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Theorem ackbij1lem14 10264
Description: Lemma for ackbij1 10269. (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 10258 . 2 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
3 pweq 4620 . . . . 5 (𝑎 = ∅ → 𝒫 𝑎 = 𝒫 ∅)
43fveq2d 6906 . . . 4 (𝑎 = ∅ → (card‘𝒫 𝑎) = (card‘𝒫 ∅))
5 fveq2 6902 . . . . 5 (𝑎 = ∅ → (𝐹𝑎) = (𝐹‘∅))
6 suceq 6440 . . . . 5 ((𝐹𝑎) = (𝐹‘∅) → suc (𝐹𝑎) = suc (𝐹‘∅))
75, 6syl 17 . . . 4 (𝑎 = ∅ → suc (𝐹𝑎) = suc (𝐹‘∅))
84, 7eqeq12d 2744 . . 3 (𝑎 = ∅ → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 ∅) = suc (𝐹‘∅)))
9 pweq 4620 . . . . 5 (𝑎 = 𝑏 → 𝒫 𝑎 = 𝒫 𝑏)
109fveq2d 6906 . . . 4 (𝑎 = 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 𝑏))
11 fveq2 6902 . . . . 5 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
12 suceq 6440 . . . . 5 ((𝐹𝑎) = (𝐹𝑏) → suc (𝐹𝑎) = suc (𝐹𝑏))
1311, 12syl 17 . . . 4 (𝑎 = 𝑏 → suc (𝐹𝑎) = suc (𝐹𝑏))
1410, 13eqeq12d 2744 . . 3 (𝑎 = 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 𝑏) = suc (𝐹𝑏)))
15 pweq 4620 . . . . 5 (𝑎 = suc 𝑏 → 𝒫 𝑎 = 𝒫 suc 𝑏)
1615fveq2d 6906 . . . 4 (𝑎 = suc 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 suc 𝑏))
17 fveq2 6902 . . . . 5 (𝑎 = suc 𝑏 → (𝐹𝑎) = (𝐹‘suc 𝑏))
18 suceq 6440 . . . . 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 2744 . . 3 (𝑎 = suc 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
21 pweq 4620 . . . . 5 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
2221fveq2d 6906 . . . 4 (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
23 fveq2 6902 . . . . 5 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
24 suceq 6440 . . . . 5 ((𝐹𝑎) = (𝐹𝐴) → suc (𝐹𝑎) = suc (𝐹𝐴))
2523, 24syl 17 . . . 4 (𝑎 = 𝐴 → suc (𝐹𝑎) = suc (𝐹𝐴))
2622, 25eqeq12d 2744 . . 3 (𝑎 = 𝐴 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 𝐴) = suc (𝐹𝐴)))
27 df-1o 8493 . . . 4 1o = suc ∅
28 pw0 4820 . . . . . 6 𝒫 ∅ = {∅}
2928fveq2i 6905 . . . . 5 (card‘𝒫 ∅) = (card‘{∅})
30 0ex 5311 . . . . . 6 ∅ ∈ V
31 cardsn 10000 . . . . . 6 (∅ ∈ V → (card‘{∅}) = 1o)
3230, 31ax-mp 5 . . . . 5 (card‘{∅}) = 1o
3329, 32eqtri 2756 . . . 4 (card‘𝒫 ∅) = 1o
341ackbij1lem13 10263 . . . . 5 (𝐹‘∅) = ∅
35 suceq 6440 . . . . 5 ((𝐹‘∅) = ∅ → suc (𝐹‘∅) = suc ∅)
3634, 35ax-mp 5 . . . 4 suc (𝐹‘∅) = suc ∅
3727, 33, 363eqtr4i 2766 . . 3 (card‘𝒫 ∅) = suc (𝐹‘∅)
38 oveq2 7434 . . . . . 6 ((card‘𝒫 𝑏) = suc (𝐹𝑏) → ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)) = ((card‘𝒫 𝑏) +o suc (𝐹𝑏)))
3938adantl 480 . . . . 5 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)) = ((card‘𝒫 𝑏) +o suc (𝐹𝑏)))
40 ackbij1lem5 10255 . . . . . 6 (𝑏 ∈ ω → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
4140adantr 479 . . . . 5 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
42 df-suc 6380 . . . . . . . . . 10 suc 𝑏 = (𝑏 ∪ {𝑏})
4342equncomi 4156 . . . . . . . . 9 suc 𝑏 = ({𝑏} ∪ 𝑏)
4443fveq2i 6905 . . . . . . . 8 (𝐹‘suc 𝑏) = (𝐹‘({𝑏} ∪ 𝑏))
45 ackbij1lem4 10254 . . . . . . . . . . 11 (𝑏 ∈ ω → {𝑏} ∈ (𝒫 ω ∩ Fin))
4645adantr 479 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → {𝑏} ∈ (𝒫 ω ∩ Fin))
47 ackbij1lem3 10253 . . . . . . . . . . 11 (𝑏 ∈ ω → 𝑏 ∈ (𝒫 ω ∩ Fin))
