Theorem bdayn0sf1o 28783

Description: The birthday function restricted to the non-negative surreal integers is a bijection with the finite ordinals. (Contributed by Scott Fenton, 7-Nov-2025.)

Assertion

Ref	Expression
bdayn0sf1o	⊢ (bday ↾ ℕ0s):ℕ0s⟶1-1ontoω

Proof of Theorem bdayn0sf1o

Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdayfun 28160	. . . . . . 7 ⊢ Fun bday
2		funres 6599	. . . . . . 7 ⊢ (Fun bday → Fun (bday ↾ ℕ0s))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun (bday ↾ ℕ0s)
4		dmres 6014	. . . . . . 7 ⊢ dom (bday ↾ ℕ0s) = (ℕ0s ∩ dom bday)
5		bdaydm 28162	. . . . . . . 8 ⊢ dom bday = No
6	5	ineq2i 4117	. . . . . . 7 ⊢ (ℕ0s ∩ dom bday) = (ℕ0s ∩ No)
7		n0ssno 28733	. . . . . . . 8 ⊢ ℕ0s ⊆ No
8		dfss2 4097	. . . . . . . 8 ⊢ (ℕ0s ⊆ No ↔ (ℕ0s ∩ No) = ℕ0s)
9	7, 8	mpbi 233	. . . . . . 7 ⊢ (ℕ0s ∩ No) = ℕ0s
10	4, 6, 9	3eqtri 2790	. . . . . 6 ⊢ dom (bday ↾ ℕ0s) = ℕ0s
11		df-fn 6558	. . . . . 6 ⊢ ((bday ↾ ℕ0s) Fn ℕ0s ↔ (Fun (bday ↾ ℕ0s) ∧ dom (bday ↾ ℕ0s) = ℕ0s))
12	3, 10, 11	mpbir2an 724	. . . . 5 ⊢ (bday ↾ ℕ0s) Fn ℕ0s
13		fvres 6924	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → ((bday ↾ ℕ0s)‘𝑥) = (bday‘𝑥))
14		n0bday 28765	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → (bday‘𝑥) ∈ ω)
15	13, 14	eqeltrd 2864	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → ((bday ↾ ℕ0s)‘𝑥) ∈ ω)
16	15	rgen 3082	. . . . . . 7 ⊢ ∀𝑥 ∈ ℕ0s ((bday ↾ ℕ0s)‘𝑥) ∈ ω
17		fnfvrnss 7142	. . . . . . 7 ⊢ (((bday ↾ ℕ0s) Fn ℕ0s ∧ ∀𝑥 ∈ ℕ0s ((bday ↾ ℕ0s)‘𝑥) ∈ ω) → ran (bday ↾ ℕ0s) ⊆ ω)
18	12, 16, 17	mp2an 705	. . . . . 6 ⊢ ran (bday ↾ ℕ0s) ⊆ ω
19		eqeq2 2775	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → ((bday‘𝑦) = 𝑏 ↔ (bday‘𝑦) = ∅))
20	19	rexbidv 3190	. . . . . . . . 9 ⊢ (𝑏 = ∅ → (∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s (bday‘𝑦) = ∅))
21		eqeq2 2775	. . . . . . . . . 10 ⊢ (𝑏 = 𝑎 → ((bday‘𝑦) = 𝑏 ↔ (bday‘𝑦) = 𝑎))
22	21	rexbidv 3190	. . . . . . . . 9 ⊢ (𝑏 = 𝑎 → (∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑎))
23		eqeq2 2775	. . . . . . . . . . 11 ⊢ (𝑏 = suc 𝑎 → ((bday‘𝑦) = 𝑏 ↔ (bday‘𝑦) = suc 𝑎))
24	23	rexbidv 3190	. . . . . . . . . 10 ⊢ (𝑏 = suc 𝑎 → (∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s (bday‘𝑦) = suc 𝑎))
25		fveqeq2 6914	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((bday‘𝑦) = suc 𝑎 ↔ (bday‘𝑧) = suc 𝑎))
26	25	cbvrexvw 3245	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ ℕ0s (bday‘𝑦) = suc 𝑎 ↔ ∃𝑧 ∈ ℕ0s (bday‘𝑧) = suc 𝑎)
27	24, 26	bitrdi 290	. . . . . . . . 9 ⊢ (𝑏 = suc 𝑎 → (∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑏 ↔ ∃𝑧 ∈ ℕ0s (bday‘𝑧) = suc 𝑎))
