MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ackbij1b Structured version   Visualization version   GIF version

Theorem ackbij1b 9995
Description: The Ackermann bijection, part 1b: the bijection from ackbij1 9994 restricts naturally to the powers of particular naturals. (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
ackbij1b (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦

Proof of Theorem ackbij1b
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 ackbij.f . . . . . 6 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
21ackbij1lem17 9992 . . . . 5 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3 ackbij2lem1 9975 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin))
4 pwexg 5301 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ∈ V)
5 f1imaeng 8800 . . . . 5 ((𝐹:(𝒫 ω ∩ Fin)–1-1→ω ∧ 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin) ∧ 𝒫 𝐴 ∈ V) → (𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴)
62, 3, 4, 5mp3an2i 1465 . . . 4 (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴)
7 nnfi 8950 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ Fin)
8 pwfi 8961 . . . . . 6 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
97, 8sylib 217 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ∈ Fin)
10 ficardid 9720 . . . . 5 (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
11 ensym 8789 . . . . 5 ((card‘𝒫 𝐴) ≈ 𝒫 𝐴 → 𝒫 𝐴 ≈ (card‘𝒫 𝐴))
129, 10, 113syl 18 . . . 4 (𝐴 ∈ ω → 𝒫 𝐴 ≈ (card‘𝒫 𝐴))
13 entr 8792 . . . 4 (((𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴 ∧ 𝒫 𝐴 ≈ (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
146, 12, 13syl2anc 584 . . 3 (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
15 onfin2 9014 . . . . . . 7 ω = (On ∩ Fin)
16 inss2 4163 . . . . . . 7 (On ∩ Fin) ⊆ Fin
1715, 16eqsstri 3955 . . . . . 6 ω ⊆ Fin
18 ficardom 9719 . . . . . . 7 (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ∈ ω)
199, 18syl 17 . . . . . 6 (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ ω)
2017, 19sselid 3919 . . . . 5 (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ Fin)
21 php3 8995 . . . . . 6 (((card‘𝒫 𝐴) ∈ Fin ∧ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴))
2221ex 413 . . . . 5 ((card‘𝒫 𝐴) ∈ Fin → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) → (𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴)))
2320, 22syl 17 . . . 4 (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) → (𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴)))
24 sdomnen 8769 . . . 4 ((𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴) → ¬ (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
2523, 24syl6 35 . . 3 (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) → ¬ (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴)))
2614, 25mt2d 136 . 2 (𝐴 ∈ ω → ¬ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴))
27 fvex 6787 . . . . . 6 (𝐹𝑎) ∈ V
28 ackbij1lem3 9978 . . . . . . . . 9 (𝐴 ∈ ω → 𝐴 ∈ (𝒫 ω ∩ Fin))
29 elpwi 4542 . . . . . . . . 9 (𝑎 ∈ 𝒫 𝐴𝑎𝐴)
301ackbij1lem12 9987 . . . . . . . . 9 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎𝐴) → (𝐹𝑎) ⊆ (𝐹𝐴))
3128, 29, 30syl2an 596 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → (𝐹𝑎) ⊆ (𝐹𝐴))
321ackbij1lem10 9985 . . . . . . . . . . 11 𝐹:(𝒫 ω ∩ Fin)⟶ω
33 peano1 7735 . . . . . . . . . . 11 ∅ ∈ ω
3432, 33f0cli 6974 . . . . . . . . . 10 (𝐹𝑎) ∈ ω
35 nnord 7720 . . . . . . . . . 10 ((𝐹𝑎) ∈ ω → Ord (𝐹𝑎))
3634, 35ax-mp 5 . . . . . . . . 9 Ord (𝐹𝑎)
3732, 33f0cli 6974 . . . . . . . . . 10 (𝐹𝐴) ∈ ω
38 nnord 7720 . . . . . . . . . 10 ((𝐹𝐴) ∈ ω → Ord (𝐹𝐴))
3937, 38ax-mp 5 . . . . . . . . 9 Ord (𝐹𝐴)
40 ordsucsssuc 7670 . . . . . . . . 9 ((Ord (𝐹𝑎) ∧ Ord (𝐹𝐴)) → ((𝐹𝑎) ⊆ (𝐹𝐴) ↔ suc (𝐹𝑎) ⊆ suc (𝐹𝐴)))
4136, 39, 40mp2an 689 . . . . . . . 8 ((𝐹𝑎) ⊆ (𝐹𝐴) ↔ suc (𝐹𝑎) ⊆ suc (𝐹𝐴))
4231, 41sylib 217 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹𝑎) ⊆ suc (𝐹𝐴))
431ackbij1lem14 9989 . . . . . . . . 9 (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹𝐴))
441ackbij1lem8 9983 . . . . . . . . 9 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
4543, 44eqtr3d 2780 . . . . . . . 8 (𝐴 ∈ ω → suc (𝐹𝐴) = (card‘𝒫 𝐴))
4645adantr 481 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹𝐴) = (card‘𝒫 𝐴))
4742, 46sseqtrd 3961 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹𝑎) ⊆ (card‘𝒫 𝐴))
48 sucssel 6358 . . . . . 6 ((𝐹𝑎) ∈ V → (suc (𝐹𝑎) ⊆ (card‘𝒫 𝐴) → (𝐹𝑎) ∈ (card‘𝒫 𝐴)))
4927, 47, 48mpsyl 68 . . . . 5 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → (𝐹𝑎) ∈ (card‘𝒫 𝐴))
5049ralrimiva 3103 . . . 4 (𝐴 ∈ ω → ∀𝑎 ∈ 𝒫 𝐴(𝐹𝑎) ∈ (card‘𝒫 𝐴))
51 f1fun 6672 . . . . . 6 (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → Fun 𝐹)
522, 51ax-mp 5 . . . . 5 Fun 𝐹
53 f1dm 6674 . . . . . . 7 (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → dom 𝐹 = (𝒫 ω ∩ Fin))
542, 53ax-mp 5 . . . . . 6 dom 𝐹 = (𝒫 ω ∩ Fin)
553, 54sseqtrrdi 3972 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ⊆ dom 𝐹)
56 funimass4 6834 . . . . 5 ((Fun 𝐹 ∧ 𝒫 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ∀𝑎 ∈ 𝒫 𝐴(𝐹𝑎) ∈ (card‘𝒫 𝐴)))
5752, 55, 56sylancr 587 . . . 4 (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ∀𝑎 ∈ 𝒫 𝐴(𝐹𝑎) ∈ (card‘𝒫 𝐴)))
5850, 57mpbird 256 . . 3 (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴))
59 sspss 4034 . . 3 ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) ∨ (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)))
6058, 59sylib 217 . 2 (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) ∨ (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)))
61 orel1 886 . 2 (¬ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) → (((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) ∨ (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)))
6226, 60, 61sylc 65 1 (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844   = wceq 1539  wcel 2106  wral 3064  Vcvv 3432  cin 3886  wss 3887  wpss 3888  𝒫 cpw 4533  {csn 4561   ciun 4924   class class class wbr 5074  cmpt 5157   × cxp 5587  dom cdm 5589  cima 5592  Ord word 6265  Oncon0 6266  suc csuc 6268  Fun wfun 6427  1-1wf1 6430  cfv 6433  ωcom 7712  cen 8730  csdm 8732  Fincfn 8733  cardccrd 9693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697
This theorem is referenced by:  ackbij2lem2  9996
  Copyright terms: Public domain W3C validator