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Theorem ackbij1b 9263
Description: The Ackermann bijection, part 1b: the bijection from ackbij1 9262 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 9243 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin))
2 pwexg 4980 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ∈ V)
3 ackbij.f . . . . . . 7 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
43ackbij1lem17 9260 . . . . . 6 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
5 f1imaeng 8169 . . . . . 6 ((𝐹:(𝒫 ω ∩ Fin)–1-1→ω ∧ 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin) ∧ 𝒫 𝐴 ∈ V) → (𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴)
64, 5mp3an1 1559 . . . . 5 ((𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin) ∧ 𝒫 𝐴 ∈ V) → (𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴)
71, 2, 6syl2anc 573 . . . 4 (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴)
8 nnfi 8309 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ Fin)
9 pwfi 8417 . . . . . 6 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
108, 9sylib 208 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ∈ Fin)
11 ficardid 8988 . . . . 5 (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
12 ensym 8158 . . . . 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 8161 . . . 4 (((𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴 ∧ 𝒫 𝐴 ≈ (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
157, 13, 14syl2anc 573 . . 3 (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
16 onfin2 8308 . . . . . . 7 ω = (On ∩ Fin)
17 inss2 3982 . . . . . . 7 (On ∩ Fin) ⊆ Fin
1816, 17eqsstri 3784 . . . . . 6 ω ⊆ Fin
19 ficardom 8987 . . . . . . 7 (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ∈ ω)
2010, 19syl 17 . . . . . 6 (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ ω)
2118, 20sseldi 3750 . . . . 5 (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ Fin)
22 php3 8302 . . . . . 6 (((card‘𝒫 𝐴) ∈ Fin ∧ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴))
2322ex 397 . . . . 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 8138 . . . 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 133 . 2 (𝐴 ∈ ω → ¬ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴))
28 fvex 6342 . . . . . 6 (𝐹𝑎) ∈ V
29 ackbij1lem3 9246 . . . . . . . . 9 (𝐴 ∈ ω → 𝐴 ∈ (𝒫 ω ∩ Fin))
30 elpwi 4307 . . . . . . . . 9 (𝑎 ∈ 𝒫 𝐴𝑎𝐴)
313ackbij1lem12 9255 . . . . . . . . 9 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎𝐴) → (𝐹𝑎) ⊆ (𝐹𝐴))
3229, 30, 31syl2an 583 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → (𝐹𝑎) ⊆ (𝐹𝐴))
333ackbij1lem10 9253 . . . . . . . . . . 11 𝐹:(𝒫 ω ∩ Fin)⟶ω
34 peano1 7232 . . . . . . . . . . 11 ∅ ∈ ω
3533, 34f0cli 6513 . . . . . . . . . 10 (𝐹𝑎) ∈ ω
36 nnord 7220 . . . . . . . . . 10 ((𝐹𝑎) ∈ ω → Ord (𝐹𝑎))
3735, 36ax-mp 5 . . . . . . . . 9 Ord (𝐹𝑎)
3833, 34f0cli 6513 . . . . . . . . . 10 (𝐹𝐴) ∈ ω
39 nnord 7220 . . . . . . . . . 10 ((𝐹𝐴) ∈ ω → Ord (𝐹𝐴))
4038, 39ax-mp 5 . . . . . . . . 9 Ord (𝐹𝐴)
41 ordsucsssuc 7170 . . . . . . . . 9 ((Ord (𝐹𝑎) ∧ Ord (𝐹𝐴)) → ((𝐹𝑎) ⊆ (𝐹𝐴) ↔ suc (𝐹𝑎) ⊆ suc (𝐹𝐴)))
4237, 40, 41mp2an 672 . . . . . . . 8 ((𝐹𝑎) ⊆ (𝐹𝐴) ↔ suc (𝐹𝑎) ⊆ suc (𝐹𝐴))
4332, 42sylib 208 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹𝑎) ⊆ suc (𝐹𝐴))
443ackbij1lem14 9257 . . . . . . . . 9 (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹𝐴))
453ackbij1lem8 9251 . . . . . . . . 9 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
4644, 45eqtr3d 2807 . . . . . . . 8 (𝐴 ∈ ω → suc (𝐹𝐴) = (card‘𝒫 𝐴))
4746adantr 466 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹𝐴) = (card‘𝒫 𝐴))
4843, 47sseqtrd 3790 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹𝑎) ⊆ (card‘𝒫 𝐴))
49 sucssel 5962 . . . . . 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 3115 . . . 4 (𝐴 ∈ ω → ∀𝑎 ∈ 𝒫 𝐴(𝐹𝑎) ∈ (card‘𝒫 𝐴))
52 f1fun 6243 . . . . . 6 (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → Fun 𝐹)
534, 52ax-mp 5 . . . . 5 Fun 𝐹
54 f1dm 6245 . . . . . . 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 3801 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ⊆ dom 𝐹)
57 funimass4 6389 . . . . 5 ((Fun 𝐹 ∧ 𝒫 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ∀𝑎 ∈ 𝒫 𝐴(𝐹𝑎) ∈ (card‘𝒫 𝐴)))
5853, 56, 57sylancr 575 . . . 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 3856 . . 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 873 . 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  wa 382  wo 834   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cin 3722  wss 3723  wpss 3724  𝒫 cpw 4297  {csn 4316   ciun 4654   class class class wbr 4786  cmpt 4863   × cxp 5247  dom cdm 5249  cima 5252  Ord word 5865  Oncon0 5866  suc csuc 5868  Fun wfun 6025  1-1wf1 6028  cfv 6031  ωcom 7212  cen 8106  csdm 8108  Fincfn 8109  cardccrd 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-cda 9192
This theorem is referenced by:  ackbij2lem2  9264
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