Theorem ackbij1 10325

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 10323	. 2 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3		f1f 6786	. . . 4 ⊢ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → 𝐹:(𝒫 ω ∩ Fin)⟶ω)
4		frn 6725	. . . 4 ⊢ (𝐹:(𝒫 ω ∩ Fin)⟶ω → ran 𝐹 ⊆ ω)
5	2, 3, 4	mp2b 10	. . 3 ⊢ ran 𝐹 ⊆ ω
6		eleq1 2851	. . . . 5 ⊢ (𝑏 = ∅ → (𝑏 ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
7		eleq1 2851	. . . . 5 ⊢ (𝑏 = 𝑎 → (𝑏 ∈ ran 𝐹 ↔ 𝑎 ∈ ran 𝐹))
8		eleq1 2851	. . . . 5 ⊢ (𝑏 = suc 𝑎 → (𝑏 ∈ ran 𝐹 ↔ suc 𝑎 ∈ ran 𝐹))
9		peano1 7907	. . . . . . . 8 ⊢ ∅ ∈ ω
10		ackbij1lem3 10309	. . . . . . . 8 ⊢ (∅ ∈ ω → ∅ ∈ (𝒫 ω ∩ Fin))
11	9, 10	ax-mp 5	. . . . . . 7 ⊢ ∅ ∈ (𝒫 ω ∩ Fin)
12	1	ackbij1lem13 10319	. . . . . . 7 ⊢ (𝐹‘∅) = ∅
13		fveqeq2 6902	. . . . . . . 8 ⊢ (𝑎 = ∅ → ((𝐹‘𝑎) = ∅ ↔ (𝐹‘∅) = ∅))
14	13	rspcev 3579	. . . . . . 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 6787	. . . . . . . 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 6953	. . . . . . 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 10324	. . . . . . . . 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 6437	. . . . . . . . . 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 3189	. . . . . . . 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 3166	. . . . . 6 ⊢ (𝑎 ∈ ω → (∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc 𝑎))
28		fvelrnb 6953	. . . . . . 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 6953	. . . . . . 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 7915	. . . 4 ⊢ (𝑎 ∈ ω → 𝑎 ∈ ran 𝐹)
34	33	ssriv 3916	. . 3 ⊢ ω ⊆ ran 𝐹
35	5, 34	eqssi 3928	. 2 ⊢ ran 𝐹 = ω
36		dff1o5 6842	. 2 ⊢ (𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω ↔ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω ∧ ran 𝐹 = ω))
37	2, 35, 36	mpbir2an 724	1 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∃wrex 3089   ⊆ wss 3899   ∩ cin 4091  ∅c0 4280  𝒫 cpw 4559  {csn 4587  ∪ ciun 4953   ↦ cmpt 5188   × cxp 5652  ran crn 5655  suc csuc 6370   Fn wfn 6541  ⟶wf 6542  –1-1→wf1 6543  –1-1-onto→wf1o 6545  ‘cfv 6546  ωcom 7884  Fincfn 8984  cardccrd 10026

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-2o 8488  df-oadd 8491  df-er 8728  df-map 8860  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9992  df-card 10030

This theorem is used by:  fictb  10332  ackbijnn  16009