Theorem bdayn0sf1o 28362

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 27740	. . . . . . 7 ⊢ Fun bday
2		funres 6540	. . . . . . 7 ⊢ (Fun bday → Fun ( bday ↾ ℕ0s))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun ( bday ↾ ℕ0s)
4		dmres 5977	. . . . . . 7 ⊢ dom ( bday ↾ ℕ0s) = (ℕ0s ∩ dom bday )
5		bdaydm 27742	. . . . . . . 8 ⊢ dom bday = No
6	5	ineq2i 4157	. . . . . . 7 ⊢ (ℕ0s ∩ dom bday ) = (ℕ0s ∩ No )
7		n0ssno 28312	. . . . . . . 8 ⊢ ℕ0s ⊆ No
8		dfss2 3907	. . . . . . . 8 ⊢ (ℕ0s ⊆ No ↔ (ℕ0s ∩ No ) = ℕ0s)
9	7, 8	mpbi 230	. . . . . . 7 ⊢ (ℕ0s ∩ No ) = ℕ0s
10	4, 6, 9	3eqtri 2763	. . . . . 6 ⊢ dom ( bday ↾ ℕ0s) = ℕ0s
11		df-fn 6501	. . . . . 6 ⊢ (( bday ↾ ℕ0s) Fn ℕ0s ↔ (Fun ( bday ↾ ℕ0s) ∧ dom ( bday ↾ ℕ0s) = ℕ0s))
12	3, 10, 11	mpbir2an 712	. . . . 5 ⊢ ( bday ↾ ℕ0s) Fn ℕ0s
13		fvres 6859	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → (( bday ↾ ℕ0s)‘𝑥) = ( bday ‘𝑥))
14		n0bday 28344	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → ( bday ‘𝑥) ∈ ω)
15	13, 14	eqeltrd 2836	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → (( bday ↾ ℕ0s)‘𝑥) ∈ ω)
16	15	rgen 3053	. . . . . . 7 ⊢ ∀𝑥 ∈ ℕ0s (( bday ↾ ℕ0s)‘𝑥) ∈ ω
17		fnfvrnss 7073	. . . . . . 7 ⊢ ((( bday ↾ ℕ0s) Fn ℕ0s ∧ ∀𝑥 ∈ ℕ0s (( bday ↾ ℕ0s)‘𝑥) ∈ ω) → ran ( bday ↾ ℕ0s) ⊆ ω)
18	12, 16, 17	mp2an 693	. . . . . 6 ⊢ ran ( bday ↾ ℕ0s) ⊆ ω
19		eqeq2 2748	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → (( bday ‘𝑦) = 𝑏 ↔ ( bday ‘𝑦) = ∅))
20	19	rexbidv 3161	. . . . . . . . 9 ⊢ (𝑏 = ∅ → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = ∅))
21		eqeq2 2748	. . . . . . . . . 10 ⊢ (𝑏 = 𝑎 → (( bday ‘𝑦) = 𝑏 ↔ ( bday ‘𝑦) = 𝑎))
22	21	rexbidv 3161	. . . . . . . . 9 ⊢ (𝑏 = 𝑎 → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑎))
23		eqeq2 2748	. . . . . . . . . . 11 ⊢ (𝑏 = suc 𝑎 → (( bday ‘𝑦) = 𝑏 ↔ ( bday ‘𝑦) = suc 𝑎))
24	23	rexbidv 3161	. . . . . . . . . 10 ⊢ (𝑏 = suc 𝑎 → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = suc 𝑎))
25		fveqeq2 6849	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (( bday ‘𝑦) = suc 𝑎 ↔ ( bday ‘𝑧) = suc 𝑎))
26	25	cbvrexvw 3216	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = suc 𝑎 ↔ ∃𝑧 ∈ ℕ0s ( bday ‘𝑧) = suc 𝑎)
27	24, 26	bitrdi 287	. . . . . . . . 9 ⊢ (𝑏 = suc 𝑎 → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑏 ↔ ∃𝑧 ∈ ℕ0s ( bday ‘𝑧) = suc 𝑎))
28		eqeq2 2748	. . . . . . . . . 10 ⊢ (𝑏 = 𝑥 → (( bday ‘𝑦) = 𝑏 ↔ ( bday ‘𝑦) = 𝑥))
29	28	rexbidv 3161	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑥))
30		0n0s 28321	. . . . . . . . . 10 ⊢ 0s ∈ ℕ0s
31		bday0 27803	. . . . . . . . . 10 ⊢ ( bday ‘ 0s ) = ∅
32		fveqeq2 6849	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → (( bday ‘𝑦) = ∅ ↔ ( bday ‘ 0s ) = ∅))
33	32	rspcev 3564	. . . . . . . . . 10 ⊢ (( 0s ∈ ℕ0s ∧ ( bday ‘ 0s ) = ∅) → ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = ∅)
34	30, 31, 33	mp2an 693	. . . . . . . . 9 ⊢ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = ∅
35		fveqeq2 6849	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑦 +s 1s ) → (( bday ‘𝑧) = suc ( bday ‘𝑦) ↔ ( bday ‘(𝑦 +s 1s )) = suc ( bday ‘𝑦)))
36		peano2n0s 28322	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
37		bdayn0p1 28361	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0s → ( bday ‘(𝑦 +s 1s )) = suc ( bday ‘𝑦))
38	35, 36, 37	rspcedvdw 3567	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0s → ∃𝑧 ∈ ℕ0s ( bday ‘𝑧) = suc ( bday ‘𝑦))
39	38	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ω ∧ 𝑦 ∈ ℕ0s) → ∃𝑧 ∈ ℕ0s ( bday ‘𝑧) = suc ( bday ‘𝑦))