28		eqeq2 2775	. . . . . . . . . 10 ⊢ (𝑏 = 𝑥 → ((bday‘𝑦) = 𝑏 ↔ (bday‘𝑦) = 𝑥))
29	28	rexbidv 3190	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → (∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑥))
30		0n0s 28742	. . . . . . . . . 10 ⊢ 0s ∈ ℕ0s
31		bday0 28224	. . . . . . . . . 10 ⊢ (bday‘ 0s ) = ∅
32		fveqeq2 6914	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → ((bday‘𝑦) = ∅ ↔ (bday‘ 0s ) = ∅))
33	32	rspcev 3580	. . . . . . . . . 10 ⊢ (( 0s ∈ ℕ0s ∧ (bday‘ 0s ) = ∅) → ∃𝑦 ∈ ℕ0s (bday‘𝑦) = ∅)
34	30, 31, 33	mp2an 705	. . . . . . . . 9 ⊢ ∃𝑦 ∈ ℕ0s (bday‘𝑦) = ∅
35		fveqeq2 6914	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑦 +s 1s ) → ((bday‘𝑧) = suc (bday‘𝑦) ↔ (bday‘(𝑦 +s 1s )) = suc (bday‘𝑦)))
36		peano2n0s 28743	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
37		bdayn0p1 28782	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0s → (bday‘(𝑦 +s 1s )) = suc (bday‘𝑦))
38	35, 36, 37	rspcedvdw 3583	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0s → ∃𝑧 ∈ ℕ0s (bday‘𝑧) = suc (bday‘𝑦))
39	38	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ω ∧ 𝑦 ∈ ℕ0s) → ∃𝑧 ∈ ℕ0s (bday‘𝑧) = suc (bday‘𝑦))
40		suceq 6446	. . . . . . . . . . . . 13 ⊢ ((bday‘𝑦) = 𝑎 → suc (bday‘𝑦) = suc 𝑎)
41	40	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ ((bday‘𝑦) = 𝑎 → ((bday‘𝑧) = suc (bday‘𝑦) ↔ (bday‘𝑧) = suc 𝑎))
42	41	rexbidv 3190	. . . . . . . . . . 11 ⊢ ((bday‘𝑦) = 𝑎 → (∃𝑧 ∈ ℕ0s (bday‘𝑧) = suc (bday‘𝑦) ↔ ∃𝑧 ∈ ℕ0s (bday‘𝑧) = suc 𝑎))
43	39, 42	syl5ibcom 248	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ω ∧ 𝑦 ∈ ℕ0s) → ((bday‘𝑦) = 𝑎 → ∃𝑧 ∈ ℕ0s (bday‘𝑧) = suc 𝑎))
44	43	rexlimdva 3167	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑎 → ∃𝑧 ∈ ℕ0s (bday‘𝑧) = suc 𝑎))
45	20, 22, 27, 29, 34, 44	finds 7929	. . . . . . . 8 ⊢ (𝑥 ∈ ω → ∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑥)
46		fvelrnb 6965	. . . . . . . . . 10 ⊢ ((bday ↾ ℕ0s) Fn ℕ0s → (𝑥 ∈ ran (bday ↾ ℕ0s) ↔ ∃𝑦 ∈ ℕ0s ((bday ↾ ℕ0s)‘𝑦) = 𝑥))
47	12, 46	ax-mp 5	. . . . . . . . 9 ⊢ (𝑥 ∈ ran (bday ↾ ℕ0s) ↔ ∃𝑦 ∈ ℕ0s ((bday ↾ ℕ0s)‘𝑦) = 𝑥)
48		fvres 6924	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → ((bday ↾ ℕ0s)‘𝑦) = (bday‘𝑦))
49	48	eqeq1d 2765	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → (((bday ↾ ℕ0s)‘𝑦) = 𝑥 ↔ (bday‘𝑦) = 𝑥))
50	49	rexbiia 3111	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℕ0s ((bday ↾ ℕ0s)‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑥)
