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Theorem axdc3lem 9864
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 9861 . . . 4 ω ∈ V
2 axdc3lem.1 . . . 4 𝐴 ∈ V
31, 2xpex 7468 . . 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 7594 . . . . . . . . . 10 (𝑛 ∈ ω → suc 𝑛 ∈ ω)
8 omelon2 7584 . . . . . . . . . . . 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 3978 . . . . . . . 8 ((𝑠:suc 𝑛𝐴𝑛 ∈ ω) → 𝑠 ⊆ (ω × 𝐴))
14 velpw 4545 . . . . . . . 8 (𝑠 ∈ 𝒫 (ω × 𝐴) ↔ 𝑠 ⊆ (ω × 𝐴))
1513, 14sylibr 236 . . . . . . 7 ((𝑠:suc 𝑛𝐴𝑛 ∈ ω) → 𝑠 ∈ 𝒫 (ω × 𝐴))
1615ancoms 461 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑠:suc 𝑛𝐴) → 𝑠 ∈ 𝒫 (ω × 𝐴))
17163ad2antr1 1182 . . . . 5 ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))) → 𝑠 ∈ 𝒫 (ω × 𝐴))
1817rexlimiva 3279 . . . 4 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → 𝑠 ∈ 𝒫 (ω × 𝐴))
1918abssi 4044 . . 3 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ 𝒫 (ω × 𝐴)
205, 19eqsstri 3999 . 2 𝑆 ⊆ 𝒫 (ω × 𝐴)
214, 20ssexi 5217 1 𝑆 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 398  w3a 1081   = wceq 1530  wcel 2107  {cab 2797  wral 3136  wrex 3137  Vcvv 3493  wss 3934  c0 4289  𝒫 cpw 4537   × cxp 5546  Oncon0 6184  suc csuc 6186  wf 6344  cfv 6348  ωcom 7572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-dc 9860
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  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 7573  df-1o 8094
This theorem is referenced by:  axdc3lem2  9865  axdc3lem4  9867
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