Theorem bdayn0sf1o 28756

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–1-1-onto→ω

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 28133	. . . . . . 7 ⊢ Fun bday
2		funres 6582	. . . . . . 7 ⊢ (Fun bday → Fun (bday ↾ ℕ0s))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun (bday ↾ ℕ0s)
4		dmres 6002	. . . . . . 7 ⊢ dom (bday ↾ ℕ0s) = (ℕ0s ∩ dom bday)
5		bdaydm 28135	. . . . . . . 8 ⊢ dom bday = No
6	5	ineq2i 4114	. . . . . . 7 ⊢ (ℕ0s ∩ dom bday) = (ℕ0s ∩ No)
7		n0ssno 28706	. . . . . . . 8 ⊢ ℕ0s ⊆ No
8		dfss2 4094	. . . . . . . 8 ⊢ (ℕ0s ⊆ No ↔ (ℕ0s ∩ No) = ℕ0s)
9	7, 8	mpbi 233	. . . . . . 7 ⊢ (ℕ0s ∩ No) = ℕ0s
10	4, 6, 9	3eqtri 2787	. . . . . 6 ⊢ dom (bday ↾ ℕ0s) = ℕ0s
11		df-fn 6541	. . . . . 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 6904	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → ((bday ↾ ℕ0s)‘𝑥) = (bday‘𝑥))
14		n0bday 28738	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → (bday‘𝑥) ∈ ω)
15	13, 14	eqeltrd 2860	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → ((bday ↾ ℕ0s)‘𝑥) ∈ ω)
16	15	rgen 3078	. . . . . . 7 ⊢ ∀𝑥 ∈ ℕ0s ((bday ↾ ℕ0s)‘𝑥) ∈ ω
17		fnfvrnss 7121	. . . . . . 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 2772	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → ((bday‘𝑦) = 𝑏 ↔ (bday‘𝑦) = ∅))
20	19	rexbidv 3186	. . . . . . . . 9 ⊢ (𝑏 = ∅ → (∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s (bday‘𝑦) = ∅))
21		eqeq2 2772	. . . . . . . . . 10 ⊢ (𝑏 = 𝑎 → ((bday‘𝑦) = 𝑏 ↔ (bday‘𝑦) = 𝑎))
22	21	rexbidv 3186	. . . . . . . . 9 ⊢ (𝑏 = 𝑎 → (∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑎))
23		eqeq2 2772	. . . . . . . . . . 11 ⊢ (𝑏 = suc 𝑎 → ((bday‘𝑦) = 𝑏 ↔ (bday‘𝑦) = suc 𝑎))
24	23	rexbidv 3186	. . . . . . . . . 10 ⊢ (𝑏 = suc 𝑎 → (∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s (bday‘𝑦) = suc 𝑎))
25		fveqeq2 6894	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((bday‘𝑦) = suc 𝑎 ↔ (bday‘𝑧) = suc 𝑎))
26	25	cbvrexvw 3241	. . . . . . . . . 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 2772	. . . . . . . . . 10 ⊢ (𝑏 = 𝑥 → ((bday‘𝑦) = 𝑏 ↔ (bday‘𝑦) = 𝑥))
29	28	rexbidv 3186	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → (∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑥))
30		0n0s 28715	. . . . . . . . . 10 ⊢ 0s ∈ ℕ0s
31		bday0 28197	. . . . . . . . . 10 ⊢ (bday‘ 0s ) = ∅
32		fveqeq2 6894	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → ((bday‘𝑦) = ∅ ↔ (bday‘ 0s ) = ∅))
33	32	rspcev 3576	. . . . . . . . . 10 ⊢ (( 0s ∈ ℕ0s ∧ (bday‘ 0s ) = ∅) → ∃𝑦 ∈ ℕ0s (bday‘𝑦) = ∅)
34	30, 31, 33	mp2an 705	. . . . . . . . 9 ⊢ ∃𝑦 ∈ ℕ0s (bday‘𝑦) = ∅
35		fveqeq2 6894	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑦 +s 1s ) → ((bday‘𝑧) = suc (bday‘𝑦) ↔ (bday‘(𝑦 +s 1s )) = suc (bday‘𝑦)))
36		peano2n0s 28716	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
