Theorem ackbij1 10131

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 10129	. 2 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3		f1f 6720	. . . 4 ⊢ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → 𝐹:(𝒫 ω ∩ Fin)⟶ω)
4		frn 6659	. . . 4 ⊢ (𝐹:(𝒫 ω ∩ Fin)⟶ω → ran 𝐹 ⊆ ω)
5	2, 3, 4	mp2b 10	. . 3 ⊢ ran 𝐹 ⊆ ω
6		eleq1 2816	. . . . 5 ⊢ (𝑏 = ∅ → (𝑏 ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
7		eleq1 2816	. . . . 5 ⊢ (𝑏 = 𝑎 → (𝑏 ∈ ran 𝐹 ↔ 𝑎 ∈ ran 𝐹))
8		eleq1 2816	. . . . 5 ⊢ (𝑏 = suc 𝑎 → (𝑏 ∈ ran 𝐹 ↔ suc 𝑎 ∈ ran 𝐹))
9		peano1 7822	. . . . . . . 8 ⊢ ∅ ∈ ω
10		ackbij1lem3 10115	. . . . . . . 8 ⊢ (∅ ∈ ω → ∅ ∈ (𝒫 ω ∩ Fin))
11	9, 10	ax-mp 5	. . . . . . 7 ⊢ ∅ ∈ (𝒫 ω ∩ Fin)
12	1	ackbij1lem13 10125	. . . . . . 7 ⊢ (𝐹‘∅) = ∅
13		fveqeq2 6831	. . . . . . . 8 ⊢ (𝑎 = ∅ → ((𝐹‘𝑎) = ∅ ↔ (𝐹‘∅) = ∅))
14	13	rspcev 3577	. . . . . . 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 6721	. . . . . . . 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 6883	. . . . . . 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 10130	. . . . . . . . 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 6375	. . . . . . . . . 10 ⊢ ((𝐹‘𝑐) = 𝑎 → suc (𝐹‘𝑐) = suc 𝑎)
24	23	eqeq2d 2740	. . . . . . . . 9 ⊢ ((𝐹‘𝑐) = 𝑎 → ((𝐹‘𝑏) = suc (𝐹‘𝑐) ↔ (𝐹‘𝑏) = suc 𝑎))
25	24	rexbidv 3153	. . . . . . . 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 3130	. . . . . 6 ⊢ (𝑎 ∈ ω → (∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc 𝑎))
28		fvelrnb 6883	. . . . . . 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 6883	. . . . . . 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 7829	. . . 4 ⊢ (𝑎 ∈ ω → 𝑎 ∈ ran 𝐹)
34	33	ssriv 3939	. . 3 ⊢ ω ⊆ ran 𝐹
35	5, 34	eqssi 3952	. 2 ⊢ ran 𝐹 = ω
36		dff1o5 6773	. 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 3053   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  {csn 4577  ∪ ciun 4941   ↦ cmpt 5173   × cxp 5617  ran crn 5620  suc csuc 6309   Fn wfn 6477  ⟶wf 6478  –1-1→wf1 6479  –1-1-onto→wf1o 6481  ‘cfv 6482  ωcom 7799  Fincfn 8872  cardccrd 9831

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-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-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-op 4584  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-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-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-oadd 8392  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-dju 9797  df-card 9835

This theorem is referenced by:  fictb  10138  ackbijnn  15735