Theorem bdayn0sf1o 28347

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 27746	. . . . . . 7 ⊢ Fun bday
2		funres 6533	. . . . . . 7 ⊢ (Fun bday → Fun ( bday ↾ ℕ0s))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun ( bday ↾ ℕ0s)
4		dmres 5970	. . . . . . 7 ⊢ dom ( bday ↾ ℕ0s) = (ℕ0s ∩ dom bday )
5		bdaydm 27748	. . . . . . . 8 ⊢ dom bday = No
6	5	ineq2i 4168	. . . . . . 7 ⊢ (ℕ0s ∩ dom bday ) = (ℕ0s ∩ No )
7		n0ssno 28299	. . . . . . . 8 ⊢ ℕ0s ⊆ No
8		dfss2 3918	. . . . . . . 8 ⊢ (ℕ0s ⊆ No ↔ (ℕ0s ∩ No ) = ℕ0s)
9	7, 8	mpbi 230	. . . . . . 7 ⊢ (ℕ0s ∩ No ) = ℕ0s
10	4, 6, 9	3eqtri 2762	. . . . . 6 ⊢ dom ( bday ↾ ℕ0s) = ℕ0s
11		df-fn 6494	. . . . . 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 6852	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → (( bday ↾ ℕ0s)‘𝑥) = ( bday ‘𝑥))
14		n0sbday 28330	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → ( bday ‘𝑥) ∈ ω)
15	13, 14	eqeltrd 2835	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → (( bday ↾ ℕ0s)‘𝑥) ∈ ω)
16	15	rgen 3052	. . . . . . 7 ⊢ ∀𝑥 ∈ ℕ0s (( bday ↾ ℕ0s)‘𝑥) ∈ ω
17		fnfvrnss 7066	. . . . . . 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 2747	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → (( bday ‘𝑦) = 𝑏 ↔ ( bday ‘𝑦) = ∅))
20	19	rexbidv 3159	. . . . . . . . 9 ⊢ (𝑏 = ∅ → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = ∅))
21		eqeq2 2747	. . . . . . . . . 10 ⊢ (𝑏 = 𝑎 → (( bday ‘𝑦) = 𝑏 ↔ ( bday ‘𝑦) = 𝑎))
22	21	rexbidv 3159	. . . . . . . . 9 ⊢ (𝑏 = 𝑎 → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑎))
23		eqeq2 2747	. . . . . . . . . . 11 ⊢ (𝑏 = suc 𝑎 → (( bday ‘𝑦) = 𝑏 ↔ ( bday ‘𝑦) = suc 𝑎))
24	23	rexbidv 3159	. . . . . . . . . 10 ⊢ (𝑏 = suc 𝑎 → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = suc 𝑎))
25		fveqeq2 6842	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (( bday ‘𝑦) = suc 𝑎 ↔ ( bday ‘𝑧) = suc 𝑎))
26	25	cbvrexvw 3214	. . . . . . . . . 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 2747	. . . . . . . . . 10 ⊢ (𝑏 = 𝑥 → (( bday ‘𝑦) = 𝑏 ↔ ( bday ‘𝑦) = 𝑥))
29	28	rexbidv 3159	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑏 ↔ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑥))
30		0n0s 28308	. . . . . . . . . 10 ⊢ 0s ∈ ℕ0s
31		bday0s 27807	. . . . . . . . . 10 ⊢ ( bday ‘ 0s ) = ∅
32		fveqeq2 6842	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → (( bday ‘𝑦) = ∅ ↔ ( bday ‘ 0s ) = ∅))
33	32	rspcev 3575	. . . . . . . . . 10 ⊢ (( 0s ∈ ℕ0s ∧ ( bday ‘ 0s ) = ∅) → ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = ∅)
34	30, 31, 33	mp2an 693	. . . . . . . . 9 ⊢ ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = ∅
35		fveqeq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑦 +s 1s ) → (( bday ‘𝑧) = suc ( bday ‘𝑦) ↔ ( bday ‘(𝑦 +s 1s )) = suc ( bday ‘𝑦)))
36		peano2n0s 28309	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
37		bdayn0p1 28346	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0s → ( bday ‘(𝑦 +s 1s )) = suc ( bday ‘𝑦))
38	35, 36, 37	rspcedvdw 3578	. . . . . . . . . . . 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 6384	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑦) = 𝑎 → suc ( bday ‘𝑦) = suc 𝑎)
41	40	eqeq2d 2746	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑦) = 𝑎 → (( bday ‘𝑧) = suc ( bday ‘𝑦) ↔ ( bday ‘𝑧) = suc 𝑎))
42	41	rexbidv 3159	. . . . . . . . . . 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 3136	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑎 → ∃𝑧 ∈ ℕ0s ( bday ‘𝑧) = suc 𝑎))
45	20, 22, 27, 29, 34, 44	finds 7838	. . . . . . . 8 ⊢ (𝑥 ∈ ω → ∃𝑦 ∈ ℕ0s ( bday ‘𝑦) = 𝑥)
46		fvelrnb 6893	. . . . . . . . . 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 6852	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → (( bday ↾ ℕ0s)‘𝑦) = ( bday ‘𝑦))
49	48	eqeq1d 2737	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → ((( bday ↾ ℕ0s)‘𝑦) = 𝑥 ↔ ( bday ‘𝑦) = 𝑥))
50	49	rexbiia 3080	. . . . . . . . 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 3936	. . . . . 6 ⊢ ω ⊆ ran ( bday ↾ ℕ0s)
54	18, 53	eqssi 3949	. . . . 5 ⊢ ran ( bday ↾ ℕ0s) = ω
55		df-fo 6497	. . . . 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 6745	. . . 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 2749	. . . . 5 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → ((( bday ↾ ℕ0s)‘𝑥) = (( bday ↾ ℕ0s)‘𝑦) ↔ ( bday ‘𝑥) = ( bday ‘𝑦)))
60		n0ons 28314	. . . . . 6 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ Ons)
61		n0ons 28314	. . . . . 6 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ Ons)
62		bday11on 28244	. . . . . . 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 3175	. . 3 ⊢ ∀𝑥 ∈ ℕ0s ∀𝑦 ∈ ℕ0s ((( bday ↾ ℕ0s)‘𝑥) = (( bday ↾ ℕ0s)‘𝑦) → 𝑥 = 𝑦)
67		dff13 7200	. . 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 6498	. 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 3050  ∃wrex 3059   ∩ cin 3899   ⊆ wss 3900  ∅c0 4284  dom cdm 5623  ran crn 5624   ↾ cres 5625  suc csuc 6318  Fun wfun 6485   Fn wfn 6486  ⟶wf 6487  –1-1→wf1 6488  –onto→wfo 6489  –1-1-onto→wf1o 6490  ‘cfv 6491  (class class class)co 7358  ωcom 7808   No csur 27609   bday cbday 27611   0_s c0s 27801   1_s c1s 27802   +_s cadds 27939  On_scons 28230  ℕ_0scnn0s 28291

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-nadd 8594  df-no 27612  df-slt 27613  df-bday 27614  df-sle 27715  df-sslt 27756  df-scut 27758  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27918  df-norec2 27929  df-adds 27940  df-negs 28001  df-subs 28002  df-ons 28231  df-n0s 28293

This theorem is referenced by:  oldfib  28354  bdayfinlem  28463