Theorem noseqrdgfn 28499

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 2849	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝑆 ↔ 𝑧 ∈ ran 𝑅))
3		frfnom 8418	. . . . . . . . 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 6628	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω))
6	3, 5	mpbiri 261	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn ω)
7		fvelrnb 6941	. . . . . . . 8 ⊢ (𝑅 Fn ω → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧))
8	6, 7	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧))
9	2, 8	bitrd 282	. . . . . 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 28497	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
15	10, 11, 12	om2noseqfo 28491	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ω–onto→𝑍)
16		fof 6792	. . . . . . . . . . . 12 ⊢ (𝐺:ω–onto→𝑍 → 𝐺:ω⟶𝑍)
17	15, 16	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ω⟶𝑍)
18	17	ffvelcdmda 7079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝐺‘𝑤) ∈ 𝑍)
19		fvex 6894	. . . . . . . . . 10 ⊢ (2nd ‘(𝑅‘𝑤)) ∈ V
20		opelxpi 5698	. . . . . . . . . 10 ⊢ (((𝐺‘𝑤) ∈ 𝑍 ∧ (2nd ‘(𝑅‘𝑤)) ∈ V) → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ (𝑍 × V))
21	18, 19, 20	sylancl 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ (𝑍 × V))
22	14, 21	eqeltrd 2863	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) ∈ (𝑍 × V))
23		eleq1 2851	. . . . . . . 8 ⊢ ((𝑅‘𝑤) = 𝑧 → ((𝑅‘𝑤) ∈ (𝑍 × V) ↔ 𝑧 ∈ (𝑍 × V)))
24	22, 23	syl5ibcom 248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = 𝑧 → 𝑧 ∈ (𝑍 × V)))
25	24	rexlimdva 3166	. . . . . 6 ⊢ (𝜑 → (∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧 → 𝑧 ∈ (𝑍 × V)))
26	9, 25	sylbid 243	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ 𝑆 → 𝑧 ∈ (𝑍 × V)))
27	26	ssrdv 3943	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑍 × V))
28		relxp 5679	. . . 4 ⊢ Rel (𝑍 × V)
29		relss 5768	. . . 4 ⊢ (𝑆 ⊆ (𝑍 × V) → (Rel (𝑍 × V) → Rel 𝑆))
30	27, 28, 29	mpisyl 22	. . 3 ⊢ (𝜑 → Rel 𝑆)
31	1	eleq2d 2849	. . . . . . . 8 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅))
32		fvelrnb 6941	. . . . . . . . 9 ⊢ (𝑅 Fn ω → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
33	6, 32	syl 18	. . . . . . . 8 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
34	31, 33	bitrd 282	. . . . . . 7 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
35	14	eqeq1d 2765	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩))
36	35	biimpd 232	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩))
37	36	impr 459	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩)
38		fvex 6894	. . . . . . . . . . . . . 14 ⊢ (𝐺‘𝑤) ∈ V
39	38, 19	opth1 5457	. . . . . . . . . . . . 13 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣)
40	37, 39	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → (𝐺‘𝑤) = 𝑣)
41	10, 11, 12	om2noseqf1o 28494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:ω–1-1-onto→𝑍)
42		f1ocnvfv 7276	. . . . . . . . . . . . . 14 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
43	41, 42	sylan 591	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
44	43	adantrr 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
45	40, 44	mpd 16	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → (◡𝐺‘𝑣) = 𝑤)
46	45	fveq2d 6885	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → (𝑅‘(◡𝐺‘𝑣)) = (𝑅‘𝑤))
47	46	fveq2d 6885	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
48		vex 3459	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
49		vex 3459	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
50	48, 49	op2ndd 7993	. . . . . . . . . 10 ⊢ ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘𝑤)) = 𝑧)
51	50	ad2antll 741	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → (2nd ‘(𝑅‘𝑤)) = 𝑧)
52	47, 51	eqtr2d 2799	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
53	52	rexlimdvaa 3167	. . . . . . 7 ⊢ (𝜑 → (∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
54	34, 53	sylbid 243	. . . . . 6 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
55	54	alrimiv 1957	. . . . 5 ⊢ (𝜑 → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
56		fvex 6894	. . . . . 6 ⊢ (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ V
57		eqeq2 2775	. . . . . . . 8 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (𝑧 = 𝑤 ↔ 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
58	57	imbi2d 343	. . . . . . 7 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
59	58	albidv 1950	. . . . . 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 3565	. . . . 5 ⊢ (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤))
61	55, 60	syl 18	. . . 4 ⊢ (𝜑 → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤))
62	61	alrimiv 1957	. . 3 ⊢ (𝜑 → ∀𝑣∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤))
63		dffun5 6550	. . 3 ⊢ (Fun 𝑆 ↔ (Rel 𝑆 ∧ ∀𝑣∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤)))
64	30, 62, 63	sylanbrc 594	. 2 ⊢ (𝜑 → Fun 𝑆)
65		dmss 5892	. . . . 5 ⊢ (𝑆 ⊆ (𝑍 × V) → dom 𝑆 ⊆ dom (𝑍 × V))
66	27, 65	syl 18	. . . 4 ⊢ (𝜑 → dom 𝑆 ⊆ dom (𝑍 × V))
67		dmxpss 6169	. . . 4 ⊢ dom (𝑍 × V) ⊆ 𝑍
68	66, 67	sstrdi 3949	. . 3 ⊢ (𝜑 → dom 𝑆 ⊆ 𝑍)
69	10, 11, 12, 13, 4	noseqrdglem 28498	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑍) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅)
70	1	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑍) → 𝑆 = ran 𝑅)
71	69, 70	eleqtrrd 2866	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑍) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑆)
72	48, 56	opeldm 5897	. . . 4 ⊢ (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑆 → 𝑣 ∈ dom 𝑆)
73	71, 72	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑍) → 𝑣 ∈ dom 𝑆)
74	68, 73	eqelssd 3958	. 2 ⊢ (𝜑 → dom 𝑆 = 𝑍)
75		df-fn 6539	. 2 ⊢ (𝑆 Fn 𝑍 ↔ (Fun 𝑆 ∧ dom 𝑆 = 𝑍))
76	64, 74, 75	sylanbrc 594	1 ⊢ (𝜑 → 𝑆 Fn 𝑍)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455   ⊆ wss 3905  ⟨cop 4595   ↦ cmpt 5192   × cxp 5659  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664  Rel wrel 5666  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  –onto→wfo 6534  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  ωcom 7858  2^nd c2nd 7981  reccrdg 8392   No csur 27804   1_s c1s 27999   +_s cadds 28152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-oadd 8453  df-nadd 8648  df-no 27807  df-lts 27808  df-bday 27809  df-les 27909  df-slts 27951  df-cuts 27953  df-0s 28000  df-1s 28001  df-made 28020  df-old 28021  df-left 28023  df-right 28024  df-norec2 28142  df-adds 28153

This theorem is referenced by:  noseqrdg0  28500  noseqrdgsuc  28501  seqsfn  28502