Theorem bdayn0sf1o 28266

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 27682	. . . . . . 7 ⊢ Fun bday
2		funres 6524	. . . . . . 7 ⊢ (Fun bday → Fun ( bday ↾ ℕ0s))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun ( bday ↾ ℕ0s)
4		dmres 5963	. . . . . . 7 ⊢ dom ( bday ↾ ℕ0s) = (ℕ0s ∩ dom bday )
5		bdaydm 27684	. . . . . . . 8 ⊢ dom bday = No
6	5	ineq2i 4168	. . . . . . 7 ⊢ (ℕ0s ∩ dom bday ) = (ℕ0s ∩ No )
7		n0ssno 28220	. . . . . . . 8 ⊢ ℕ0s ⊆ No
8		dfss2 3921	. . . . . . . 8 ⊢ (ℕ0s ⊆ No ↔ (ℕ0s ∩ No ) = ℕ0s)
9	7, 8	mpbi 230	. . . . . . 7 ⊢ (ℕ0s ∩ No ) = ℕ0s
10	4, 6, 9	3eqtri 2756	. . . . . 6 ⊢ dom ( bday ↾ ℕ0s) = ℕ0s
11		df-fn 6485	. . . . . 6 ⊢ (( bday ↾ ℕ0s) Fn ℕ0s ↔ (Fun ( bday ↾ ℕ0s) ∧ dom ( bday ↾ ℕ0s) = ℕ0s))
12	3, 10, 11	mpbir2an 711	. . . . 5 ⊢ ( bday ↾ ℕ0s) Fn ℕ0s
13		fvres 6841	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → (( bday ↾ ℕ0s)‘𝑥) = ( bday ‘𝑥))
14		n0sbday 28251	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → ( bday ‘𝑥) ∈ ω)
15	13, 14	eqeltrd 2828	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → (( bday ↾ ℕ0s)‘𝑥) ∈ ω)
16	15	rgen 3046	. . . . . . 7 ⊢ ∀𝑥 ∈ ℕ0s (( bday ↾ ℕ0s)‘𝑥) ∈ ω
17		fnfvrnss 7055	. . . . . . 7 ⊢ ((( bday ↾ ℕ0s) Fn ℕ0s ∧ ∀𝑥 ∈ ℕ0s (( bday ↾ ℕ0s)‘𝑥) ∈ ω) → ran ( bday ↾ ℕ0s) ⊆ ω)
18	12, 16, 17	mp2an 692	. . . . . 6 ⊢ ran ( bday ↾ ℕ0s) ⊆ ω
19		eqeq2 2741	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → (( bday ‘𝑦) = 𝑏 ↔ ( bday ‘𝑦) = ∅))
20	19	rexbidv 3153	. . . . . . . . 9 ⊢ (𝑏 = ∅ → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = ∅))
21		eqeq2 2741	. . . . . . . . . 10 ⊢ (𝑏 = 𝑎 → (( bday ‘𝑦) = 𝑏 ↔ ( bday ‘𝑦) = 𝑎))
22	21	rexbidv 3153	. . . . . . . . 9 ⊢ (𝑏 = 𝑎 → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑎))
23		eqeq2 2741	. . . . . . . . . . 11 ⊢ (𝑏 = suc 𝑎 → (( bday ‘𝑦) = 𝑏 ↔ ( bday ‘𝑦) = suc 𝑎))
24	23	rexbidv 3153	. . . . . . . . . 10 ⊢ (𝑏 = suc 𝑎 → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = suc 𝑎))
25		fveqeq2 6831	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (( bday ‘𝑦) = suc 𝑎 ↔ ( bday ‘𝑧) = suc 𝑎))
26	25	cbvrexvw 3208	. . . . . . . . . 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 2741	. . . . . . . . . 10 ⊢ (𝑏 = 𝑥 → (( bday ‘𝑦) = 𝑏 ↔ ( bday ‘𝑦) = 𝑥))
29	28	rexbidv 3153	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑥))
30		0n0s 28229	. . . . . . . . . 10 ⊢ 0s ∈ ℕ0s
31		bday0s 27743	. . . . . . . . . 10 ⊢ ( bday ‘ 0s ) = ∅
32		fveqeq2 6831	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → (( bday ‘𝑦) = ∅ ↔ ( bday ‘ 0s ) = ∅))
33	32	rspcev 3577	. . . . . . . . . 10 ⊢ (( 0s ∈ ℕ0s ∧ ( bday ‘ 0s ) = ∅) → ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = ∅)
34	30, 31, 33	mp2an 692	. . . . . . . . 9 ⊢ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = ∅
35		fveqeq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑦 +s 1s ) → (( bday ‘𝑧) = suc ( bday ‘𝑦) ↔ ( bday ‘(𝑦 +s 1s )) = suc ( bday ‘𝑦)))
36		peano2n0s 28230	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
37		bdayn0p1 28265	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0s → ( bday ‘(𝑦 +s 1s )) = suc ( bday ‘𝑦))