4847adantr 479 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → 𝑏 ∈ (𝒫 ω ∩ Fin))
49 incom 4203 . . . . . . . . . . . 12 ({𝑏} ∩ 𝑏) = (𝑏 ∩ {𝑏})
50 nnord 7884 . . . . . . . . . . . . 13 (𝑏 ∈ ω → Ord 𝑏)
51 orddisj 6412 . . . . . . . . . . . . 13 (Ord 𝑏 → (𝑏 ∩ {𝑏}) = ∅)
5250, 51syl 17 . . . . . . . . . . . 12 (𝑏 ∈ ω → (𝑏 ∩ {𝑏}) = ∅)
5349, 52eqtrid 2780 . . . . . . . . . . 11 (𝑏 ∈ ω → ({𝑏} ∩ 𝑏) = ∅)
5453adantr 479 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ({𝑏} ∩ 𝑏) = ∅)
551ackbij1lem9 10259 . . . . . . . . . 10 (({𝑏} ∈ (𝒫 ω ∩ Fin) ∧ 𝑏 ∈ (𝒫 ω ∩ Fin) ∧ ({𝑏} ∩ 𝑏) = ∅) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +o (𝐹𝑏)))
5646, 48, 54, 55syl3anc 1368 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +o (𝐹𝑏)))
571ackbij1lem8 10258 . . . . . . . . . . 11 (𝑏 ∈ ω → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
5857adantr 479 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
5958oveq1d 7441 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ((𝐹‘{𝑏}) +o (𝐹𝑏)) = ((card‘𝒫 𝑏) +o (𝐹𝑏)))
6056, 59eqtrd 2768 . . . . . . . 8 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((card‘𝒫 𝑏) +o (𝐹𝑏)))
6144, 60eqtrid 2780 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o (𝐹𝑏)))
62 suceq 6440 . . . . . . 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 9198 . . . . . . . . . 10 (𝑏 ∈ ω → 𝑏 ∈ Fin)
65 pwfi 9209 . . . . . . . . . 10 (𝑏 ∈ Fin ↔ 𝒫 𝑏 ∈ Fin)
6664, 65sylib 217 . . . . . . . . 9 (𝑏 ∈ ω → 𝒫 𝑏 ∈ Fin)
6766adantr 479 . . . . . . . 8 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → 𝒫 𝑏 ∈ Fin)
68 ficardom 9992 . . . . . . . 8 (𝒫 𝑏 ∈ Fin → (card‘𝒫 𝑏) ∈ ω)
6967, 68syl 17 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 𝑏) ∈ ω)
701ackbij1lem10 10260 . . . . . . . . 9 𝐹:(𝒫 ω ∩ Fin)⟶ω
7170ffvelcdmi 7098 . . . . . . . 8 (𝑏 ∈ (𝒫 ω ∩ Fin) → (𝐹𝑏) ∈ ω)
7248, 71syl 17 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹𝑏) ∈ ω)
73 nnasuc 8633 . . . . . . 7 (((card‘𝒫 𝑏) ∈ ω ∧ (𝐹𝑏) ∈ ω) → ((card‘𝒫 𝑏) +o suc (𝐹𝑏)) = suc ((card‘𝒫 𝑏) +o (𝐹𝑏)))
7469, 72, 73syl2anc 582 . . . . . 6 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ((card‘𝒫 𝑏) +o suc (𝐹𝑏)) = suc ((card‘𝒫 𝑏) +o (𝐹𝑏)))
7563, 74eqtr4d 2771 . . . . 5 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → suc (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o suc (𝐹𝑏)))
7639, 41, 753eqtr4d 2778 . . . 4 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏))
7776ex 411 . . 3 (𝑏 ∈ ω → ((card‘𝒫 𝑏) = suc (𝐹𝑏) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
788, 14, 20, 26, 37, 77finds 7910 . 2 (𝐴 ∈ ω → (card‘𝒫 𝐴) = suc (𝐹𝐴))
792, 78eqtrd 2768 1 (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3473  cun 3947  cin 3948  c0 4326  𝒫 cpw 4606  {csn 4632   ciun 5000  cmpt 5235   × cxp 5680  Ord word 6373  suc csuc 6376  cfv 6553  (class class class)co 7426  ωcom 7876  1oc1o 8486   +o coa 8490  Fincfn 8970  cardccrd 9966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-oadd 8497  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-dju 9932  df-card 9970
This theorem is referenced by:  ackbij1lem15  10265  ackbij1lem18  10268  ackbij1b  10270
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