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Theorem noseqrdg0 28313
Description: Initial value of a recursive definition generator on surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)
Hypotheses
Ref Expression
om2noseq.1 (𝜑𝐶 No )
om2noseq.2 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
om2noseq.3 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
noseqrdg.1 (𝜑𝐴𝑉)
noseqrdg.2 (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
noseqrdg.3 (𝜑𝑆 = ran 𝑅)
Assertion
Ref Expression
noseqrdg0 (𝜑 → (𝑆𝐶) = 𝐴)
Distinct variable groups:   𝑥,𝐶   𝑥,𝐹,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem noseqrdg0
StepHypRef Expression
1 om2noseq.1 . . . 4 (𝜑𝐶 No )
2 om2noseq.2 . . . 4 (𝜑𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
3 om2noseq.3 . . . 4 (𝜑𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
4 noseqrdg.1 . . . 4 (𝜑𝐴𝑉)
5 noseqrdg.2 . . . 4 (𝜑𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
6 noseqrdg.3 . . . 4 (𝜑𝑆 = ran 𝑅)
71, 2, 3, 4, 5, 6noseqrdgfn 28312 . . 3 (𝜑𝑆 Fn 𝑍)
87fnfund 6669 . 2 (𝜑 → Fun 𝑆)
9 frfnom 8475 . . . . 5 (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω
105fneq1d 6661 . . . . 5 (𝜑 → (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω))
119, 10mpbiri 258 . . . 4 (𝜑𝑅 Fn ω)
12 peano1 7910 . . . 4 ∅ ∈ ω
13 fnfvelrn 7100 . . . 4 ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅)
1411, 12, 13sylancl 586 . . 3 (𝜑 → (𝑅‘∅) ∈ ran 𝑅)
155fveq1d 6908 . . . 4 (𝜑 → (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅))
16 opex 5469 . . . . 5 𝐶, 𝐴⟩ ∈ V
17 fr0g 8476 . . . . 5 (⟨𝐶, 𝐴⟩ ∈ V → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅) = ⟨𝐶, 𝐴⟩)
1816, 17ax-mp 5 . . . 4 ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅) = ⟨𝐶, 𝐴
1915, 18eqtr2di 2794 . . 3 (𝜑 → ⟨𝐶, 𝐴⟩ = (𝑅‘∅))
2014, 19, 63eltr4d 2856 . 2 (𝜑 → ⟨𝐶, 𝐴⟩ ∈ 𝑆)
21 funopfv 6958 . 2 (Fun 𝑆 → (⟨𝐶, 𝐴⟩ ∈ 𝑆 → (𝑆𝐶) = 𝐴))
228, 20, 21sylc 65 1 (𝜑 → (𝑆𝐶) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cop 4632  cmpt 5225  ran crn 5686  cres 5687  cima 5688  Fun wfun 6555   Fn wfn 6556  cfv 6561  (class class class)co 7431  cmpo 7433  ωcom 7887  reccrdg 8449   No csur 27684   1s c1s 27868   +s cadds 27992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec2 27982  df-adds 27993
This theorem is referenced by:  seqs1  28316
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