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Theorem ackbij1lem14 9643
Description: Lemma for ackbij1 9648. (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
ackbij1lem14 (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹𝐴))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦

Proof of Theorem ackbij1lem14
Dummy variables 𝑎 𝑏 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)))
21ackbij1lem8 9637 . 2 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
3 pweq 4538 . . . . 5 (𝑎 = ∅ → 𝒫 𝑎 = 𝒫 ∅)
43fveq2d 6667 . . . 4 (𝑎 = ∅ → (card‘𝒫 𝑎) = (card‘𝒫 ∅))
5 fveq2 6663 . . . . 5 (𝑎 = ∅ → (𝐹𝑎) = (𝐹‘∅))
6 suceq 6249 . . . . 5 ((𝐹𝑎) = (𝐹‘∅) → suc (𝐹𝑎) = suc (𝐹‘∅))
75, 6syl 17 . . . 4 (𝑎 = ∅ → suc (𝐹𝑎) = suc (𝐹‘∅))
84, 7eqeq12d 2834 . . 3 (𝑎 = ∅ → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 ∅) = suc (𝐹‘∅)))
9 pweq 4538 . . . . 5 (𝑎 = 𝑏 → 𝒫 𝑎 = 𝒫 𝑏)
109fveq2d 6667 . . . 4 (𝑎 = 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 𝑏))
11 fveq2 6663 . . . . 5 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
12 suceq 6249 . . . . 5 ((𝐹𝑎) = (𝐹𝑏) → suc (𝐹𝑎) = suc (𝐹𝑏))
1311, 12syl 17 . . . 4 (𝑎 = 𝑏 → suc (𝐹𝑎) = suc (𝐹𝑏))
1410, 13eqeq12d 2834 . . 3 (𝑎 = 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 𝑏) = suc (𝐹𝑏)))
15 pweq 4538 . . . . 5 (𝑎 = suc 𝑏 → 𝒫 𝑎 = 𝒫 suc 𝑏)
1615fveq2d 6667 . . . 4 (𝑎 = suc 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 suc 𝑏))
17 fveq2 6663 . . . . 5 (𝑎 = suc 𝑏 → (𝐹𝑎) = (𝐹‘suc 𝑏))
18 suceq 6249 . . . . 5 ((𝐹𝑎) = (𝐹‘suc 𝑏) → suc (𝐹𝑎) = suc (𝐹‘suc 𝑏))
1917, 18syl 17 . . . 4 (𝑎 = suc 𝑏 → suc (𝐹𝑎) = suc (𝐹‘suc 𝑏))
2016, 19eqeq12d 2834 . . 3 (𝑎 = suc 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
21 pweq 4538 . . . . 5 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
2221fveq2d 6667 . . . 4 (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
23 fveq2 6663 . . . . 5 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
24 suceq 6249 . . . . 5 ((𝐹𝑎) = (𝐹𝐴) → suc (𝐹𝑎) = suc (𝐹𝐴))
2523, 24syl 17 . . . 4 (𝑎 = 𝐴 → suc (𝐹𝑎) = suc (𝐹𝐴))
2622, 25eqeq12d 2834 . . 3 (𝑎 = 𝐴 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 𝐴) = suc (𝐹𝐴)))
27 df-1o 8091 . . . 4 1o = suc ∅
28 pw0 4737 . . . . . 6 𝒫 ∅ = {∅}
2928fveq2i 6666 . . . . 5 (card‘𝒫 ∅) = (card‘{∅})
30 0ex 5202 . . . . . 6 ∅ ∈ V
31 cardsn 9386 . . . . . 6 (∅ ∈ V → (card‘{∅}) = 1o)
3230, 31ax-mp 5 . . . . 5 (card‘{∅}) = 1o
3329, 32eqtri 2841 . . . 4 (card‘𝒫 ∅) = 1o
341ackbij1lem13 9642 . . . . 5 (𝐹‘∅) = ∅
35 suceq 6249 . . . . 5 ((𝐹‘∅) = ∅ → suc (𝐹‘∅) = suc ∅)
3634, 35ax-mp 5 . . . 4 suc (𝐹‘∅) = suc ∅
3727, 33, 363eqtr4i 2851 . . 3 (card‘𝒫 ∅) = suc (𝐹‘∅)
38 oveq2 7153 . . . . . 6 ((card‘𝒫 𝑏) = suc (𝐹𝑏) → ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)) = ((card‘𝒫 𝑏) +o suc (𝐹𝑏)))
3938adantl 482 . . . . 5 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)) = ((card‘𝒫 𝑏) +o suc (𝐹𝑏)))
40 ackbij1lem5 9634 . . . . . 6 (𝑏 ∈ ω → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
4140adantr 481 . . . . 5 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
42 df-suc 6190 . . . . . . . . . 10 suc 𝑏 = (𝑏 ∪ {𝑏})
4342equncomi 4128 . . . . . . . . 9 suc 𝑏 = ({𝑏} ∪ 𝑏)
4443fveq2i 6666 . . . . . . . 8 (𝐹‘suc 𝑏) = (𝐹‘({𝑏} ∪ 𝑏))
45 ackbij1lem4 9633 . . . . . . . . . . 11 (𝑏 ∈ ω → {𝑏} ∈ (𝒫 ω ∩ Fin))
4645adantr 481 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → {𝑏} ∈ (𝒫 ω ∩ Fin))