40		suceq 6391	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑦) = 𝑎 → suc ( bday ‘𝑦) = suc 𝑎)
41	40	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑦) = 𝑎 → (( bday ‘𝑧) = suc ( bday ‘𝑦) ↔ ( bday ‘𝑧) = suc 𝑎))
42	41	rexbidv 3161	. . . . . . . . . . 11 ⊢ (( bday ‘𝑦) = 𝑎 → (∃𝑧 ∈ ℕ0s ( bday ‘𝑧) = suc ( bday ‘𝑦) ↔ ∃𝑧 ∈ ℕ0s ( bday ‘𝑧) = suc 𝑎))
43	39, 42	syl5ibcom 245	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ω ∧ 𝑦 ∈ ℕ0s) → (( bday ‘𝑦) = 𝑎 → ∃𝑧 ∈ ℕ0s ( bday ‘𝑧) = suc 𝑎))
44	43	rexlimdva 3138	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑎 → ∃𝑧 ∈ ℕ0s ( bday ‘𝑧) = suc 𝑎))
45	20, 22, 27, 29, 34, 44	finds 7847	. . . . . . . 8 ⊢ (𝑥 ∈ ω → ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑥)
46		fvelrnb 6900	. . . . . . . . . 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 6859	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → (( bday ↾ ℕ0s)‘𝑦) = ( bday ‘𝑦))
49	48	eqeq1d 2738	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → ((( bday ↾ ℕ0s)‘𝑦) = 𝑥 ↔ ( bday ‘𝑦) = 𝑥))
50	49	rexbiia 3082	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℕ0s (( bday ↾ ℕ0s)‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑥)
51	47, 50	bitri 275	. . . . . . . 8 ⊢ (𝑥 ∈ ran ( bday ↾ ℕ0s) ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑥)
52	45, 51	sylibr 234	. . . . . . 7 ⊢ (𝑥 ∈ ω → 𝑥 ∈ ran ( bday ↾ ℕ0s))
53	52	ssriv 3925	. . . . . 6 ⊢ ω ⊆ ran ( bday ↾ ℕ0s)
54	18, 53	eqssi 3938	. . . . 5 ⊢ ran ( bday ↾ ℕ0s) = ω
55		df-fo 6504	. . . . 5 ⊢ (( bday ↾ ℕ0s):ℕ0s–onto→ω ↔ (( bday ↾ ℕ0s) Fn ℕ0s ∧ ran ( bday ↾ ℕ0s) = ω))
56	12, 54, 55	mpbir2an 712	. . . 4 ⊢ ( bday ↾ ℕ0s):ℕ0s–onto→ω
57		fof 6752	. . . 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 2750	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → ((( bday ↾ ℕ0s)‘𝑥) = (( bday ↾ ℕ0s)‘𝑦) ↔ ( bday ‘𝑥) = ( bday ‘𝑦)))
60		n0on 28328	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ Ons)
61		n0on 28328	. . . . . 6 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ Ons)
62		bday11on 28257	. . . . . . 7 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons ∧ ( bday ‘𝑥) = ( bday ‘𝑦)) → 𝑥 = 𝑦)
63	62	3expia 1122	. . . . . 6 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (( bday ‘𝑥) = ( bday ‘𝑦) → 𝑥 = 𝑦))
64	60, 61, 63	syl2an 597	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (( bday ‘𝑥) = ( bday ‘𝑦) → 𝑥 = 𝑦))
65	59, 64	sylbid 240	. . . 4 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → ((( bday ↾ ℕ0s)‘𝑥) = (( bday ↾ ℕ0s)‘𝑦) → 𝑥 = 𝑦))
66	65	rgen2 3177	. . 3 ⊢ ∀𝑥 ∈ ℕ0s ∀𝑦 ∈ ℕ0s ((( bday ↾ ℕ0s)‘𝑥) = (( bday ↾ ℕ0s)‘𝑦) → 𝑥 = 𝑦)
67		dff13 7209	. . 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 712	. 2 ⊢ ( bday ↾ ℕ0s):ℕ0s–1-1→ω
69		df-f1o 6505	. 2 ⊢ (( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω ↔ (( bday ↾ ℕ0s):ℕ0s–1-1→ω ∧ ( bday ↾ ℕ0s):ℕ0s–onto→ω))
70	68, 56, 69	mpbir2an 712	1 ⊢ ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  dom cdm 5631  ran crn 5632   ↾ cres 5633  suc csuc 6325  Fun wfun 6492   Fn wfn 6493  ⟶wf 6494  –1-1→wf1 6495  –onto→wfo 6496  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  ωcom 7817   No csur 27603   bday cbday 27605   0_s c0s 27797   1_s c1s 27798   +_s cadds 27951  On_scons 28243  ℕ_0scn0s 28304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-nadd 8602  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-1s 27800  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-norec2 27941  df-adds 27952  df-negs 28013  df-subs 28014  df-ons 28244  df-n0s 28306

This theorem is referenced by:  oldfib  28369  bdayfinlem  28478