38	35, 36, 37	rspcedvdw 3580	. . . . . . . . . . . 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 6375	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑦) = 𝑎 → suc ( bday ‘𝑦) = suc 𝑎)
41	40	eqeq2d 2740	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑦) = 𝑎 → (( bday ‘𝑧) = suc ( bday ‘𝑦) ↔ ( bday ‘𝑧) = suc 𝑎))
42	41	rexbidv 3153	. . . . . . . . . . 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 3130	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑎 → ∃𝑧 ∈ ℕ0s ( bday ‘𝑧) = suc 𝑎))
45	20, 22, 27, 29, 34, 44	finds 7829	. . . . . . . 8 ⊢ (𝑥 ∈ ω → ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑥)
46		fvelrnb 6883	. . . . . . . . . 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 6841	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → (( bday ↾ ℕ0s)‘𝑦) = ( bday ‘𝑦))
49	48	eqeq1d 2731	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → ((( bday ↾ ℕ0s)‘𝑦) = 𝑥 ↔ ( bday ‘𝑦) = 𝑥))
50	49	rexbiia 3074	. . . . . . . . 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 3939	. . . . . 6 ⊢ ω ⊆ ran ( bday ↾ ℕ0s)
54	18, 53	eqssi 3952	. . . . 5 ⊢ ran ( bday ↾ ℕ0s) = ω
55		df-fo 6488	. . . . 5 ⊢ (( bday ↾ ℕ0s):ℕ0s–onto→ω ↔ (( bday ↾ ℕ0s) Fn ℕ0s ∧ ran ( bday ↾ ℕ0s) = ω))
56	12, 54, 55	mpbir2an 711	. . . 4 ⊢ ( bday ↾ ℕ0s):ℕ0s–onto→ω
57		fof 6736	. . . 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 2743	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → ((( bday ↾ ℕ0s)‘𝑥) = (( bday ↾ ℕ0s)‘𝑦) ↔ ( bday ‘𝑥) = ( bday ‘𝑦)))
60		n0ons 28235	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ Ons)
61		n0ons 28235	. . . . . 6 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ Ons)
62		bday11on 28173	. . . . . . 7 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons ∧ ( bday ‘𝑥) = ( bday ‘𝑦)) → 𝑥 = 𝑦)
63	62	3expia 1121	. . . . . 6 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (( bday ‘𝑥) = ( bday ‘𝑦) → 𝑥 = 𝑦))
64	60, 61, 63	syl2an 596	. . . . 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 3169	. . 3 ⊢ ∀𝑥 ∈ ℕ0s ∀𝑦 ∈ ℕ0s ((( bday ↾ ℕ0s)‘𝑥) = (( bday ↾ ℕ0s)‘𝑦) → 𝑥 = 𝑦)
67		dff13 7191	. . 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 711	. 2 ⊢ ( bday ↾ ℕ0s):ℕ0s–1-1→ω
69		df-f1o 6489	. 2 ⊢ (( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω ↔ (( bday ↾ ℕ0s):ℕ0s–1-1→ω ∧ ( bday ↾ ℕ0s):ℕ0s–onto→ω))
70	68, 56, 69	mpbir2an 711	1 ⊢ ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  dom cdm 5619  ran crn 5620   ↾ cres 5621  suc csuc 6309  Fun wfun 6476   Fn wfn 6477  ⟶wf 6478  –1-1→wf1 6479  –onto→wfo 6480  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349  ωcom 7799   No csur 27549   bday cbday 27551   0_s c0s 27737   1_s c1s 27738   +_s cadds 27873  On_scons 28159  ℕ_0scnn0s 28213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-nadd 8584  df-no 27552  df-slt 27553  df-bday 27554  df-sle 27655  df-sslt 27692  df-scut 27694  df-0s 27739  df-1s 27740  df-made 27759  df-old 27760  df-left 27762  df-right 27763  df-norec 27852  df-norec2 27863  df-adds 27874  df-negs 27934  df-subs 27935  df-ons 28160  df-n0s 28215

This theorem is referenced by: (None)