Theorem ackbij1 10354

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)⟶1-1ontoω

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 10352	. 2 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶1-1ω
3		f1f 6808	. . . 4 ⊢ (𝐹:(𝒫 ω ∩ Fin)⟶1-1ω → 𝐹:(𝒫 ω ∩ Fin)⟶ω)
4		frn 6747	. . . 4 ⊢ (𝐹:(𝒫 ω ∩ Fin)⟶ω → ran 𝐹 ⊆ ω)
5	2, 3, 4	mp2b 10	. . 3 ⊢ ran 𝐹 ⊆ ω
6		eleq1 2852	. . . . 5 ⊢ (𝑏 = ∅ → (𝑏 ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
7		eleq1 2852	. . . . 5 ⊢ (𝑏 = 𝑎 → (𝑏 ∈ ran 𝐹 ↔ 𝑎 ∈ ran 𝐹))
8		eleq1 2852	. . . . 5 ⊢ (𝑏 = suc 𝑎 → (𝑏 ∈ ran 𝐹 ↔ suc 𝑎 ∈ ran 𝐹))
9		peano1 7934	. . . . . . . 8 ⊢ ∅ ∈ ω
10		ackbij1lem3 10338	. . . . . . . 8 ⊢ (∅ ∈ ω → ∅ ∈ (𝒫 ω ∩ Fin))
11	9, 10	ax-mp 5	. . . . . . 7 ⊢ ∅ ∈ (𝒫 ω ∩ Fin)
12	1	ackbij1lem13 10348	. . . . . . 7 ⊢ (𝐹‘∅) = ∅
13		fveqeq2 6925	. . . . . . . 8 ⊢ (𝑎 = ∅ → ((𝐹‘𝑎) = ∅ ↔ (𝐹‘∅) = ∅))
14	13	rspcev 3580	. . . . . . 7 ⊢ ((∅ ∈ (𝒫 ω ∩ Fin) ∧ (𝐹‘∅) = ∅) → ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑎) = ∅)
15	11, 12, 14	mp2an 705	. . . . . 6 ⊢ ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑎) = ∅
16		f1fn 6809	. . . . . . . 8 ⊢ (𝐹:(𝒫 ω ∩ Fin)⟶1-1ω → 𝐹 Fn (𝒫 ω ∩ Fin))
17	2, 16	ax-mp 5	. . . . . . 7 ⊢ 𝐹 Fn (𝒫 ω ∩ Fin)
18		fvelrnb 6976	. . . . . . 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 234	. . . . 5 ⊢ ∅ ∈ ran 𝐹
21	1	ackbij1lem18 10353	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝑐))
22	21	adantl 487	. . . . . . . 8 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ (𝒫 ω ∩ Fin)) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝑐))
23		suceq 6457	. . . . . . . . . 10 ⊢ ((𝐹‘𝑐) = 𝑎 → suc (𝐹‘𝑐) = suc 𝑎)
24	23	eqeq2d 2774	. . . . . . . . 9 ⊢ ((𝐹‘𝑐) = 𝑎 → ((𝐹‘𝑏) = suc (𝐹‘𝑐) ↔ (𝐹‘𝑏) = suc 𝑎))
25	24	rexbidv 3190	. . . . . . . 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 248	. . . . . . 7 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc 𝑎))
27	26	rexlimdva 3167	. . . . . 6 ⊢ (𝑎 ∈ ω → (∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc 𝑎))
28		fvelrnb 6976	. . . . . . 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 6976	. . . . . . 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 299	. . . . 5 ⊢ (𝑎 ∈ ω → (𝑎 ∈ ran 𝐹 → suc 𝑎 ∈ ran 𝐹))
33	6, 7, 8, 7, 20, 32	finds 7942	. . . 4 ⊢ (𝑎 ∈ ω → 𝑎 ∈ ran 𝐹)
34	33	ssriv 3917	. . 3 ⊢ ω ⊆ ran 𝐹
35	5, 34	eqssi 3929	. 2 ⊢ ran 𝐹 = ω
36		dff1o5 6864	. 2 ⊢ (𝐹:(𝒫 ω ∩ Fin)⟶1-1ontoω ↔ (𝐹:(𝒫 ω ∩ Fin)⟶1-1ω ∧ ran 𝐹 = ω))
37	2, 35, 36	mpbir2an 724	1 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶1-1ontoω

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  𝒫 cpw 4560  {csn 4588  ∪ ciun 4956   ↦ cmpt 5192   × cxp 5656  ran crn 5659  suc csuc 6390   Fn wfn 6561  ⟶wf 6562  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565  ‘cfv 6566  ωcom 7911  Fincfn 9011  cardccrd 10053

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-oadd 8518  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-dju 10019  df-card 10057

This theorem is used by:  fictb  10361  ackbijnn  16041