Theorem noseqrdglem 28251

Description: A helper lemma for the value of a recursive defintion 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))

Assertion

Ref	Expression
noseqrdglem	⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem noseqrdglem

Step	Hyp	Ref	Expression
1		om2noseq.1	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ No )
2		om2noseq.2	. . . . . 6 ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
3		om2noseq.3	. . . . . 6 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
4	1, 2, 3	om2noseqf1o 28247	. . . . 5 ⊢ (𝜑 → 𝐺:ω–1-1-onto→𝑍)
5		f1ocnvdm 7278	. . . . 5 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝐵 ∈ 𝑍) → (◡𝐺‘𝐵) ∈ ω)
6	4, 5	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (◡𝐺‘𝐵) ∈ ω)
7		noseqrdg.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		noseqrdg.2	. . . . 5 ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
9	1, 2, 3, 7, 8	om2noseqrdg 28250	. . . 4 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘(◡𝐺‘𝐵)) = ⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
10	6, 9	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑅‘(◡𝐺‘𝐵)) = ⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
11		f1ocnvfv2 7270	. . . . 5 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝐵 ∈ 𝑍) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
12	4, 11	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
13	12	opeq1d 4855	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ = ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
14	10, 13	eqtrd 2770	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑅‘(◡𝐺‘𝐵)) = ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
15		frfnom 8449	. . . . 5 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω
16	8	fneq1d 6631	. . . . 5 ⊢ (𝜑 → (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω))
17	15, 16	mpbiri 258	. . . 4 ⊢ (𝜑 → 𝑅 Fn ω)
18	17	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝑅 Fn ω)
19	18, 6	fnfvelrnd 7072	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅)
20	14, 19	eqeltrrd 2835	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ⟨cop 4607   ↦ cmpt 5201  ^◡ccnv 5653  ran crn 5655   ↾ cres 5656   “ cima 5657   Fn wfn 6526  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  ωcom 7861  2^nd c2nd 7987  reccrdg 8423   No csur 27603   1_s c1s 27787   +_s cadds 27918

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-nadd 8678  df-no 27606  df-slt 27607  df-bday 27608  df-sle 27709  df-sslt 27745  df-scut 27747  df-0s 27788  df-1s 27789  df-made 27807  df-old 27808  df-left 27810  df-right 27811  df-norec2 27908  df-adds 27919

This theorem is referenced by:  noseqrdgfn  28252  noseqrdgsuc  28254