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Theorem axdc3lem 9860
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 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 9857 . . . 4 ω ∈ V
2 axdc3lem.1 . . . 4 𝐴 ∈ V
31, 2xpex 7465 . . 3 (ω × 𝐴) ∈ V
43pwex 5272 . 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 6527 . . . . . . . . 9 (𝑠:suc 𝑛𝐴𝑠 ⊆ (suc 𝑛 × 𝐴))
7 peano2 7591 . . . . . . . . . 10 (𝑛 ∈ ω → suc 𝑛 ∈ ω)
8 omelon2 7581 . . . . . . . . . . . 12 (ω ∈ V → ω ∈ On)
91, 8ax-mp 5 . . . . . . . . . . 11 ω ∈ On
109onelssi 6292 . . . . . . . . . 10 (suc 𝑛 ∈ ω → suc 𝑛 ⊆ ω)
11 xpss1 5567 . . . . . . . . . 10 (suc 𝑛 ⊆ ω → (suc 𝑛 × 𝐴) ⊆ (ω × 𝐴))
127, 10, 113syl 18 . . . . . . . . 9 (𝑛 ∈ ω → (suc 𝑛 × 𝐴) ⊆ (ω × 𝐴))
136, 12sylan9ss 3977 . . . . . . . 8 ((𝑠:suc 𝑛𝐴𝑛 ∈ ω) → 𝑠 ⊆ (ω × 𝐴))
14 velpw 4543 . . . . . . . 8 (𝑠 ∈ 𝒫 (ω × 𝐴) ↔ 𝑠 ⊆ (ω × 𝐴))
1513, 14sylibr 235 . . . . . . 7 ((𝑠:suc 𝑛𝐴𝑛 ∈ ω) → 𝑠 ∈ 𝒫 (ω × 𝐴))
1615ancoms 459 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑠:suc 𝑛𝐴) → 𝑠 ∈ 𝒫 (ω × 𝐴))
17163ad2antr1 1180 . . . . 5 ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))) → 𝑠 ∈ 𝒫 (ω × 𝐴))
1817rexlimiva 3278 . . . 4 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → 𝑠 ∈ 𝒫 (ω × 𝐴))
1918abssi 4043 . . 3 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ 𝒫 (ω × 𝐴)
205, 19eqsstri 3998 . 2 𝑆 ⊆ 𝒫 (ω × 𝐴)
214, 20ssexi 5217 1 𝑆 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wral 3135  wrex 3136  Vcvv 3492  wss 3933  c0 4288  𝒫 cpw 4535   × cxp 5546  Oncon0 6184  suc csuc 6186  wf 6344  cfv 6348  ωcom 7569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-dc 9856
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-om 7570  df-1o 8091
This theorem is referenced by:  axdc3lem2  9861  axdc3lem4  9863
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