37		bdayn0p1 28755	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0s → (bday‘(𝑦 +s 1s )) = suc (bday‘𝑦))
38	35, 36, 37	rspcedvdw 3579	. . . . . . . . . . . 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 6431	. . . . . . . . . . . . 13 ⊢ ((bday‘𝑦) = 𝑎 → suc (bday‘𝑦) = suc 𝑎)
41	40	eqeq2d 2771	. . . . . . . . . . . 12 ⊢ ((bday‘𝑦) = 𝑎 → ((bday‘𝑧) = suc (bday‘𝑦) ↔ (bday‘𝑧) = suc 𝑎))
42	41	rexbidv 3186	. . . . . . . . . . 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 3163	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑎 → ∃𝑧 ∈ ℕ0s (bday‘𝑧) = suc 𝑎))
45	20, 22, 27, 29, 34, 44	finds 7908	. . . . . . . 8 ⊢ (𝑥 ∈ ω → ∃𝑦 ∈ ℕ0s (bday‘𝑦) = 𝑥)
46		fvelrnb 6945	. . . . . . . . . 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 6904	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → ((bday ↾ ℕ0s)‘𝑦) = (bday‘𝑦))
49	48	eqeq1d 2762	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → (((bday ↾ ℕ0s)‘𝑦) = 𝑥 ↔ (bday‘𝑦) = 𝑥))
50	49	rexbiia 3107	. . . . . . . . 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 3912	. . . . . 6 ⊢ ω ⊆ ran (bday ↾ ℕ0s)
54	18, 53	eqssi 3924	. . . . 5 ⊢ ran (bday ↾ ℕ0s) = ω
55		df-fo 6544	. . . . 5 ⊢ ((bday ↾ ℕ0s):ℕ0s–onto→ω ↔ ((bday ↾ ℕ0s) Fn ℕ0s ∧ ran (bday ↾ ℕ0s) = ω))
56	12, 54, 55	mpbir2an 724	. . . 4 ⊢ (bday ↾ ℕ0s):ℕ0s–onto→ω
57		fof 6796	. . . 4 ⊢ ((bday ↾ ℕ0s):ℕ0s–onto→ω → (bday ↾ ℕ0s):ℕ0s⟶ω)
58	56, 57	ax-mp 5	. . 3 ⊢ (bday ↾ ℕ0s):ℕ0s⟶ω
59	13, 48	eqeqan12d 2774	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (((bday ↾ ℕ0s)‘𝑥) = ((bday ↾ ℕ0s)‘𝑦) ↔ (bday‘𝑥) = (bday‘𝑦)))
60		n0on 28722	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ Ons)
61		n0on 28722	. . . . . 6 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ Ons)
62		bday11on 28651	. . . . . . 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 3202	. . 3 ⊢ ∀𝑥 ∈ ℕ0s ∀𝑦 ∈ ℕ0s (((bday ↾ ℕ0s)‘𝑥) = ((bday ↾ ℕ0s)‘𝑦) → 𝑥 = 𝑦)
67		dff13 7258	. . 3 ⊢ ((bday ↾ ℕ0s):ℕ0s–1-1→ω ↔ ((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–1-1→ω
69		df-f1o 6545	. 2 ⊢ ((bday ↾ ℕ0s):ℕ0s–1-1-onto→ω ↔ ((bday ↾ ℕ0s):ℕ0s–1-1→ω ∧ (bday ↾ ℕ0s):ℕ0s–onto→ω))
70	68, 56, 69	mpbir2an 724	1 ⊢ (bday ↾ ℕ0s):ℕ0s–1-1-onto→ω

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  dom cdm 5650  ran crn 5651   ↾ cres 5652  suc csuc 6364  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  –1-1→wf1 6535  –onto→wfo 6536  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420  ωcom 7877  Nocsur 27997  bdaycbday 27999   0_s c0s 28191   1_s c1s 28192   +_s cadds 28345  On_scons 28637  ℕ_0scn0s 28698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-nadd 8675  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-1s 28194  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-norec2 28335  df-adds 28346  df-negs 28407  df-subs 28408  df-ons 28638  df-n0s 28700

This theorem is used by:  oldfib  28763  bdayfinlem  28872