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Theorem ackbij1lem14 9002
Description: Lemma for ackbij1 9007. (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 8996 . 2 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
3 pweq 4135 . . . . 5 (𝑎 = ∅ → 𝒫 𝑎 = 𝒫 ∅)
43fveq2d 6154 . . . 4 (𝑎 = ∅ → (card‘𝒫 𝑎) = (card‘𝒫 ∅))
5 fveq2 6150 . . . . 5 (𝑎 = ∅ → (𝐹𝑎) = (𝐹‘∅))
6 suceq 5751 . . . . 5 ((𝐹𝑎) = (𝐹‘∅) → suc (𝐹𝑎) = suc (𝐹‘∅))
75, 6syl 17 . . . 4 (𝑎 = ∅ → suc (𝐹𝑎) = suc (𝐹‘∅))
84, 7eqeq12d 2636 . . 3 (𝑎 = ∅ → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 ∅) = suc (𝐹‘∅)))
9 pweq 4135 . . . . 5 (𝑎 = 𝑏 → 𝒫 𝑎 = 𝒫 𝑏)
109fveq2d 6154 . . . 4 (𝑎 = 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 𝑏))
11 fveq2 6150 . . . . 5 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
12 suceq 5751 . . . . 5 ((𝐹𝑎) = (𝐹𝑏) → suc (𝐹𝑎) = suc (𝐹𝑏))
1311, 12syl 17 . . . 4 (𝑎 = 𝑏 → suc (𝐹𝑎) = suc (𝐹𝑏))
1410, 13eqeq12d 2636 . . 3 (𝑎 = 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 𝑏) = suc (𝐹𝑏)))
15 pweq 4135 . . . . 5 (𝑎 = suc 𝑏 → 𝒫 𝑎 = 𝒫 suc 𝑏)
1615fveq2d 6154 . . . 4 (𝑎 = suc 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 suc 𝑏))
17 fveq2 6150 . . . . 5 (𝑎 = suc 𝑏 → (𝐹𝑎) = (𝐹‘suc 𝑏))
18 suceq 5751 . . . . 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 2636 . . 3 (𝑎 = suc 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
21 pweq 4135 . . . . 5 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
2221fveq2d 6154 . . . 4 (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
23 fveq2 6150 . . . . 5 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
24 suceq 5751 . . . . 5 ((𝐹𝑎) = (𝐹𝐴) → suc (𝐹𝑎) = suc (𝐹𝐴))
2523, 24syl 17 . . . 4 (𝑎 = 𝐴 → suc (𝐹𝑎) = suc (𝐹𝐴))
2622, 25eqeq12d 2636 . . 3 (𝑎 = 𝐴 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 𝐴) = suc (𝐹𝐴)))
27 df-1o 7508 . . . 4 1𝑜 = suc ∅
28 pw0 4313 . . . . . 6 𝒫 ∅ = {∅}
2928fveq2i 6153 . . . . 5 (card‘𝒫 ∅) = (card‘{∅})
30 0ex 4752 . . . . . 6 ∅ ∈ V
31 cardsn 8742 . . . . . 6 (∅ ∈ V → (card‘{∅}) = 1𝑜)
3230, 31ax-mp 5 . . . . 5 (card‘{∅}) = 1𝑜
3329, 32eqtri 2643 . . . 4 (card‘𝒫 ∅) = 1𝑜
341ackbij1lem13 9001 . . . . 5 (𝐹‘∅) = ∅
35 suceq 5751 . . . . 5 ((𝐹‘∅) = ∅ → suc (𝐹‘∅) = suc ∅)
3634, 35ax-mp 5 . . . 4 suc (𝐹‘∅) = suc ∅
3727, 33, 363eqtr4i 2653 . . 3 (card‘𝒫 ∅) = suc (𝐹‘∅)
38 oveq2 6615 . . . . . 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 8993 . . . . . 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 5690 . . . . . . . . . 10 suc 𝑏 = (𝑏 ∪ {𝑏})
4342equncomi 3739 . . . . . . . . 9 suc 𝑏 = ({𝑏} ∪ 𝑏)
4443fveq2i 6153 . . . . . . . 8 (𝐹‘suc 𝑏) = (𝐹‘({𝑏} ∪ 𝑏))
45 ackbij1lem4 8992 . . . . . . . . . . 11 (𝑏 ∈ ω → {𝑏} ∈ (𝒫 ω ∩ Fin))
4645adantr 481 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → {𝑏} ∈ (𝒫 ω ∩ Fin))
47 ackbij1lem3 8991 . . . . . . . . . . 11 (𝑏 ∈ ω → 𝑏 ∈ (𝒫 ω ∩ Fin))
4847adantr 481 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → 𝑏 ∈ (𝒫 ω ∩ Fin))
49 incom 3785 . . . . . . . . . . . 12 ({𝑏} ∩ 𝑏) = (𝑏 ∩ {𝑏})
