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	⊢ (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 28312	. . 3 ⊢ (𝜑 → 𝑆 Fn 𝑍)
8	7	fnfund 6593	. 2 ⊢ (𝜑 → Fun 𝑆)
9		frfnom 8367	. . . . 5 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω
10	5	fneq1d 6585	. . . . 5 ⊢ (𝜑 → (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω))
11	9, 10	mpbiri 258	. . . 4 ⊢ (𝜑 → 𝑅 Fn ω)
12		peano1 7833	. . . 4 ⊢ ∅ ∈ ω
13		fnfvelrn 7026	. . . 4 ⊢ ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅)
14	11, 12, 13	sylancl 587	. . 3 ⊢ (𝜑 → (𝑅‘∅) ∈ ran 𝑅)
15	5	fveq1d 6836	. . . 4 ⊢ (𝜑 → (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅))
16		opex 5411	. . . . 5 ⊢ ⟨𝐶, 𝐴⟩ ∈ V
17		fr0g 8368	. . . . 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 2789	. . 3 ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ = (𝑅‘∅))
20	14, 19, 6	3eltr4d 2852	. 2 ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ ∈ 𝑆)
21		funopfv 6883	. 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 1542   ∈ wcel 2114  Vcvv 3430  ∅c0 4274  ⟨cop 4574   ↦ cmpt 5167  ran crn 5625   ↾ cres 5626   “ cima 5627  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  ωcom 7810  reccrdg 8341   No csur 27617   1_s c1s 27812   +_s cadds 27965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-oadd 8402  df-nadd 8595  df-no 27620  df-lts 27621  df-bday 27622  df-les 27723  df-slts 27764  df-cuts 27766  df-0s 27813  df-1s 27814  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec2 27955  df-adds 27966

This theorem is referenced by:  seqs1  28316