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Theorem axdc3lem 9155
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 9152 . . . 4 ω ∈ V
2 axdc3lem.1 . . . 4 𝐴 ∈ V
31, 2xpex 6860 . . 3 (ω × 𝐴) ∈ V
43pwex 4774 . 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 5973 . . . . . . . . 9 (𝑠:suc 𝑛𝐴𝑠 ⊆ (suc 𝑛 × 𝐴))
7 peano2 6978 . . . . . . . . . 10 (𝑛 ∈ ω → suc 𝑛 ∈ ω)
8 omelon2 6969 . . . . . . . . . . . 12 (ω ∈ V → ω ∈ On)
91, 8ax-mp 5 . . . . . . . . . . 11 ω ∈ On
109onelssi 5753 . . . . . . . . . 10 (suc 𝑛 ∈ ω → suc 𝑛 ⊆ ω)
11 xpss1 5151 . . . . . . . . . 10 (suc 𝑛 ⊆ ω → (suc 𝑛 × 𝐴) ⊆ (ω × 𝐴))
127, 10, 113syl 18 . . . . . . . . 9 (𝑛 ∈ ω → (suc 𝑛 × 𝐴) ⊆ (ω × 𝐴))
136, 12sylan9ss 3581 . . . . . . . 8 ((𝑠:suc 𝑛𝐴𝑛 ∈ ω) → 𝑠 ⊆ (ω × 𝐴))
14 selpw 4115 . . . . . . . 8 (𝑠 ∈ 𝒫 (ω × 𝐴) ↔ 𝑠 ⊆ (ω × 𝐴))
1513, 14sylibr 223 . . . . . . 7 ((𝑠:suc 𝑛𝐴𝑛 ∈ ω) → 𝑠 ∈ 𝒫 (ω × 𝐴))
1615ancoms 468 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑠:suc 𝑛𝐴) → 𝑠 ∈ 𝒫 (ω × 𝐴))
17163ad2antr1 1219 . . . . 5 ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))) → 𝑠 ∈ 𝒫 (ω × 𝐴))
1817rexlimiva 3010 . . . 4 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → 𝑠 ∈ 𝒫 (ω × 𝐴))
1918abssi 3640 . . 3 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ 𝒫 (ω × 𝐴)
205, 19eqsstri 3598 . 2 𝑆 ⊆ 𝒫 (ω × 𝐴)
214, 20ssexi 4731 1 𝑆 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 383  w3a 1031   = wceq 1475  wcel 1977  {cab 2596  wral 2896  wrex 2897  Vcvv 3173  wss 3540  c0 3874  𝒫 cpw 4108   × cxp 5036  Oncon0 5640  suc csuc 5642  wf 5800  cfv 5804  ωcom 6957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-dc 9151
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-om 6958  df-1o 7447
This theorem is referenced by:  axdc3lem2  9156  axdc3lem4  9158
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