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Theorem ackbij1lem14 9390
 Description: Lemma for ackbij1 9395. (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 9384 . 2 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
3 pweq 4382 . . . . 5 (𝑎 = ∅ → 𝒫 𝑎 = 𝒫 ∅)
43fveq2d 6450 . . . 4 (𝑎 = ∅ → (card‘𝒫 𝑎) = (card‘𝒫 ∅))
5 fveq2 6446 . . . . 5 (𝑎 = ∅ → (𝐹𝑎) = (𝐹‘∅))
6 suceq 6041 . . . . 5 ((𝐹𝑎) = (𝐹‘∅) → suc (𝐹𝑎) = suc (𝐹‘∅))
75, 6syl 17 . . . 4 (𝑎 = ∅ → suc (𝐹𝑎) = suc (𝐹‘∅))
84, 7eqeq12d 2793 . . 3 (𝑎 = ∅ → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 ∅) = suc (𝐹‘∅)))
9 pweq 4382 . . . . 5 (𝑎 = 𝑏 → 𝒫 𝑎 = 𝒫 𝑏)
109fveq2d 6450 . . . 4 (𝑎 = 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 𝑏))
11 fveq2 6446 . . . . 5 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
12 suceq 6041 . . . . 5 ((𝐹𝑎) = (𝐹𝑏) → suc (𝐹𝑎) = suc (𝐹𝑏))
1311, 12syl 17 . . . 4 (𝑎 = 𝑏 → suc (𝐹𝑎) = suc (𝐹𝑏))
1410, 13eqeq12d 2793 . . 3 (𝑎 = 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 𝑏) = suc (𝐹𝑏)))
15 pweq 4382 . . . . 5 (𝑎 = suc 𝑏 → 𝒫 𝑎 = 𝒫 suc 𝑏)
1615fveq2d 6450 . . . 4 (𝑎 = suc 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 suc 𝑏))
17 fveq2 6446 . . . . 5 (𝑎 = suc 𝑏 → (𝐹𝑎) = (𝐹‘suc 𝑏))
18 suceq 6041 . . . . 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 2793 . . 3 (𝑎 = suc 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
21 pweq 4382 . . . . 5 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
2221fveq2d 6450 . . . 4 (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
23 fveq2 6446 . . . . 5 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
24 suceq 6041 . . . . 5 ((𝐹𝑎) = (𝐹𝐴) → suc (𝐹𝑎) = suc (𝐹𝐴))
2523, 24syl 17 . . . 4 (𝑎 = 𝐴 → suc (𝐹𝑎) = suc (𝐹𝐴))
2622, 25eqeq12d 2793 . . 3 (𝑎 = 𝐴 → ((card‘𝒫 𝑎) = suc (𝐹𝑎) ↔ (card‘𝒫 𝐴) = suc (𝐹𝐴)))
27 df-1o 7843 . . . 4 1o = suc ∅
28 pw0 4574 . . . . . 6 𝒫 ∅ = {∅}
2928fveq2i 6449 . . . . 5 (card‘𝒫 ∅) = (card‘{∅})
30 0ex 5026 . . . . . 6 ∅ ∈ V
31 cardsn 9128 . . . . . 6 (∅ ∈ V → (card‘{∅}) = 1o)
3230, 31ax-mp 5 . . . . 5 (card‘{∅}) = 1o
3329, 32eqtri 2802 . . . 4 (card‘𝒫 ∅) = 1o
341ackbij1lem13 9389 . . . . 5 (𝐹‘∅) = ∅
35 suceq 6041 . . . . 5 ((𝐹‘∅) = ∅ → suc (𝐹‘∅) = suc ∅)
3634, 35ax-mp 5 . . . 4 suc (𝐹‘∅) = suc ∅
3727, 33, 363eqtr4i 2812 . . 3 (card‘𝒫 ∅) = suc (𝐹‘∅)
38 oveq2 6930 . . . . . 6 ((card‘𝒫 𝑏) = suc (𝐹𝑏) → ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)) = ((card‘𝒫 𝑏) +o suc (𝐹𝑏)))
3938adantl 475 . . . . 5 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)) = ((card‘𝒫 𝑏) +o suc (𝐹𝑏)))
40 ackbij1lem5 9381 . . . . . 6 (𝑏 ∈ ω → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
4140adantr 474 . . . . 5 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
42 df-suc 5982 . . . . . . . . . 10 suc 𝑏 = (𝑏 ∪ {𝑏})
4342equncomi 3982 . . . . . . . . 9 suc 𝑏 = ({𝑏} ∪ 𝑏)
4443fveq2i 6449 . . . . . . . 8 (𝐹‘suc 𝑏) = (𝐹‘({𝑏} ∪ 𝑏))
45 ackbij1lem4 9380 . . . . . . . . . . 11 (𝑏 ∈ ω → {𝑏} ∈ (𝒫 ω ∩ Fin))
4645adantr 474 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → {𝑏} ∈ (𝒫 ω ∩ Fin))
47 ackbij1lem3 9379 . . . . . . . . . . 11 (𝑏 ∈ ω → 𝑏 ∈ (𝒫 ω ∩ Fin))