51	47, 50	bitri 278	. . . . . . . 8 ⊢ (𝑥 ∈ ran (bday ↾ ℕ0s) ↔ ∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑥)
52	45, 51	sylibr 237	. . . . . . 7 ⊢ (𝑥 ∈ ω → 𝑥 ∈ ran (bday ↾ ℕ0s))
53	52	ssriv 3917	. . . . . 6 ⊢ ω ⊆ ran (bday ↾ ℕ0s)
54	18, 53	eqssi 3929	. . . . 5 ⊢ ran (bday ↾ ℕ0s) = ω
55		df-fo 6561	. . . . 5 ⊢ ((bday ↾ ℕ0s):ℕ0s⟶ontoω ↔ ((bday ↾ ℕ0s) Fn ℕ0s ∧ ran (bday ↾ ℕ0s) = ω))
56	12, 54, 55	mpbir2an 724	. . . 4 ⊢ (bday ↾ ℕ0s):ℕ0s⟶ontoω
57		fof 6815	. . . 4 ⊢ ((bday ↾ ℕ0s):ℕ0s⟶ontoω → (bday ↾ ℕ0s):ℕ0s⟶ω)
58	56, 57	ax-mp 5	. . 3 ⊢ (bday ↾ ℕ0s):ℕ0s⟶ω
59	13, 48	eqeqan12d 2777	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (((bday ↾ ℕ0s)‘𝑥) = ((bday ↾ ℕ0s)‘𝑦) ↔ (bday‘𝑥) = (bday‘𝑦)))
60		n0on 28749	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ Ons)
61		n0on 28749	. . . . . 6 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ Ons)
62		bday11on 28678	. . . . . . 7 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons ∧ (bday‘𝑥) = (bday‘𝑦)) → 𝑥 = 𝑦)
63	62	3expia 1139	. . . . . 6 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → ((bday‘𝑥) = (bday‘𝑦) → 𝑥 = 𝑦))
64	60, 61, 63	syl2an 608	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → ((bday‘𝑥) = (bday‘𝑦) → 𝑥 = 𝑦))
65	59, 64	sylbid 243	. . . 4 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (((bday ↾ ℕ0s)‘𝑥) = ((bday ↾ ℕ0s)‘𝑦) → 𝑥 = 𝑦))
66	65	rgen2 3206	. . 3 ⊢ ∀𝑥 ∈ ℕ0s ∀𝑦 ∈ ℕ0s (((bday ↾ ℕ0s)‘𝑥) = ((bday ↾ ℕ0s)‘𝑦) → 𝑥 = 𝑦)
67		dff13 7278	. . 3 ⊢ ((bday ↾ ℕ0s):ℕ0s⟶1-1ω ↔ ((bday ↾ ℕ0s):ℕ0s⟶ω ∧ ∀𝑥 ∈ ℕ0s ∀𝑦 ∈ ℕ0s (((bday ↾ ℕ0s)‘𝑥) = ((bday ↾ ℕ0s)‘𝑦) → 𝑥 = 𝑦)))
68	58, 66, 67	mpbir2an 724	. 2 ⊢ (bday ↾ ℕ0s):ℕ0s⟶1-1ω
69		df-f1o 6562	. 2 ⊢ ((bday ↾ ℕ0s):ℕ0s⟶1-1ontoω ↔ ((bday ↾ ℕ0s):ℕ0s⟶1-1ω ∧ (bday ↾ ℕ0s):ℕ0s⟶ontoω))
70	68, 56, 69	mpbir2an 724	1 ⊢ (bday ↾ ℕ0s):ℕ0s⟶1-1ontoω

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  dom cdm 5655  ran crn 5656   ↾ cres 5657  suc csuc 6379  Fun wfun 6549   Fn wfn 6550  ⟶wf 6551  ⟶1-1wf1 6552  ⟶ontowfo 6553  ⟶1-1ontowf1o 6554  ‘cfv 6555  (class class class)co 7440  ωcom 7898  Nocsur 28024  bdaycbday 28026   0_s c0s 28218   1_s c1s 28219   +_s cadds 28372  On_scons 28664  ℕ_0scn0s 28725

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-nadd 8700  df-no 28027  df-lts 28028  df-bday 28029  df-les 28129  df-slts 28171  df-cuts 28173  df-0s 28220  df-1s 28221  df-made 28240  df-old 28241  df-left 28243  df-right 28244  df-norec 28351  df-norec2 28362  df-adds 28373  df-negs 28434  df-subs 28435  df-ons 28665  df-n0s 28727

This theorem is used by:  oldfib  28790  bdayfinlem  28899