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‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))

Assertion

Ref	Expression
ackbij1lem14	⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹‘𝐴))

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.

Step	Hyp	Ref	Expression
1		ackbij.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
2	1	ackbij1lem8 10267	. 2 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
3		pweq 4613	. . . . 5 ⊢ (𝑎 = ∅ → 𝒫 𝑎 = 𝒫 ∅)
4	3	fveq2d 6909	. . . 4 ⊢ (𝑎 = ∅ → (card‘𝒫 𝑎) = (card‘𝒫 ∅))
5		fveq2 6905	. . . . 5 ⊢ (𝑎 = ∅ → (𝐹‘𝑎) = (𝐹‘∅))
6		suceq 6449	. . . . 5 ⊢ ((𝐹‘𝑎) = (𝐹‘∅) → suc (𝐹‘𝑎) = suc (𝐹‘∅))
7	5, 6	syl 17	. . . 4 ⊢ (𝑎 = ∅ → suc (𝐹‘𝑎) = suc (𝐹‘∅))
8	4, 7	eqeq12d 2752	. . 3 ⊢ (𝑎 = ∅ → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 ∅) = suc (𝐹‘∅)))
9		pweq 4613	. . . . 5 ⊢ (𝑎 = 𝑏 → 𝒫 𝑎 = 𝒫 𝑏)
10	9	fveq2d 6909	. . . 4 ⊢ (𝑎 = 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 𝑏))
11		fveq2 6905	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
12		suceq 6449	. . . . 5 ⊢ ((𝐹‘𝑎) = (𝐹‘𝑏) → suc (𝐹‘𝑎) = suc (𝐹‘𝑏))
13	11, 12	syl 17	. . . 4 ⊢ (𝑎 = 𝑏 → suc (𝐹‘𝑎) = suc (𝐹‘𝑏))
14	10, 13	eqeq12d 2752	. . 3 ⊢ (𝑎 = 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)))
15		pweq 4613	. . . . 5 ⊢ (𝑎 = suc 𝑏 → 𝒫 𝑎 = 𝒫 suc 𝑏)
16	15	fveq2d 6909	. . . 4 ⊢ (𝑎 = suc 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 suc 𝑏))
17		fveq2 6905	. . . . 5 ⊢ (𝑎 = suc 𝑏 → (𝐹‘𝑎) = (𝐹‘suc 𝑏))
18		suceq 6449	. . . . 5 ⊢ ((𝐹‘𝑎) = (𝐹‘suc 𝑏) → suc (𝐹‘𝑎) = suc (𝐹‘suc 𝑏))
19	17, 18	syl 17	. . . 4 ⊢ (𝑎 = suc 𝑏 → suc (𝐹‘𝑎) = suc (𝐹‘suc 𝑏))
20	16, 19	eqeq12d 2752	. . 3 ⊢ (𝑎 = suc 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
21		pweq 4613	. . . . 5 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
22	21	fveq2d 6909	. . . 4 ⊢ (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
23		fveq2 6905	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
24		suceq 6449	. . . . 5 ⊢ ((𝐹‘𝑎) = (𝐹‘𝐴) → suc (𝐹‘𝑎) = suc (𝐹‘𝐴))
25	23, 24	syl 17	. . . 4 ⊢ (𝑎 = 𝐴 → suc (𝐹‘𝑎) = suc (𝐹‘𝐴))
26	22, 25	eqeq12d 2752	. . 3 ⊢ (𝑎 = 𝐴 → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 𝐴) = suc (𝐹‘𝐴)))
27		df-1o 8507	. . . 4 ⊢ 1o = suc ∅
28		pw0 4811	. . . . . 6 ⊢ 𝒫 ∅ = {∅}
29	28	fveq2i 6908	. . . . 5 ⊢ (card‘𝒫 ∅) = (card‘{∅})
30		0ex 5306	. . . . . 6 ⊢ ∅ ∈ V
31		cardsn 10010	. . . . . 6 ⊢ (∅ ∈ V → (card‘{∅}) = 1o)
32	30, 31	ax-mp 5	. . . . 5 ⊢ (card‘{∅}) = 1o
33	29, 32	eqtri 2764	. . . 4 ⊢ (card‘𝒫 ∅) = 1o
34	1	ackbij1lem13 10272	. . . . 5 ⊢ (𝐹‘∅) = ∅
35		suceq 6449	. . . . 5 ⊢ ((𝐹‘∅) = ∅ → suc (𝐹‘∅) = suc ∅)
36	34, 35	ax-mp 5	. . . 4 ⊢ suc (𝐹‘∅) = suc ∅
37	27, 33, 36	3eqtr4i 2774	. . 3 ⊢ (card‘𝒫 ∅) = suc (𝐹‘∅)
38		oveq2 7440	. . . . . 6 ⊢ ((card‘𝒫 𝑏) = suc (𝐹‘𝑏) → ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)) = ((card‘𝒫 𝑏) +o suc (𝐹‘𝑏)))
39	38	adantl 481	. . . . 5 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)) = ((card‘𝒫 𝑏) +o suc (𝐹‘𝑏)))
40		ackbij1lem5 10264	. . . . . 6 ⊢ (𝑏 ∈ ω → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
41	40	adantr 480	. . . . 5 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
42		df-suc 6389	. . . . . . . . . 10 ⊢ suc 𝑏 = (𝑏 ∪ {𝑏})
43	42	equncomi 4159	. . . . . . . . 9 ⊢ suc 𝑏 = ({𝑏} ∪ 𝑏)
44	43	fveq2i 6908	. . . . . . . 8 ⊢ (𝐹‘suc 𝑏) = (𝐹‘({𝑏} ∪ 𝑏))
45		ackbij1lem4 10263	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → {𝑏} ∈ (𝒫 ω ∩ Fin))
