Theorem noseqrdgfn 28236

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 2817	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝑆 ↔ 𝑧 ∈ ran 𝑅))
3		frfnom 8354	. . . . . . . . 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 6574	. . . . . . . . 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 6882	. . . . . . . 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 28234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
15	10, 11, 12	om2noseqfo 28228	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ω–onto→𝑍)
16		fof 6735	. . . . . . . . . . . 12 ⊢ (𝐺:ω–onto→𝑍 → 𝐺:ω⟶𝑍)
17	15, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ω⟶𝑍)
18	17	ffvelcdmda 7017	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝐺‘𝑤) ∈ 𝑍)
19		fvex 6835	. . . . . . . . . 10 ⊢ (2nd ‘(𝑅‘𝑤)) ∈ V
20		opelxpi 5651	. . . . . . . . . 10 ⊢ (((𝐺‘𝑤) ∈ 𝑍 ∧ (2nd ‘(𝑅‘𝑤)) ∈ V) → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ (𝑍 × V))
21	18, 19, 20	sylancl 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ (𝑍 × V))
22	14, 21	eqeltrd 2831	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) ∈ (𝑍 × V))
23		eleq1 2819	. . . . . . . 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 3133	. . . . . 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 3935	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑍 × V))
28		relxp 5632	. . . 4 ⊢ Rel (𝑍 × V)
29		relss 5721	. . . 4 ⊢ (𝑆 ⊆ (𝑍 × V) → (Rel (𝑍 × V) → Rel 𝑆))
30	27, 28, 29	mpisyl 21	. . 3 ⊢ (𝜑 → Rel 𝑆)
31	1	eleq2d 2817	. . . . . . . 8 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅))
32		fvelrnb 6882	. . . . . . . . 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 2733	. . . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . 14 ⊢ (𝐺‘𝑤) ∈ V
39	38, 19	opth1 5413	. . . . . . . . . . . . 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 28231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:ω–1-1-onto→𝑍)
42		f1ocnvfv 7212	. . . . . . . . . . . . . 14 ⊢ ((𝐺:ω–1-1-onto→𝑍 ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
43	41, 42	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
44	43	adantrr 717	. . . . . . . . . . . 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 6826	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → (𝑅‘(◡𝐺‘𝑣)) = (𝑅‘𝑤))
47	46	fveq2d 6826	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
48		vex 3440	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
49		vex 3440	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
50	48, 49	op2ndd 7932	. . . . . . . . . 10 ⊢ ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘𝑤)) = 𝑧)
51	50	ad2antll 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → (2nd ‘(𝑅‘𝑤)) = 𝑧)
52	47, 51	eqtr2d 2767	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)) → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
53	52	rexlimdvaa 3134	. . . . . . 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 1928	. . . . 5 ⊢ (𝜑 → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
56		fvex 6835	. . . . . 6 ⊢ (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ V
57		eqeq2 2743	. . . . . . . 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 1921	. . . . . 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 3556	. . . . 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 1928	. . 3 ⊢ (𝜑 → ∀𝑣∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤))
63		dffun5 6495	. . 3 ⊢ (Fun 𝑆 ↔ (Rel 𝑆 ∧ ∀𝑣∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤)))
64	30, 62, 63	sylanbrc 583	. 2 ⊢ (𝜑 → Fun 𝑆)
65		dmss 5841	. . . . 5 ⊢ (𝑆 ⊆ (𝑍 × V) → dom 𝑆 ⊆ dom (𝑍 × V))
66	27, 65	syl 17	. . . 4 ⊢ (𝜑 → dom 𝑆 ⊆ dom (𝑍 × V))
67		dmxpss 6118	. . . 4 ⊢ dom (𝑍 × V) ⊆ 𝑍
68	66, 67	sstrdi 3942	. . 3 ⊢ (𝜑 → dom 𝑆 ⊆ 𝑍)
69	10, 11, 12, 13, 4	noseqrdglem 28235	. . . . 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 2834	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑍) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑆)
72	48, 56	opeldm 5846	. . . 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 3951	. 2 ⊢ (𝜑 → dom 𝑆 = 𝑍)
75		df-fn 6484	. 2 ⊢ (𝑆 Fn 𝑍 ↔ (Fun 𝑆 ∧ dom 𝑆 = 𝑍))
76	64, 74, 75	sylanbrc 583	1 ⊢ (𝜑 → 𝑆 Fn 𝑍)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436   ⊆ wss 3897  ⟨cop 4579   ↦ cmpt 5170   × cxp 5612  ^◡ccnv 5613  dom cdm 5614  ran crn 5615   ↾ cres 5616   “ cima 5617  Rel wrel 5619  Fun wfun 6475   Fn wfn 6476  ⟶wf 6477  –onto→wfo 6479  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  ωcom 7796  2^nd c2nd 7920  reccrdg 8328   No csur 27578   1_s c1s 27767   +_s cadds 27902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-ot 4582  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-nadd 8581  df-no 27581  df-slt 27582  df-bday 27583  df-sle 27684  df-sslt 27721  df-scut 27723  df-0s 27768  df-1s 27769  df-made 27788  df-old 27789  df-left 27791  df-right 27792  df-norec2 27892  df-adds 27903

This theorem is referenced by:  noseqrdg0  28237  noseqrdgsuc  28238  seqsfn  28239