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Theorem ackbij1 9098
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‘ 𝑦𝑥 ({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.
StepHypRef Expression
1 ackbij.f . . 3 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
21ackbij1lem17 9096 . 2 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3 f1f 6139 . . . 4 (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → 𝐹:(𝒫 ω ∩ Fin)⟶ω)
4 frn 6091 . . . 4 (𝐹:(𝒫 ω ∩ Fin)⟶ω → ran 𝐹 ⊆ ω)
52, 3, 4mp2b 10 . . 3 ran 𝐹 ⊆ ω
6 eleq1 2718 . . . . 5 (𝑏 = ∅ → (𝑏 ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
7 eleq1 2718 . . . . 5 (𝑏 = 𝑎 → (𝑏 ∈ ran 𝐹𝑎 ∈ ran 𝐹))
8 eleq1 2718 . . . . 5 (𝑏 = suc 𝑎 → (𝑏 ∈ ran 𝐹 ↔ suc 𝑎 ∈ ran 𝐹))
9 peano1 7127 . . . . . . . 8 ∅ ∈ ω
10 ackbij1lem3 9082 . . . . . . . 8 (∅ ∈ ω → ∅ ∈ (𝒫 ω ∩ Fin))
119, 10ax-mp 5 . . . . . . 7 ∅ ∈ (𝒫 ω ∩ Fin)
121ackbij1lem13 9092 . . . . . . 7 (𝐹‘∅) = ∅
13 fveq2 6229 . . . . . . . . 9 (𝑎 = ∅ → (𝐹𝑎) = (𝐹‘∅))
1413eqeq1d 2653 . . . . . . . 8 (𝑎 = ∅ → ((𝐹𝑎) = ∅ ↔ (𝐹‘∅) = ∅))
1514rspcev 3340 . . . . . . 7 ((∅ ∈ (𝒫 ω ∩ Fin) ∧ (𝐹‘∅) = ∅) → ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹𝑎) = ∅)
1611, 12, 15mp2an 708 . . . . . 6 𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹𝑎) = ∅
17 f1fn 6140 . . . . . . . 8 (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → 𝐹 Fn (𝒫 ω ∩ Fin))
182, 17ax-mp 5 . . . . . . 7 𝐹 Fn (𝒫 ω ∩ Fin)
19 fvelrnb 6282 . . . . . . 7 (𝐹 Fn (𝒫 ω ∩ Fin) → (∅ ∈ ran 𝐹 ↔ ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹𝑎) = ∅))
2018, 19ax-mp 5 . . . . . 6 (∅ ∈ ran 𝐹 ↔ ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹𝑎) = ∅)
2116, 20mpbir 221 . . . . 5 ∅ ∈ ran 𝐹
221ackbij1lem18 9097 . . . . . . . . 9 (𝑐 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝑐))
2322adantl 481 . . . . . . . 8 ((𝑎 ∈ ω ∧ 𝑐 ∈ (𝒫 ω ∩ Fin)) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝑐))
24 suceq 5828 . . . . . . . . . 10 ((𝐹𝑐) = 𝑎 → suc (𝐹𝑐) = suc 𝑎)
2524eqeq2d 2661 . . . . . . . . 9 ((𝐹𝑐) = 𝑎 → ((𝐹𝑏) = suc (𝐹𝑐) ↔ (𝐹𝑏) = suc 𝑎))
2625rexbidv 3081 . . . . . . . 8 ((𝐹𝑐) = 𝑎 → (∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝑐) ↔ ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc 𝑎))
2723, 26syl5ibcom 235 . . . . . . 7 ((𝑎 ∈ ω ∧ 𝑐 ∈ (𝒫 ω ∩ Fin)) → ((𝐹𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc 𝑎))
2827rexlimdva 3060 . . . . . 6 (𝑎 ∈ ω → (∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc 𝑎))
29 fvelrnb 6282 . . . . . . 7 (𝐹 Fn (𝒫 ω ∩ Fin) → (𝑎 ∈ ran 𝐹 ↔ ∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹𝑐) = 𝑎))
3018, 29ax-mp 5 . . . . . 6 (𝑎 ∈ ran 𝐹 ↔ ∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹𝑐) = 𝑎)
31 fvelrnb 6282 . . . . . . 7 (𝐹 Fn (𝒫 ω ∩ Fin) → (suc 𝑎 ∈ ran 𝐹 ↔ ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc 𝑎))
3218, 31ax-mp 5 . . . . . 6 (suc 𝑎 ∈ ran 𝐹 ↔ ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc 𝑎)
3328, 30, 323imtr4g 285 . . . . 5 (𝑎 ∈ ω → (𝑎 ∈ ran 𝐹 → suc 𝑎 ∈ ran 𝐹))
346, 7, 8, 7, 21, 33finds 7134 . . . 4 (𝑎 ∈ ω → 𝑎 ∈ ran 𝐹)
3534ssriv 3640 . . 3 ω ⊆ ran 𝐹
365, 35eqssi 3652 . 2 ran 𝐹 = ω
37 dff1o5 6184 . 2 (𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω ↔ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω ∧ ran 𝐹 = ω))
382, 36, 37mpbir2an 975 1 𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   ciun 4552  cmpt 4762   × cxp 5141  ran crn 5144  suc csuc 5763   Fn wfn 5921  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  ωcom 7107  Fincfn 7997  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028
This theorem is referenced by:  fictb  9105  ackbijnn  14604
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