Theorem ackbij1 10272

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 10270	. 2 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3		f1f 6790	. . . 4 ⊢ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → 𝐹:(𝒫 ω ∩ Fin)⟶ω)
4		frn 6727	. . . 4 ⊢ (𝐹:(𝒫 ω ∩ Fin)⟶ω → ran 𝐹 ⊆ ω)
5	2, 3, 4	mp2b 10	. . 3 ⊢ ran 𝐹 ⊆ ω
6		eleq1 2814	. . . . 5 ⊢ (𝑏 = ∅ → (𝑏 ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
7		eleq1 2814	. . . . 5 ⊢ (𝑏 = 𝑎 → (𝑏 ∈ ran 𝐹 ↔ 𝑎 ∈ ran 𝐹))
8		eleq1 2814	. . . . 5 ⊢ (𝑏 = suc 𝑎 → (𝑏 ∈ ran 𝐹 ↔ suc 𝑎 ∈ ran 𝐹))
9		peano1 7892	. . . . . . . 8 ⊢ ∅ ∈ ω
10		ackbij1lem3 10256	. . . . . . . 8 ⊢ (∅ ∈ ω → ∅ ∈ (𝒫 ω ∩ Fin))
11	9, 10	ax-mp 5	. . . . . . 7 ⊢ ∅ ∈ (𝒫 ω ∩ Fin)
12	1	ackbij1lem13 10266	. . . . . . 7 ⊢ (𝐹‘∅) = ∅
13		fveqeq2 6902	. . . . . . . 8 ⊢ (𝑎 = ∅ → ((𝐹‘𝑎) = ∅ ↔ (𝐹‘∅) = ∅))
14	13	rspcev 3607	. . . . . . 7 ⊢ ((∅ ∈ (𝒫 ω ∩ Fin) ∧ (𝐹‘∅) = ∅) → ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑎) = ∅)
15	11, 12, 14	mp2an 690	. . . . . 6 ⊢ ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑎) = ∅
16		f1fn 6791	. . . . . . . 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 6955	. . . . . . 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 230	. . . . 5 ⊢ ∅ ∈ ran 𝐹
21	1	ackbij1lem18 10271	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝑐))
22	21	adantl 480	. . . . . . . 8 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ (𝒫 ω ∩ Fin)) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝑐))
23		suceq 6434	. . . . . . . . . 10 ⊢ ((𝐹‘𝑐) = 𝑎 → suc (𝐹‘𝑐) = suc 𝑎)
24	23	eqeq2d 2737	. . . . . . . . 9 ⊢ ((𝐹‘𝑐) = 𝑎 → ((𝐹‘𝑏) = suc (𝐹‘𝑐) ↔ (𝐹‘𝑏) = suc 𝑎))
25	24	rexbidv 3169	. . . . . . . 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 244	. . . . . . 7 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc 𝑎))
27	26	rexlimdva 3145	. . . . . 6 ⊢ (𝑎 ∈ ω → (∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc 𝑎))
28		fvelrnb 6955	. . . . . . 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 6955	. . . . . . 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 295	. . . . 5 ⊢ (𝑎 ∈ ω → (𝑎 ∈ ran 𝐹 → suc 𝑎 ∈ ran 𝐹))
33	6, 7, 8, 7, 20, 32	finds 7901	. . . 4 ⊢ (𝑎 ∈ ω → 𝑎 ∈ ran 𝐹)
34	33	ssriv 3982	. . 3 ⊢ ω ⊆ ran 𝐹
35	5, 34	eqssi 3995	. 2 ⊢ ran 𝐹 = ω
36		dff1o5 6844	. 2 ⊢ (𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω ↔ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω ∧ ran 𝐹 = ω))
37	2, 35, 36	mpbir2an 709	1 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∃wrex 3060   ∩ cin 3945   ⊆ wss 3946  ∅c0 4322  𝒫 cpw 4597  {csn 4623  ∪ ciun 4993   ↦ cmpt 5228   × cxp 5672  ran crn 5675  suc csuc 6370   Fn wfn 6541  ⟶wf 6542  –1-1→wf1 6543  –1-1-onto→wf1o 6545  ‘cfv 6546  ωcom 7868  Fincfn 8966  cardccrd 9971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-oadd 8492  df-er 8726  df-map 8849  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-dju 9937  df-card 9975

This theorem is referenced by:  fictb  10279  ackbijnn  15827