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Theorem ackbij1 10275
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 10273 . 2 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3 f1f 6805 . . . 4 (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → 𝐹:(𝒫 ω ∩ Fin)⟶ω)
4 frn 6744 . . . 4 (𝐹:(𝒫 ω ∩ Fin)⟶ω → ran 𝐹 ⊆ ω)
52, 3, 4mp2b 10 . . 3 ran 𝐹 ⊆ ω
6 eleq1 2827 . . . . 5 (𝑏 = ∅ → (𝑏 ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
7 eleq1 2827 . . . . 5 (𝑏 = 𝑎 → (𝑏 ∈ ran 𝐹𝑎 ∈ ran 𝐹))
8 eleq1 2827 . . . . 5 (𝑏 = suc 𝑎 → (𝑏 ∈ ran 𝐹 ↔ suc 𝑎 ∈ ran 𝐹))
9 peano1 7911 . . . . . . . 8 ∅ ∈ ω
10 ackbij1lem3 10259 . . . . . . . 8 (∅ ∈ ω → ∅ ∈ (𝒫 ω ∩ Fin))
119, 10ax-mp 5 . . . . . . 7 ∅ ∈ (𝒫 ω ∩ Fin)
121ackbij1lem13 10269 . . . . . . 7 (𝐹‘∅) = ∅
13 fveqeq2 6916 . . . . . . . 8 (𝑎 = ∅ → ((𝐹𝑎) = ∅ ↔ (𝐹‘∅) = ∅))
1413rspcev 3622 . . . . . . 7 ((∅ ∈ (𝒫 ω ∩ Fin) ∧ (𝐹‘∅) = ∅) → ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹𝑎) = ∅)
1511, 12, 14mp2an 692 . . . . . 6 𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹𝑎) = ∅
16 f1fn 6806 . . . . . . . 8 (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → 𝐹 Fn (𝒫 ω ∩ Fin))
172, 16ax-mp 5 . . . . . . 7 𝐹 Fn (𝒫 ω ∩ Fin)
18 fvelrnb 6969 . . . . . . 7 (𝐹 Fn (𝒫 ω ∩ Fin) → (∅ ∈ ran 𝐹 ↔ ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹𝑎) = ∅))
1917, 18ax-mp 5 . . . . . 6 (∅ ∈ ran 𝐹 ↔ ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹𝑎) = ∅)
2015, 19mpbir 231 . . . . 5 ∅ ∈ ran 𝐹
211ackbij1lem18 10274 . . . . . . . . 9 (𝑐 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝑐))
2221adantl 481 . . . . . . . 8 ((𝑎 ∈ ω ∧ 𝑐 ∈ (𝒫 ω ∩ Fin)) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝑐))
23 suceq 6452 . . . . . . . . . 10 ((𝐹𝑐) = 𝑎 → suc (𝐹𝑐) = suc 𝑎)
2423eqeq2d 2746 . . . . . . . . 9 ((𝐹𝑐) = 𝑎 → ((𝐹𝑏) = suc (𝐹𝑐) ↔ (𝐹𝑏) = suc 𝑎))
2524rexbidv 3177 . . . . . . . 8 ((𝐹𝑐) = 𝑎 → (∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝑐) ↔ ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc 𝑎))
2622, 25syl5ibcom 245 . . . . . . 7 ((𝑎 ∈ ω ∧ 𝑐 ∈ (𝒫 ω ∩ Fin)) → ((𝐹𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc 𝑎))
2726rexlimdva 3153 . . . . . 6 (𝑎 ∈ ω → (∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc 𝑎))
28 fvelrnb 6969 . . . . . . 7 (𝐹 Fn (𝒫 ω ∩ Fin) → (𝑎 ∈ ran 𝐹 ↔ ∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹𝑐) = 𝑎))
2917, 28ax-mp 5 . . . . . 6 (𝑎 ∈ ran 𝐹 ↔ ∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹𝑐) = 𝑎)
30 fvelrnb 6969 . . . . . . 7 (𝐹 Fn (𝒫 ω ∩ Fin) → (suc 𝑎 ∈ ran 𝐹 ↔ ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc 𝑎))
3117, 30ax-mp 5 . . . . . 6 (suc 𝑎 ∈ ran 𝐹 ↔ ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc 𝑎)
3227, 29, 313imtr4g 296 . . . . 5 (𝑎 ∈ ω → (𝑎 ∈ ran 𝐹 → suc 𝑎 ∈ ran 𝐹))
336, 7, 8, 7, 20, 32finds 7919 . . . 4 (𝑎 ∈ ω → 𝑎 ∈ ran 𝐹)
3433ssriv 3999 . . 3 ω ⊆ ran 𝐹
355, 34eqssi 4012 . 2 ran 𝐹 = ω
36 dff1o5 6858 . 2 (𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω ↔ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω ∧ ran 𝐹 = ω))
372, 35, 36mpbir2an 711 1 𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   ciun 4996  cmpt 5231   × cxp 5687  ran crn 5690  suc csuc 6388   Fn wfn 6558  wf 6559  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  ωcom 7887  Fincfn 8984  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977
This theorem is referenced by:  fictb  10282  ackbijnn  15861
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