4847adantr 474 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → 𝑏 ∈ (𝒫 ω ∩ Fin))
49 incom 4028 . . . . . . . . . . . 12 ({𝑏} ∩ 𝑏) = (𝑏 ∩ {𝑏})
50 nnord 7351 . . . . . . . . . . . . 13 (𝑏 ∈ ω → Ord 𝑏)
51 orddisj 6014 . . . . . . . . . . . . 13 (Ord 𝑏 → (𝑏 ∩ {𝑏}) = ∅)
5250, 51syl 17 . . . . . . . . . . . 12 (𝑏 ∈ ω → (𝑏 ∩ {𝑏}) = ∅)
5349, 52syl5eq 2826 . . . . . . . . . . 11 (𝑏 ∈ ω → ({𝑏} ∩ 𝑏) = ∅)
5453adantr 474 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ({𝑏} ∩ 𝑏) = ∅)
551ackbij1lem9 9385 . . . . . . . . . 10 (({𝑏} ∈ (𝒫 ω ∩ Fin) ∧ 𝑏 ∈ (𝒫 ω ∩ Fin) ∧ ({𝑏} ∩ 𝑏) = ∅) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +o (𝐹𝑏)))
5646, 48, 54, 55syl3anc 1439 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +o (𝐹𝑏)))
571ackbij1lem8 9384 . . . . . . . . . . 11 (𝑏 ∈ ω → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
5857adantr 474 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
5958oveq1d 6937 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ((𝐹‘{𝑏}) +o (𝐹𝑏)) = ((card‘𝒫 𝑏) +o (𝐹𝑏)))
6056, 59eqtrd 2814 . . . . . . . 8 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((card‘𝒫 𝑏) +o (𝐹𝑏)))
6144, 60syl5eq 2826 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o (𝐹𝑏)))
62 suceq 6041 . . . . . . 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 8441 . . . . . . . . . 10 (𝑏 ∈ ω → 𝑏 ∈ Fin)
65 pwfi 8549 . . . . . . . . . 10 (𝑏 ∈ Fin ↔ 𝒫 𝑏 ∈ Fin)
6664, 65sylib 210 . . . . . . . . 9 (𝑏 ∈ ω → 𝒫 𝑏 ∈ Fin)
6766adantr 474 . . . . . . . 8 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → 𝒫 𝑏 ∈ Fin)
68 ficardom 9120 . . . . . . . 8 (𝒫 𝑏 ∈ Fin → (card‘𝒫 𝑏) ∈ ω)
6967, 68syl 17 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 𝑏) ∈ ω)
701ackbij1lem10 9386 . . . . . . . . 9 𝐹:(𝒫 ω ∩ Fin)⟶ω
7170ffvelrni 6622 . . . . . . . 8 (𝑏 ∈ (𝒫 ω ∩ Fin) → (𝐹𝑏) ∈ ω)
7248, 71syl 17 . . . . . . 7 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (𝐹𝑏) ∈ ω)
73 nnasuc 7970 . . . . . . 7 (((card‘𝒫 𝑏) ∈ ω ∧ (𝐹𝑏) ∈ ω) → ((card‘𝒫 𝑏) +o suc (𝐹𝑏)) = suc ((card‘𝒫 𝑏) +o (𝐹𝑏)))
7469, 72, 73syl2anc 579 . . . . . 6 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → ((card‘𝒫 𝑏) +o suc (𝐹𝑏)) = suc ((card‘𝒫 𝑏) +o (𝐹𝑏)))
7563, 74eqtr4d 2817 . . . . 5 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → suc (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o suc (𝐹𝑏)))
7639, 41, 753eqtr4d 2824 . . . 4 ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹𝑏)) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏))
7776ex 403 . . 3 (𝑏 ∈ ω → ((card‘𝒫 𝑏) = suc (𝐹𝑏) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
788, 14, 20, 26, 37, 77finds 7370 . 2 (𝐴 ∈ ω → (card‘𝒫 𝐴) = suc (𝐹𝐴))
792, 78eqtrd 2814 1 (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ∪ cun 3790   ∩ cin 3791  ∅c0 4141  𝒫 cpw 4379  {csn 4398  ∪ ciun 4753   ↦ cmpt 4965   × cxp 5353  Ord word 5975  suc csuc 5978  ‘cfv 6135  (class class class)co 6922  ωcom 7343  1oc1o 7836   +o coa 7840  Fincfn 8241  cardccrd 9094 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-cda 9325 This theorem is referenced by:  ackbij1lem15  9391  ackbij1lem18  9394  ackbij1b  9396
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