47 ackbij1lem3 9632 . . . . . . . . . . 11 (𝑏 ∈ ω → 𝑏 ∈ (𝒫 ω ∩ Fin))
4847adantr 481 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → 𝑏 ∈ (𝒫 ω ∩ Fin))
49 incom 4175 . . . . . . . . . . . 12 ({𝑏} ∩ 𝑏) = (𝑏 ∩ {𝑏})
50 nnord 7577 . . . . . . . . . . . . 13 (𝑏 ∈ ω → Ord 𝑏)
51 orddisj 6222 . . . . . . . . . . . . 13 (Ord 𝑏 → (𝑏 ∩ {𝑏}) = ∅)
5250, 51syl 17 . . . . . . . . . . . 12 (𝑏 ∈ ω → (𝑏 ∩ {𝑏}) = ∅)
5349, 52syl5eq 2865 . . . . . . . . . . 11 (𝑏 ∈ ω → ({𝑏} ∩ 𝑏) = ∅)
5453adantr 481 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ({𝑏} ∩ 𝑏) = ∅)
551ackbij1lem9 9638 . . . . . . . . . 10 (({𝑏} ∈ (𝒫 ω ∩ Fin) ∧ 𝑏 ∈ (𝒫 ω ∩ Fin) ∧ ({𝑏} ∩ 𝑏) = ∅) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +o (𝐹𝑏)))
5646, 48, 54, 55syl3anc 1363 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +o (𝐹𝑏)))
571ackbij1lem8 9637 . . . . . . . . . . 11 (𝑏 ∈ ω → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
5857adantr 481 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
5958oveq1d 7160 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ((𝐹‘{𝑏}) +o (𝐹𝑏)) = ((card‘𝒫 𝑏) +o (𝐹𝑏)))
6056, 59eqtrd 2853 . . . . . . . 8 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((card‘𝒫 𝑏) +o (𝐹𝑏)))
6144, 60syl5eq 2865 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o (𝐹𝑏)))
62 suceq 6249 . . . . . . 7 ((𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o (𝐹𝑏)) → suc (𝐹‘suc 𝑏) = suc ((card‘𝒫 𝑏) +o (𝐹𝑏)))
6361, 62syl 17 . . . . . 6 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → suc (𝐹‘suc 𝑏) = suc ((card‘𝒫 𝑏) +o (𝐹𝑏)))
64 nnfi 8699 . . . . . . . . . 10 (𝑏 ∈ ω → 𝑏 ∈ Fin)
65 pwfi 8807 . . . . . . . . . 10 (𝑏 ∈ Fin ↔ 𝒫 𝑏 ∈ Fin)
6664, 65sylib 219 . . . . . . . . 9 (𝑏 ∈ ω → 𝒫 𝑏 ∈ Fin)
6766adantr 481 . . . . . . . 8 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → 𝒫 𝑏 ∈ Fin)
68 ficardom 9378 . . . . . . . 8 (𝒫 𝑏 ∈ Fin → (card‘𝒫 𝑏) ∈ ω)
6967, 68syl 17 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 𝑏) ∈ ω)
701ackbij1lem10 9639 . . . . . . . . 9 𝐹:(𝒫 ω ∩ Fin)⟶ω
7170ffvelrni 6842 . . . . . . . 8 (𝑏 ∈ (𝒫 ω ∩ Fin) → (𝐹𝑏) ∈ ω)
7248, 71syl 17 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹𝑏) ∈ ω)
73 nnasuc 8221 . . . . . . 7 (((card‘𝒫 𝑏) ∈ ω ∧ (𝐹𝑏) ∈ ω) → ((card‘𝒫 𝑏) +o suc (𝐹𝑏)) = suc ((card‘𝒫 𝑏) +o (𝐹𝑏)))
7469, 72, 73syl2anc 584 . . . . . 6 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ((card‘𝒫 𝑏) +o suc (𝐹𝑏)) = suc ((card‘𝒫 𝑏) +o (𝐹𝑏)))
7563, 74eqtr4d 2856 . . . . 5 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → suc (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o suc (𝐹𝑏)))
7639, 41, 753eqtr4d 2863 . . . 4 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏))
7776ex 413 . . 3 (𝑏 ∈ ω → ((card‘𝒫 𝑏) = suc (𝐹𝑏) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
788, 14, 20, 26, 37, 77finds 7597 . 2 (𝐴 ∈ ω → (card‘𝒫 𝐴) = suc (𝐹𝐴))
792, 78eqtrd 2853 1 (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cun 3931  cin 3932  c0 4288  𝒫 cpw 4535  {csn 4557   ciun 4910  cmpt 5137   × cxp 5546  Ord word 6183  suc csuc 6186  cfv 6348  (class class class)co 7145  ωcom 7569  1oc1o 8084   +o coa 8088  Fincfn 8497  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  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 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356
This theorem is referenced by:  ackbij1lem15  9644  ackbij1lem18  9647  ackbij1b  9649
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