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Theorem ackbij1b 9046
Description: The Ackermann bijection, part 1b: the bijection from ackbij1 9045 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 ackbij2lem1 9026 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin))
2 pwexg 4841 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ∈ V)
3 ackbij.f . . . . . . 7 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
43ackbij1lem17 9043 . . . . . 6 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
5 f1imaeng 8001 . . . . . 6 ((𝐹:(𝒫 ω ∩ Fin)–1-1→ω ∧ 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin) ∧ 𝒫 𝐴 ∈ V) → (𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴)
64, 5mp3an1 1409 . . . . 5 ((𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin) ∧ 𝒫 𝐴 ∈ V) → (𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴)
71, 2, 6syl2anc 692 . . . 4 (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴)
8 nnfi 8138 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ Fin)
9 pwfi 8246 . . . . . 6 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
108, 9sylib 208 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ∈ Fin)
11 ficardid 8773 . . . . 5 (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
12 ensym 7990 . . . . 5 ((card‘𝒫 𝐴) ≈ 𝒫 𝐴 → 𝒫 𝐴 ≈ (card‘𝒫 𝐴))
1310, 11, 123syl 18 . . . 4 (𝐴 ∈ ω → 𝒫 𝐴 ≈ (card‘𝒫 𝐴))
14 entr 7993 . . . 4 (((𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴 ∧ 𝒫 𝐴 ≈ (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
157, 13, 14syl2anc 692 . . 3 (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
16 onfin2 8137 . . . . . . 7 ω = (On ∩ Fin)
17 inss2 3826 . . . . . . 7 (On ∩ Fin) ⊆ Fin
1816, 17eqsstri 3627 . . . . . 6 ω ⊆ Fin
19 ficardom 8772 . . . . . . 7 (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ∈ ω)
2010, 19syl 17 . . . . . 6 (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ ω)
2118, 20sseldi 3593 . . . . 5 (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ Fin)
22 php3 8131 . . . . . 6 (((card‘𝒫 𝐴) ∈ Fin ∧ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴))
2322ex 450 . . . . 5 ((card‘𝒫 𝐴) ∈ Fin → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) → (𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴)))
2421, 23syl 17 . . . 4 (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) → (𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴)))
25 sdomnen 7969 . . . 4 ((𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴) → ¬ (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
2624, 25syl6 35 . . 3 (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) → ¬ (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴)))
2715, 26mt2d 131 . 2 (𝐴 ∈ ω → ¬ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴))
28 fvex 6188 . . . . . 6 (𝐹𝑎) ∈ V
29 ackbij1lem3 9029 . . . . . . . . 9 (𝐴 ∈ ω → 𝐴 ∈ (𝒫 ω ∩ Fin))
30 elpwi 4159 . . . . . . . . 9 (𝑎 ∈ 𝒫 𝐴𝑎𝐴)
313ackbij1lem12 9038 . . . . . . . . 9 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎𝐴) → (𝐹𝑎) ⊆ (𝐹𝐴))
3229, 30, 31syl2an 494 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → (𝐹𝑎) ⊆ (𝐹𝐴))
333ackbij1lem10 9036 . . . . . . . . . . 11 𝐹:(𝒫 ω ∩ Fin)⟶ω
34 peano1 7070 . . . . . . . . . . 11 ∅ ∈ ω
3533, 34f0cli 6356 . . . . . . . . . 10 (𝐹𝑎) ∈ ω
36 nnord 7058 . . . . . . . . . 10 ((𝐹𝑎) ∈ ω → Ord (𝐹𝑎))
3735, 36ax-mp 5 . . . . . . . . 9 Ord (𝐹𝑎)
3833, 34f0cli 6356 . . . . . . . . . 10 (𝐹𝐴) ∈ ω
39 nnord 7058 . . . . . . . . . 10 ((𝐹𝐴) ∈ ω → Ord (𝐹𝐴))
4038, 39ax-mp 5 . . . . . . . . 9 Ord (𝐹𝐴)
41 ordsucsssuc 7008 . . . . . . . . 9 ((Ord (𝐹𝑎) ∧ Ord (𝐹𝐴)) → ((𝐹𝑎) ⊆ (𝐹𝐴) ↔ suc (𝐹𝑎) ⊆ suc (𝐹𝐴)))
4237, 40, 41mp2an 707 . . . . . . . 8 ((𝐹𝑎) ⊆ (𝐹𝐴) ↔ suc (𝐹𝑎) ⊆ suc (𝐹𝐴))
4332, 42sylib 208 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹𝑎) ⊆ suc (𝐹𝐴))
443ackbij1lem14 9040 . . . . . . . . 9 (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹𝐴))
453ackbij1lem8 9034 . . . . . . . . 9 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
4644, 45eqtr3d 2656 . . . . . . . 8 (𝐴 ∈ ω → suc (𝐹𝐴) = (card‘𝒫 𝐴))
4746adantr 481 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹𝐴) = (card‘𝒫 𝐴))
4843, 47sseqtrd 3633 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹𝑎) ⊆ (card‘𝒫 𝐴))
49 sucssel 5807 . . . . . 6 ((𝐹𝑎) ∈ V → (suc (𝐹𝑎) ⊆ (card‘𝒫 𝐴) → (𝐹𝑎) ∈ (card‘𝒫 𝐴)))
5028, 48, 49mpsyl 68 . . . . 5 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → (𝐹𝑎) ∈ (card‘𝒫 𝐴))
5150ralrimiva 2963 . . . 4 (𝐴 ∈ ω → ∀𝑎 ∈ 𝒫 𝐴(𝐹𝑎) ∈ (card‘𝒫 𝐴))
52 f1fun 6090 . . . . . 6 (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → Fun 𝐹)
534, 52ax-mp 5 . . . . 5 Fun 𝐹
54 f1dm 6092 . . . . . . 7 (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → dom 𝐹 = (𝒫 ω ∩ Fin))
554, 54ax-mp 5 . . . . . 6 dom 𝐹 = (𝒫 ω ∩ Fin)
561, 55syl6sseqr 3644 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ⊆ dom 𝐹)
57 funimass4 6234 . . . . 5 ((Fun 𝐹 ∧ 𝒫 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ∀𝑎 ∈ 𝒫 𝐴(𝐹𝑎) ∈ (card‘𝒫 𝐴)))
5853, 56, 57sylancr 694 . . . 4 (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ∀𝑎 ∈ 𝒫 𝐴(𝐹𝑎) ∈ (card‘𝒫 𝐴)))
5951, 58mpbird 247 . . 3 (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴))
60 sspss 3698 . . 3 ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) ∨ (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)))
6159, 60sylib 208 . 2 (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) ∨ (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)))
62 orel1 397 . 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)))
6327, 61, 62sylc 65 1 (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  cin 3566  wss 3567  wpss 3568  𝒫 cpw 4149  {csn 4168   ciun 4511   class class class wbr 4644  cmpt 4720   × cxp 5102  dom cdm 5104  cima 5107  Ord word 5710  Oncon0 5711  suc csuc 5713  Fun wfun 5870  1-1wf1 5873  cfv 5876  ωcom 7050  cen 7937  csdm 7939  Fincfn 7940  cardccrd 8746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975
This theorem is referenced by:  ackbij2lem2  9047
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