Theorem noseqrdgfn 28330

Description: The recursive definition generator on surreal sequences is a function. (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
noseqrdgfn	⊢ (𝜑 → 𝑆 Fn 𝑍)

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 noseqrdgfn

Dummy variables 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noseqrdg.3	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = ran 𝑅)
2	1	eleq2d 2830	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝑆 ↔ 𝑧 ∈ ran 𝑅))
3		frfnom 8491	. . . . . . . . 9 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω
4		noseqrdg.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))
5	4	fneq1d 6672	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω))
6	3, 5	mpbiri 258	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn ω)
7		fvelrnb 6982	. . . . . . . 8 ⊢ (𝑅 Fn ω → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧))
8	6, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧))
9	2, 8	bitrd 279	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ 𝑆 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧))
10		om2noseq.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ No )
11		om2noseq.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))
12		om2noseq.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))
13		noseqrdg.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
14	10, 11, 12, 13, 4	om2noseqrdg 28328	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
15	10, 11, 12	om2noseqfo 28322	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ω–onto→𝑍)
16		fof 6834	. . . . . . . . . . . 12 ⊢ (𝐺:ω–onto→𝑍 → 𝐺:ω⟶𝑍)
17	15, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ω⟶𝑍)
18	17	ffvelcdmda 7118	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝐺‘𝑤) ∈ 𝑍)
19		fvex 6933	. . . . . . . . . 10 ⊢ (2nd ‘(𝑅‘𝑤)) ∈ V
20		opelxpi 5737	. . . . . . . . . 10 ⊢ (((𝐺‘𝑤) ∈ 𝑍 ∧ (2nd ‘(𝑅‘𝑤)) ∈ V) → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ (𝑍 × V))
21	18, 19, 20	sylancl 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ (𝑍 × V))
22	14, 21	eqeltrd 2844	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) ∈ (𝑍 × V))
23		eleq1 2832	. . . . . . . 8 ⊢ ((𝑅‘𝑤) = 𝑧 → ((𝑅‘𝑤) ∈ (𝑍 × V) ↔ 𝑧 ∈ (𝑍 × V)))
24	22, 23	syl5ibcom 245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = 𝑧 → 𝑧 ∈ (𝑍 × V)))
25	24	rexlimdva 3161	. . . . . 6 ⊢ (𝜑 → (∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧 → 𝑧 ∈ (𝑍 × V)))
26	9, 25	sylbid 240	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝑆 → 𝑧 ∈ (𝑍 × V)))
27	26	ssrdv 4014	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑍 × V))
28		relxp 5718	. . . 4 ⊢ Rel (𝑍 × V)
29		relss 5805	. . . 4 ⊢ (𝑆 ⊆ (𝑍 × V) → (Rel (𝑍 × V) → Rel 𝑆))
30	27, 28, 29	mpisyl 21	. . 3 ⊢ (𝜑 → Rel 𝑆)
31	1	eleq2d 2830	. . . . . . . 8 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅))
32		fvelrnb 6982	. . . . . . . . 9 ⊢ (𝑅 Fn ω → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
33	6, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
34	31, 33	bitrd 279	. . . . . . 7 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
35	14	eqeq1d 2742	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩))
36	35	biimpd 229	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩))
37	36	impr 454	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩)
38		fvex 6933	. . . . . . . . . . . . . 14 ⊢ (𝐺‘𝑤) ∈ V
39	38, 19	opth1 5495	. . . . . . . . . . . . 13 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣)
40	37, 39	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → (𝐺‘𝑤) = 𝑣)
41	10, 11, 12	om2noseqf1o 28325	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:ω–1-1-onto→𝑍)
42		f1ocnvfv 7314	. . . . . . . . . . . . . 14 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
43	41, 42	sylan 579	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
44	43	adantrr 716	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
45	40, 44	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → (◡𝐺‘𝑣) = 𝑤)
46	45	fveq2d 6924	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → (𝑅‘(◡𝐺‘𝑣)) = (𝑅‘𝑤))
47	46	fveq2d 6924	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
48		vex 3492	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
49		vex 3492	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
50	48, 49	op2ndd 8041	. . . . . . . . . 10 ⊢ ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘𝑤)) = 𝑧)
51	50	ad2antll 728	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → (2nd ‘(𝑅‘𝑤)) = 𝑧)
52	47, 51	eqtr2d 2781	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
53	52	rexlimdvaa 3162	. . . . . . 7 ⊢ (𝜑 → (∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
54	34, 53	sylbid 240	. . . . . 6 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
55	54	alrimiv 1926	. . . . 5 ⊢ (𝜑 → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
56		fvex 6933	. . . . . 6 ⊢ (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ V
57		eqeq2 2752	. . . . . . . 8 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (𝑧 = 𝑤 ↔ 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
58	57	imbi2d 340	. . . . . . 7 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
59	58	albidv 1919	. . . . . 6 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
60	56, 59	spcev 3619	. . . . 5 ⊢ (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤))
61	55, 60	syl 17	. . . 4 ⊢ (𝜑 → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤))
62	61	alrimiv 1926	. . 3 ⊢ (𝜑 → ∀𝑣∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤))
63		dffun5 6590	. . 3 ⊢ (Fun 𝑆 ↔ (Rel 𝑆 ∧ ∀𝑣∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤)))
64	30, 62, 63	sylanbrc 582	. 2 ⊢ (𝜑 → Fun 𝑆)
65		dmss 5927	. . . . 5 ⊢ (𝑆 ⊆ (𝑍 × V) → dom 𝑆 ⊆ dom (𝑍 × V))
66	27, 65	syl 17	. . . 4 ⊢ (𝜑 → dom 𝑆 ⊆ dom (𝑍 × V))
67		dmxpss 6202	. . . 4 ⊢ dom (𝑍 × V) ⊆ 𝑍
68	66, 67	sstrdi 4021	. . 3 ⊢ (𝜑 → dom 𝑆 ⊆ 𝑍)
69	10, 11, 12, 13, 4	noseqrdglem 28329	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑍) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅)
70	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑍) → 𝑆 = ran 𝑅)
71	69, 70	eleqtrrd 2847	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑍) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑆)
72	48, 56	opeldm 5932	. . . 4 ⊢ (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑆 → 𝑣 ∈ dom 𝑆)
73	71, 72	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑍) → 𝑣 ∈ dom 𝑆)
74	68, 73	eqelssd 4030	. 2 ⊢ (𝜑 → dom 𝑆 = 𝑍)
75		df-fn 6576	. 2 ⊢ (𝑆 Fn 𝑍 ↔ (Fun 𝑆 ∧ dom 𝑆 = 𝑍))
76	64, 74, 75	sylanbrc 582	1 ⊢ (𝜑 → 𝑆 Fn 𝑍)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976  ⟨cop 4654   ↦ cmpt 5249   × cxp 5698  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  Rel wrel 5705  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  –onto→wfo 6571  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  ωcom 7903  2^nd c2nd 8029  reccrdg 8465   No csur 27702   1_s c1s 27886   +_s cadds 28010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec2 28000  df-adds 28011

This theorem is referenced by:  noseqrdg0  28331  noseqrdgsuc  28332  seqsfn  28333