MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ackbij1lem14 Structured version   Visualization version   GIF version

Theorem ackbij1lem14 10273
Description: Lemma for ackbij1 10278. (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 10267 . 2 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
3 pweq 4613 . . . . 5 (𝑎 = ∅ → 𝒫 𝑎 = 𝒫 ∅)
43fveq2d 6909 . . . 4 (𝑎 = ∅ → (card‘𝒫 𝑎) = (card‘𝒫 ∅))
5 fveq2 6905 . . . . 5 (𝑎 = ∅ → (𝐹𝑎) = (𝐹‘∅))
6 suceq 6449 . . . . 5 ((𝐹𝑎) = (𝐹‘∅) → suc (𝐹𝑎) = suc (𝐹‘∅))
75, 6syl 17 . . . 4 (𝑎 = ∅ → suc (𝐹𝑎) = suc (𝐹‘∅))
84, 7eqeq12d 2752 . . 3 (𝑎 = ∅ → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 ∅) = suc (𝐹‘∅)))
9 pweq 4613 . . . . 5 (𝑎 = 𝑏 → 𝒫 𝑎 = 𝒫 𝑏)
109fveq2d 6909 . . . 4 (𝑎 = 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 𝑏))
11 fveq2 6905 . . . . 5 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
12 suceq 6449 . . . . 5 ((𝐹𝑎) = (𝐹𝑏) → suc (𝐹𝑎) = suc (𝐹𝑏))
1311, 12syl 17 . . . 4 (𝑎 = 𝑏 → suc (𝐹𝑎) = suc (𝐹𝑏))
1410, 13eqeq12d 2752 . . 3 (𝑎 = 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 𝑏) = suc (𝐹𝑏)))
15 pweq 4613 . . . . 5 (𝑎 = suc 𝑏 → 𝒫 𝑎 = 𝒫 suc 𝑏)
1615fveq2d 6909 . . . 4 (𝑎 = suc 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 suc 𝑏))
17 fveq2 6905 . . . . 5 (𝑎 = suc 𝑏 → (𝐹𝑎) = (𝐹‘suc 𝑏))
18 suceq 6449 . . . . 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 2752 . . 3 (𝑎 = suc 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
21 pweq 4613 . . . . 5 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
2221fveq2d 6909 . . . 4 (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
23 fveq2 6905 . . . . 5 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
24 suceq 6449 . . . . 5 ((𝐹𝑎) = (𝐹𝐴) → suc (𝐹𝑎) = suc (𝐹𝐴))
2523, 24syl 17 . . . 4 (𝑎 = 𝐴 → suc (𝐹𝑎) = suc (𝐹𝐴))
2622, 25eqeq12d 2752 . . 3 (𝑎 = 𝐴 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 𝐴) = suc (𝐹𝐴)))
27 df-1o 8507 . . . 4 1o = suc ∅
28 pw0 4811 . . . . . 6 𝒫 ∅ = {∅}
2928fveq2i 6908 . . . . 5 (card‘𝒫 ∅) = (card‘{∅})
30 0ex 5306 . . . . . 6 ∅ ∈ V
31 cardsn 10010 . . . . . 6 (∅ ∈ V → (card‘{∅}) = 1o)
3230, 31ax-mp 5 . . . . 5 (card‘{∅}) = 1o
3329, 32eqtri 2764 . . . 4 (card‘𝒫 ∅) = 1o
341ackbij1lem13 10272 . . . . 5 (𝐹‘∅) = ∅
35 suceq 6449 . . . . 5 ((𝐹‘∅) = ∅ → suc (𝐹‘∅) = suc ∅)
3634, 35ax-mp 5 . . . 4 suc (𝐹‘∅) = suc ∅
3727, 33, 363eqtr4i 2774 . . 3 (card‘𝒫 ∅) = suc (𝐹‘∅)
38 oveq2 7440 . . . . . 6 ((card‘𝒫 𝑏) = suc (𝐹𝑏) → ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)) = ((card‘𝒫 𝑏) +o suc (𝐹𝑏)))
3938adantl 481 . . . . 5 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)) = ((card‘𝒫 𝑏) +o suc (𝐹𝑏)))
40 ackbij1lem5 10264 . . . . . 6 (𝑏 ∈ ω → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
4140adantr 480 . . . . 5 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
42 df-suc 6389 . . . . . . . . . 10 suc 𝑏 = (𝑏 ∪ {𝑏})
4342equncomi 4159 . . . . . . . . 9 suc 𝑏 = ({𝑏} ∪ 𝑏)
4443fveq2i 6908 . . . . . . . 8 (𝐹‘suc 𝑏) = (𝐹‘({𝑏} ∪ 𝑏))
45 ackbij1lem4 10263 . . . . . . . . . . 11 (𝑏 ∈ ω → {𝑏} ∈ (𝒫 ω ∩ Fin))
