 (ω × 𝐴)
214, 20ssexi 4836 1 𝑆 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231  wss 3607  c0 3948  𝒫 cpw 4191   × cxp 5141  Oncon0 5761  suc csuc 5763  wf 5922  cfv 5926  ωcom 7107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-dc 9306
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-om 7108  df-1o 7605
This theorem is referenced by:  axdc3lem2  9311  axdc3lem4  9313
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