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Theorem ackbij1b 9653
Description: The Ackermann bijection, part 1b: the bijection from ackbij1 9652 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 9650 . . . . 5 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3 ackbij2lem1 9633 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin))
4 pwexg 5270 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ∈ V)
5 f1imaeng 8561 . . . . 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 1460 . . . 4 (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴)
7 nnfi 8703 . . . . . 6 (𝐴 ∈ ω → 𝐴 ∈ Fin)
8 pwfi 8811 . . . . . 6 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
97, 8sylib 220 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ∈ Fin)
10 ficardid 9383 . . . . 5 (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
11 ensym 8550 . . . . 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 8553 . . . 4 (((𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴 ∧ 𝒫 𝐴 ≈ (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
146, 12, 13syl2anc 586 . . 3 (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
15 onfin2 8702 . . . . . . 7 ω = (On ∩ Fin)
16 inss2 4204 . . . . . . 7 (On ∩ Fin) ⊆ Fin
1715, 16eqsstri 3999 . . . . . 6 ω ⊆ Fin
18 ficardom 9382 . . . . . . 7 (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ∈ ω)
199, 18syl 17 . . . . . 6 (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ ω)
2017, 19sseldi 3963 . . . . 5 (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ Fin)
21 php3 8695 . . . . . 6 (((card‘𝒫 𝐴) ∈ Fin ∧ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴))
2221ex 415 . . . . 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 8530 . . . 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 138 . 2 (𝐴 ∈ ω → ¬ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴))
27 fvex 6676 . . . . . 6 (𝐹𝑎) ∈ V
28 ackbij1lem3 9636 . . . . . . . . 9 (𝐴 ∈ ω → 𝐴 ∈ (𝒫 ω ∩ Fin))
29 elpwi 4549 . . . . . . . . 9 (𝑎 ∈ 𝒫 𝐴𝑎𝐴)
301ackbij1lem12 9645 . . . . . . . . 9 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎𝐴) → (𝐹𝑎) ⊆ (𝐹𝐴))
3128, 29, 30syl2an 597 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → (𝐹𝑎) ⊆ (𝐹𝐴))
321ackbij1lem10 9643 . . . . . . . . . . 11 𝐹:(𝒫 ω ∩ Fin)⟶ω
33 peano1 7593 . . . . . . . . . . 11 ∅ ∈ ω
3432, 33f0cli 6857 . . . . . . . . . 10 (𝐹𝑎) ∈ ω
35 nnord 7580 . . . . . . . . . 10 ((𝐹𝑎) ∈ ω → Ord (𝐹𝑎))
3634, 35ax-mp 5 . . . . . . . . 9 Ord (𝐹𝑎)
3732, 33f0cli 6857 . . . . . . . . . 10 (𝐹𝐴) ∈ ω
38 nnord 7580 . . . . . . . . . 10 ((𝐹𝐴) ∈ ω → Ord (𝐹𝐴))
3937, 38ax-mp 5 . . . . . . . . 9 Ord (𝐹𝐴)
40 ordsucsssuc 7530 . . . . . . . . 9 ((Ord (𝐹𝑎) ∧ Ord (𝐹𝐴)) → ((𝐹𝑎) ⊆ (𝐹𝐴) ↔ suc (𝐹𝑎) ⊆ suc (𝐹𝐴)))
4136, 39, 40mp2an 690 . . . . . . . 8 ((𝐹𝑎) ⊆ (𝐹𝐴) ↔ suc (𝐹𝑎) ⊆ suc (𝐹𝐴))
4231, 41sylib 220 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹𝑎) ⊆ suc (𝐹𝐴))
431ackbij1lem14 9647 . . . . . . . . 9 (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹𝐴))
441ackbij1lem8 9641 . . . . . . . . 9 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
4543, 44eqtr3d 2856 . . . . . . . 8 (𝐴 ∈ ω → suc (𝐹𝐴) = (card‘𝒫 𝐴))
4645adantr 483 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹𝐴) = (card‘𝒫 𝐴))
4742, 46sseqtrd 4005 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹𝑎) ⊆ (card‘𝒫 𝐴))
48 sucssel 6276 . . . . . 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 3180 . . . 4 (𝐴 ∈ ω → ∀𝑎 ∈ 𝒫 𝐴(𝐹𝑎) ∈ (card‘𝒫 𝐴))
51 f1fun 6570 . . . . . 6 (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → Fun 𝐹)
522, 51ax-mp 5 . . . . 5 Fun 𝐹
53 f1dm 6572 . . . . . . 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 4016 . . . . 5 (𝐴 ∈ ω → 𝒫 𝐴 ⊆ dom 𝐹)
56 funimass4 6723 . . . . 5 ((Fun 𝐹 ∧ 𝒫 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ∀𝑎 ∈ 𝒫 𝐴(𝐹𝑎) ∈ (card‘𝒫 𝐴)))
5752, 55, 56sylancr 589 . . . 4 (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ∀𝑎 ∈ 𝒫 𝐴(𝐹𝑎) ∈ (card‘𝒫 𝐴)))
5850, 57mpbird 259 . . 3 (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴))
59 sspss 4074 . . 3 ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) ∨ (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)))
6058, 59sylib 220 . 2 (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) ∨ (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)))
61 orel1 885 . 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 208  wa 398  wo 843   = wceq 1531  wcel 2108  wral 3136  Vcvv 3493  cin 3933  wss 3934  wpss 3935  𝒫 cpw 4537  {csn 4559   ciun 4910   class class class wbr 5057  cmpt 5137   × cxp 5546  dom cdm 5548  cima 5551  Ord word 6183  Oncon0 6184  suc csuc 6186  Fun wfun 6342  1-1wf1 6345  cfv 6348  ωcom 7572  cen 8498  csdm 8500  Fincfn 8501  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360
This theorem is referenced by:  ackbij2lem2  9654
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