4645adantr 480 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → {𝑏} ∈ (𝒫 ω ∩ Fin))
47 ackbij1lem3 10262 . . . . . . . . . . 11 (𝑏 ∈ ω → 𝑏 ∈ (𝒫 ω ∩ Fin))
4847adantr 480 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → 𝑏 ∈ (𝒫 ω ∩ Fin))
49 incom 4208 . . . . . . . . . . . 12 ({𝑏} ∩ 𝑏) = (𝑏 ∩ {𝑏})
50 nnord 7896 . . . . . . . . . . . . 13 (𝑏 ∈ ω → Ord 𝑏)
51 orddisj 6421 . . . . . . . . . . . . 13 (Ord 𝑏 → (𝑏 ∩ {𝑏}) = ∅)
5250, 51syl 17 . . . . . . . . . . . 12 (𝑏 ∈ ω → (𝑏 ∩ {𝑏}) = ∅)
5349, 52eqtrid 2788 . . . . . . . . . . 11 (𝑏 ∈ ω → ({𝑏} ∩ 𝑏) = ∅)
5453adantr 480 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ({𝑏} ∩ 𝑏) = ∅)
551ackbij1lem9 10268 . . . . . . . . . 10 (({𝑏} ∈ (𝒫 ω ∩ Fin) ∧ 𝑏 ∈ (𝒫 ω ∩ Fin) ∧ ({𝑏} ∩ 𝑏) = ∅) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +o (𝐹𝑏)))
5646, 48, 54, 55syl3anc 1372 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +o (𝐹𝑏)))
571ackbij1lem8 10267 . . . . . . . . . . 11 (𝑏 ∈ ω → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
5857adantr 480 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
5958oveq1d 7447 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ((𝐹‘{𝑏}) +o (𝐹𝑏)) = ((card‘𝒫 𝑏) +o (𝐹𝑏)))
6056, 59eqtrd 2776 . . . . . . . 8 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((card‘𝒫 𝑏) +o (𝐹𝑏)))
6144, 60eqtrid 2788 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o (𝐹𝑏)))
62 suceq 6449 . . . . . . 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 9208 . . . . . . . . . 10 (𝑏 ∈ ω → 𝑏 ∈ Fin)
65 pwfi 9358 . . . . . . . . . 10 (𝑏 ∈ Fin ↔ 𝒫 𝑏 ∈ Fin)
6664, 65sylib 218 . . . . . . . . 9 (𝑏 ∈ ω → 𝒫 𝑏 ∈ Fin)
6766adantr 480 . . . . . . . 8 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → 𝒫 𝑏 ∈ Fin)
68 ficardom 10002 . . . . . . . 8 (𝒫 𝑏 ∈ Fin → (card‘𝒫 𝑏) ∈ ω)
6967, 68syl 17 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 𝑏) ∈ ω)
701ackbij1lem10 10269 . . . . . . . . 9 𝐹:(𝒫 ω ∩ Fin)⟶ω
7170ffvelcdmi 7102 . . . . . . . 8 (𝑏 ∈ (𝒫 ω ∩ Fin) → (𝐹𝑏) ∈ ω)
7248, 71syl 17 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹𝑏) ∈ ω)
73 nnasuc 8645 . . . . . . 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 2779 . . . . 5 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → suc (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o suc (𝐹𝑏)))
7639, 41, 753eqtr4d 2786 . . . 4 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏))
7776ex 412 . . 3 (𝑏 ∈ ω → ((card‘𝒫 𝑏) = suc (𝐹𝑏) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
788, 14, 20, 26, 37, 77finds 7919 . 2 (𝐴 ∈ ω → (card‘𝒫 𝐴) = suc (𝐹𝐴))
792, 78eqtrd 2776 1 (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  cun 3948  cin 3949  c0 4332  𝒫 cpw 4599  {csn 4625   ciun 4990  cmpt 5224   × cxp 5682  Ord word 6382  suc csuc 6385  cfv 6560  (class class class)co 7432  ωcom 7888  1oc1o 8500   +o coa 8504  Fincfn 8986  cardccrd 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-dju 9942  df-card 9980
This theorem is referenced by:  ackbij1lem15  10274  ackbij1lem18  10277  ackbij1b  10279
  Copyright terms: Public domain W3C validator