46	45	adantr 480	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → {𝑏} ∈ (𝒫 ω ∩ Fin))
47		ackbij1lem3 10262	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → 𝑏 ∈ (𝒫 ω ∩ Fin))
48	47	adantr 480	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → 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}) = ∅)
52	50, 51	syl 17	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ω → (𝑏 ∩ {𝑏}) = ∅)
53	49, 52	eqtrid 2788	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → ({𝑏} ∩ 𝑏) = ∅)
54	53	adantr 480	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → ({𝑏} ∩ 𝑏) = ∅)
55	1	ackbij1lem9 10268	. . . . . . . . . 10 ⊢ (({𝑏} ∈ (𝒫 ω ∩ Fin) ∧ 𝑏 ∈ (𝒫 ω ∩ Fin) ∧ ({𝑏} ∩ 𝑏) = ∅) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +o (𝐹‘𝑏)))
56	46, 48, 54, 55	syl3anc 1372	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +o (𝐹‘𝑏)))
57	1	ackbij1lem8 10267	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
58	57	adantr 480	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
59	58	oveq1d 7447	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → ((𝐹‘{𝑏}) +o (𝐹‘𝑏)) = ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
60	56, 59	eqtrd 2776	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
61	44, 60	eqtrid 2788	. . . . . . 7 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
62		suceq 6449	. . . . . . 7 ⊢ ((𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o (𝐹‘𝑏)) → suc (𝐹‘suc 𝑏) = suc ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
63	61, 62	syl 17	. . . . . 6 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → suc (𝐹‘suc 𝑏) = suc ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
64		nnfi 9208	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → 𝑏 ∈ Fin)
65		pwfi 9358	. . . . . . . . . 10 ⊢ (𝑏 ∈ Fin ↔ 𝒫 𝑏 ∈ Fin)
66	64, 65	sylib 218	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → 𝒫 𝑏 ∈ Fin)
67	66	adantr 480	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → 𝒫 𝑏 ∈ Fin)
68		ficardom 10002	. . . . . . . 8 ⊢ (𝒫 𝑏 ∈ Fin → (card‘𝒫 𝑏) ∈ ω)
69	67, 68	syl 17	. . . . . . 7 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (card‘𝒫 𝑏) ∈ ω)
70	1	ackbij1lem10 10269	. . . . . . . . 9 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
71	70	ffvelcdmi 7102	. . . . . . . 8 ⊢ (𝑏 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝑏) ∈ ω)
72	48, 71	syl 17	. . . . . . 7 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘𝑏) ∈ ω)
73		nnasuc 8645	. . . . . . 7 ⊢ (((card‘𝒫 𝑏) ∈ ω ∧ (𝐹‘𝑏) ∈ ω) → ((card‘𝒫 𝑏) +o suc (𝐹‘𝑏)) = suc ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
74	69, 72, 73	syl2anc 584	. . . . . 6 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → ((card‘𝒫 𝑏) +o suc (𝐹‘𝑏)) = suc ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
75	63, 74	eqtr4d 2779	. . . . 5 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → suc (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o suc (𝐹‘𝑏)))
76	39, 41, 75	3eqtr4d 2786	. . . 4 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏))
77	76	ex 412	. . 3 ⊢ (𝑏 ∈ ω → ((card‘𝒫 𝑏) = suc (𝐹‘𝑏) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
78	8, 14, 20, 26, 37, 77	finds 7919	. 2 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) = suc (𝐹‘𝐴))
79	2, 78	eqtrd 2776	1 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹‘𝐴))

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  1_oc1o 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