Theorem noseqrdg0 28277

Description: Initial value of a recursive definition generator on surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)

Hypotheses

Ref	Expression
om2noseq.1	⊢ (𝜑 → 𝐶 ∈ No )
om2noseq.2	⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
om2noseq.3	⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
noseqrdg.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
noseqrdg.2	⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
noseqrdg.3	⊢ (𝜑 → 𝑆 = ran 𝑅)

Assertion

Ref	Expression
noseqrdg0	⊢ (𝜑 → (𝑆‘𝐶) = 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

Step	Hyp	Ref	Expression
1		om2noseq.1	. . . 4 ⊢ (𝜑 → 𝐶 ∈ No )
2		om2noseq.2	. . . 4 ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
3		om2noseq.3	. . . 4 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
4		noseqrdg.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		noseqrdg.2	. . . 4 ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
6		noseqrdg.3	. . . 4 ⊢ (𝜑 → 𝑆 = ran 𝑅)
7	1, 2, 3, 4, 5, 6	noseqrdgfn 28276	. . 3 ⊢ (𝜑 → 𝑆 Fn 𝑍)
8	7	fnfund 6652	. 2 ⊢ (𝜑 → Fun 𝑆)
9		frfnom 8456	. . . . 5 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω
10	5	fneq1d 6644	. . . . 5 ⊢ (𝜑 → (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω))
11	9, 10	mpbiri 257	. . . 4 ⊢ (𝜑 → 𝑅 Fn ω)
12		peano1 7891	. . . 4 ⊢ ∅ ∈ ω
13		fnfvelrn 7085	. . . 4 ⊢ ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅)
14	11, 12, 13	sylancl 584	. . 3 ⊢ (𝜑 → (𝑅‘∅) ∈ ran 𝑅)
15	5	fveq1d 6894	. . . 4 ⊢ (𝜑 → (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅))
16		opex 5462	. . . . 5 ⊢ ⟨𝐶, 𝐴⟩ ∈ V
17		fr0g 8457	. . . . 5 ⊢ (⟨𝐶, 𝐴⟩ ∈ V → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅) = ⟨𝐶, 𝐴⟩)
18	16, 17	ax-mp 5	. . . 4 ⊢ ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅) = ⟨𝐶, 𝐴⟩
19	15, 18	eqtr2di 2783	. . 3 ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ = (𝑅‘∅))
20	14, 19, 6	3eltr4d 2841	. 2 ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ ∈ 𝑆)
21		funopfv 6944	. 2 ⊢ (Fun 𝑆 → (⟨𝐶, 𝐴⟩ ∈ 𝑆 → (𝑆‘𝐶) = 𝐴))
22	8, 20, 21	sylc 65	1 ⊢ (𝜑 → (𝑆‘𝐶) = 𝐴)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3464  ∅c0 4324  ⟨cop 4631   ↦ cmpt 5228  ran crn 5675   ↾ cres 5676   “ cima 5677  Fun wfun 6539   Fn wfn 6540  ‘cfv 6545  (class class class)co 7415   ∈ cmpo 7417  ωcom 7867  reccrdg 8430   No csur 27665   1_s c1s 27849   +_s cadds 27969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3968  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4908  df-int 4949  df-iun 4997  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-oadd 8491  df-nadd 8687  df-no 27668  df-slt 27669  df-bday 27670  df-sle 27771  df-sslt 27807  df-scut 27809  df-0s 27850  df-1s 27851  df-made 27867  df-old 27868  df-left 27870  df-right 27871  df-norec2 27959  df-adds 27970

This theorem is referenced by:  seqs1  28280