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Theorem ackbij1 9649
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 9647 . 2 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3 f1f 6549 . . . 4 (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → 𝐹:(𝒫 ω ∩ Fin)⟶ω)
4 frn 6493 . . . 4 (𝐹:(𝒫 ω ∩ Fin)⟶ω → ran 𝐹 ⊆ ω)
52, 3, 4mp2b 10 . . 3 ran 𝐹 ⊆ ω
6 eleq1 2877 . . . . 5 (𝑏 = ∅ → (𝑏 ∈ ran 𝐹 ↔ ∅ ∈ ran 𝐹))
7 eleq1 2877 . . . . 5 (𝑏 = 𝑎 → (𝑏 ∈ ran 𝐹𝑎 ∈ ran 𝐹))
8 eleq1 2877 . . . . 5 (𝑏 = suc 𝑎 → (𝑏 ∈ ran 𝐹 ↔ suc 𝑎 ∈ ran 𝐹))
9 peano1 7581 . . . . . . . 8 ∅ ∈ ω
10 ackbij1lem3 9633 . . . . . . . 8 (∅ ∈ ω → ∅ ∈ (𝒫 ω ∩ Fin))
119, 10ax-mp 5 . . . . . . 7 ∅ ∈ (𝒫 ω ∩ Fin)
121ackbij1lem13 9643 . . . . . . 7 (𝐹‘∅) = ∅
13 fveqeq2 6654 . . . . . . . 8 (𝑎 = ∅ → ((𝐹𝑎) = ∅ ↔ (𝐹‘∅) = ∅))
1413rspcev 3571 . . . . . . 7 ((∅ ∈ (𝒫 ω ∩ Fin) ∧ (𝐹‘∅) = ∅) → ∃𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹𝑎) = ∅)
1511, 12, 14mp2an 691 . . . . . 6 𝑎 ∈ (𝒫 ω ∩ Fin)(𝐹𝑎) = ∅
16 f1fn 6550 . . . . . . . 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 6701 . . . . . . 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 234 . . . . 5 ∅ ∈ ran 𝐹
211ackbij1lem18 9648 . . . . . . . . 9 (𝑐 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝑐))
2221adantl 485 . . . . . . . 8 ((𝑎 ∈ ω ∧ 𝑐 ∈ (𝒫 ω ∩ Fin)) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝑐))
23 suceq 6224 . . . . . . . . . 10 ((𝐹𝑐) = 𝑎 → suc (𝐹𝑐) = suc 𝑎)
2423eqeq2d 2809 . . . . . . . . 9 ((𝐹𝑐) = 𝑎 → ((𝐹𝑏) = suc (𝐹𝑐) ↔ (𝐹𝑏) = suc 𝑎))
2524rexbidv 3256 . . . . . . . 8 ((𝐹𝑐) = 𝑎 → (∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝑐) ↔ ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc 𝑎))
2622, 25syl5ibcom 248 . . . . . . 7 ((𝑎 ∈ ω ∧ 𝑐 ∈ (𝒫 ω ∩ Fin)) → ((𝐹𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc 𝑎))
2726rexlimdva 3243 . . . . . 6 (𝑎 ∈ ω → (∃𝑐 ∈ (𝒫 ω ∩ Fin)(𝐹𝑐) = 𝑎 → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc 𝑎))
28 fvelrnb 6701 . . . . . . 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 6701 . . . . . . 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 299 . . . . 5 (𝑎 ∈ ω → (𝑎 ∈ ran 𝐹 → suc 𝑎 ∈ ran 𝐹))
336, 7, 8, 7, 20, 32finds 7589 . . . 4 (𝑎 ∈ ω → 𝑎 ∈ ran 𝐹)
3433ssriv 3919 . . 3 ω ⊆ ran 𝐹
355, 34eqssi 3931 . 2 ran 𝐹 = ω
36 dff1o5 6599 . 2 (𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω ↔ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω ∧ ran 𝐹 = ω))
372, 35, 36mpbir2an 710 1 𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2111  wrex 3107  cin 3880  wss 3881  c0 4243  𝒫 cpw 4497  {csn 4525   ciun 4881  cmpt 5110   × cxp 5517  ran crn 5520  suc csuc 6161   Fn wfn 6319  wf 6320  1-1wf1 6321  1-1-ontowf1o 6323  cfv 6324  ωcom 7560  Fincfn 8492  cardccrd 9348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352
This theorem is referenced by:  fictb  9656  ackbijnn  15175
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