50 nnord 7023 . . . . . . . . . . . . 13 (𝑏 ∈ ω → Ord 𝑏)
51 orddisj 5723 . . . . . . . . . . . . 13 (Ord 𝑏 → (𝑏 ∩ {𝑏}) = ∅)
5250, 51syl 17 . . . . . . . . . . . 12 (𝑏 ∈ ω → (𝑏 ∩ {𝑏}) = ∅)
5349, 52syl5eq 2667 . . . . . . . . . . 11 (𝑏 ∈ ω → ({𝑏} ∩ 𝑏) = ∅)
5453adantr 481 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ({𝑏} ∩ 𝑏) = ∅)
551ackbij1lem9 8997 . . . . . . . . . 10 (({𝑏} ∈ (𝒫 ω ∩ Fin) ∧ 𝑏 ∈ (𝒫 ω ∩ Fin) ∧ ({𝑏} ∩ 𝑏) = ∅) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +𝑜 (𝐹𝑏)))
5646, 48, 54, 55syl3anc 1323 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +𝑜 (𝐹𝑏)))
571ackbij1lem8 8996 . . . . . . . . . . 11 (𝑏 ∈ ω → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
5857adantr 481 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
5958oveq1d 6622 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ((𝐹‘{𝑏}) +𝑜 (𝐹𝑏)) = ((card‘𝒫 𝑏) +𝑜 (𝐹𝑏)))
6056, 59eqtrd 2655 . . . . . . . 8 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((card‘𝒫 𝑏) +𝑜 (𝐹𝑏)))
6144, 60syl5eq 2667 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +𝑜 (𝐹𝑏)))
62 suceq 5751 . . . . . . 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 8100 . . . . . . . . . 10 (𝑏 ∈ ω → 𝑏 ∈ Fin)
65 pwfi 8208 . . . . . . . . . 10 (𝑏 ∈ Fin ↔ 𝒫 𝑏 ∈ Fin)
6664, 65sylib 208 . . . . . . . . 9 (𝑏 ∈ ω → 𝒫 𝑏 ∈ Fin)
6766adantr 481 . . . . . . . 8 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → 𝒫 𝑏 ∈ Fin)
68 ficardom 8734 . . . . . . . 8 (𝒫 𝑏 ∈ Fin → (card‘𝒫 𝑏) ∈ ω)
6967, 68syl 17 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 𝑏) ∈ ω)
701ackbij1lem10 8998 . . . . . . . . 9 𝐹:(𝒫 ω ∩ Fin)⟶ω
7170ffvelrni 6316 . . . . . . . 8 (𝑏 ∈ (𝒫 ω ∩ Fin) → (𝐹𝑏) ∈ ω)
7248, 71syl 17 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹𝑏) ∈ ω)
73 nnasuc 7634 . . . . . . 7 (((card‘𝒫 𝑏) ∈ ω ∧ (𝐹𝑏) ∈ ω) → ((card‘𝒫 𝑏) +𝑜 suc (𝐹𝑏)) = suc ((card‘𝒫 𝑏) +𝑜 (𝐹𝑏)))
7469, 72, 73syl2anc 692 . . . . . 6 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ((card‘𝒫 𝑏) +𝑜 suc (𝐹𝑏)) = suc ((card‘𝒫 𝑏) +𝑜 (𝐹𝑏)))
7563, 74eqtr4d 2658 . . . . 5 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → suc (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +𝑜 suc (𝐹𝑏)))
7639, 41, 753eqtr4d 2665 . . . 4 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏))
7776ex 450 . . 3 (𝑏 ∈ ω → ((card‘𝒫 𝑏) = suc (𝐹𝑏) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
788, 14, 20, 26, 37, 77finds 7042 . 2 (𝐴 ∈ ω → (card‘𝒫 𝐴) = suc (𝐹𝐴))
792, 78eqtrd 2655 1 (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cun 3554  cin 3555  c0 3893  𝒫 cpw 4132  {csn 4150   ciun 4487  cmpt 4675   × cxp 5074  Ord word 5683  suc csuc 5686  cfv 5849  (class class class)co 6607  ωcom 7015  1𝑜c1o 7501   +𝑜 coa 7505  Fincfn 7902  cardccrd 8708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-map 7807  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-card 8712  df-cda 8937
This theorem is referenced by:  ackbij1lem15  9003  ackbij1lem18  9006  ackbij1b  9008
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