Theorem ackbij1 10256

Description: The Ackermann bijection, part 1: each natural number can be uniquely coded in binary as a finite set of natural numbers and conversely. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Hypothesis

Ref	Expression
ackbij.f	⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))

Assertion

Ref	Expression
ackbij1	⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω

Distinct variable group:   𝑥,𝐹,𝑦

Proof of Theorem ackbij1

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ackbij.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
2	1	ackbij1lem17 10254	. 2 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3		f1f 6779	. . . 4 ⊢ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → 𝐹:(𝒫 ω ∩ Fin)⟶ω)
4		frn 6718	. . . 4 ⊢ (𝐹:(𝒫 ω ∩ Fin)⟶ω → ran 𝐹 ⊆ ω)
5	2, 3, 4	mp2b 10	. . 3 ⊢ ran 𝐹 ⊆ ω
6		eleq1 2823	. . . . 5 ⊢ (𝑏 = ∅ → (𝑏 ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
7		eleq1 2823	. . . . 5 ⊢ (𝑏 = 𝑎 → (𝑏 ∈ ran 𝐹 ↔ 𝑎 ∈ ran 𝐹))
8		eleq1 2823	. . . . 5 ⊢ (𝑏 = suc 𝑎 → (𝑏 ∈ ran 𝐹 ↔ suc 𝑎 ∈ ran 𝐹))
9		peano1 7889	. . . . . . . 8 ⊢ ∅ ∈ ω
10		ackbij1lem3 10240	. . . . . . . 8 ⊢ (∅ ∈ ω → ∅ ∈ (𝒫 ω ∩ Fin))
11	9, 10	ax-mp 5	. . . . . . 7 ⊢ ∅ ∈ (𝒫 ω ∩ Fin)
12	1	ackbij1lem13 10250	. . . . . . 7 ⊢ (𝐹‘∅) = ∅
13		fveqeq2 6890	. . . . . . . 8 ⊢ (𝑎 = ∅ → ((𝐹‘𝑎) = ∅ ↔ (𝐹‘∅) = ∅))
14	13	rspcev 3606	. . . . . . 7 ⊢ ((∅ ∈ (𝒫 ω ∩ Fin) ∧ (𝐹‘∅) = ∅) → ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑎) = ∅)
15	11, 12, 14	mp2an 692	. . . . . 6 ⊢ ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑎) = ∅
16		f1fn 6780	. . . . . . . 8 ⊢ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → 𝐹 Fn (𝒫 ω ∩ Fin))
17	2, 16	ax-mp 5	. . . . . . 7 ⊢ 𝐹 Fn (𝒫 ω ∩ Fin)
18		fvelrnb 6944	. . . . . . 7 ⊢ (𝐹 Fn (𝒫 ω ∩ Fin) → (∅ ∈ ran 𝐹 ↔ ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑎) = ∅))
19	17, 18	ax-mp 5	. . . . . 6 ⊢ (∅ ∈ ran 𝐹 ↔ ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑎) = ∅)
20	15, 19	mpbir 231	. . . . 5 ⊢ ∅ ∈ ran 𝐹
21	1	ackbij1lem18 10255	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝑐))
22	21	adantl 481	. . . . . . . 8 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ (𝒫 ω ∩ Fin)) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝑐))
23		suceq 6424	. . . . . . . . . 10 ⊢ ((𝐹‘𝑐) = 𝑎 → suc (𝐹‘𝑐) = suc 𝑎)
24	23	eqeq2d 2747	. . . . . . . . 9 ⊢ ((𝐹‘𝑐) = 𝑎 → ((𝐹‘𝑏) = suc (𝐹‘𝑐) ↔ (𝐹‘𝑏) = suc 𝑎))
25	24	rexbidv 3165	. . . . . . . 8 ⊢ ((𝐹‘𝑐) = 𝑎 → (∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝑐) ↔ ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc 𝑎))
26	22, 25	syl5ibcom 245	. . . . . . 7 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc 𝑎))
27	26	rexlimdva 3142	. . . . . 6 ⊢ (𝑎 ∈ ω → (∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc 𝑎))
28		fvelrnb 6944	. . . . . . 7 ⊢ (𝐹 Fn (𝒫 ω ∩ Fin) → (𝑎 ∈ ran 𝐹 ↔ ∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑐) = 𝑎))
29	17, 28	ax-mp 5	. . . . . 6 ⊢ (𝑎 ∈ ran 𝐹 ↔ ∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑐) = 𝑎)
30		fvelrnb 6944	. . . . . . 7 ⊢ (𝐹 Fn (𝒫 ω ∩ Fin) → (suc 𝑎 ∈ ran 𝐹 ↔ ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc 𝑎))
31	17, 30	ax-mp 5	. . . . . 6 ⊢ (suc 𝑎 ∈ ran 𝐹 ↔ ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc 𝑎)
32	27, 29, 31	3imtr4g 296	. . . . 5 ⊢ (𝑎 ∈ ω → (𝑎 ∈ ran 𝐹 → suc 𝑎 ∈ ran 𝐹))
33	6, 7, 8, 7, 20, 32	finds 7897	. . . 4 ⊢ (𝑎 ∈ ω → 𝑎 ∈ ran 𝐹)
34	33	ssriv 3967	. . 3 ⊢ ω ⊆ ran 𝐹
35	5, 34	eqssi 3980	. 2 ⊢ ran 𝐹 = ω
36		dff1o5 6832	. 2 ⊢ (𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω ↔ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω ∧ ran 𝐹 = ω))
37	2, 35, 36	mpbir2an 711	1 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3061   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  {csn 4606  ∪ ciun 4972   ↦ cmpt 5206   × cxp 5657  ran crn 5660  suc csuc 6359   Fn wfn 6531  ⟶wf 6532  –1-1→wf1 6533  –1-1-onto→wf1o 6535  ‘cfv 6536  ωcom 7866  Fincfn 8964  cardccrd 9954

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-dju 9920  df-card 9958

This theorem is referenced by:  fictb  10263  ackbijnn  15849