Theorem ackbij1lem14 10270

Description: Lemma for ackbij1 10275. (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 10264	. 2 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
3		pweq 4619	. . . . 5 ⊢ (𝑎 = ∅ → 𝒫 𝑎 = 𝒫 ∅)
4	3	fveq2d 6911	. . . 4 ⊢ (𝑎 = ∅ → (card‘𝒫 𝑎) = (card‘𝒫 ∅))
5		fveq2 6907	. . . . 5 ⊢ (𝑎 = ∅ → (𝐹‘𝑎) = (𝐹‘∅))
6		suceq 6452	. . . . 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 2751	. . 3 ⊢ (𝑎 = ∅ → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 ∅) = suc (𝐹‘∅)))
9		pweq 4619	. . . . 5 ⊢ (𝑎 = 𝑏 → 𝒫 𝑎 = 𝒫 𝑏)
10	9	fveq2d 6911	. . . 4 ⊢ (𝑎 = 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 𝑏))
11		fveq2 6907	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
12		suceq 6452	. . . . 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 2751	. . 3 ⊢ (𝑎 = 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)))
15		pweq 4619	. . . . 5 ⊢ (𝑎 = suc 𝑏 → 𝒫 𝑎 = 𝒫 suc 𝑏)
16	15	fveq2d 6911	. . . 4 ⊢ (𝑎 = suc 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 suc 𝑏))
17		fveq2 6907	. . . . 5 ⊢ (𝑎 = suc 𝑏 → (𝐹‘𝑎) = (𝐹‘suc 𝑏))
18		suceq 6452	. . . . 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 2751	. . 3 ⊢ (𝑎 = suc 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
21		pweq 4619	. . . . 5 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
22	21	fveq2d 6911	. . . 4 ⊢ (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
23		fveq2 6907	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
24		suceq 6452	. . . . 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 2751	. . 3 ⊢ (𝑎 = 𝐴 → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 𝐴) = suc (𝐹‘𝐴)))
27		df-1o 8505	. . . 4 ⊢ 1o = suc ∅
28		pw0 4817	. . . . . 6 ⊢ 𝒫 ∅ = {∅}
29	28	fveq2i 6910	. . . . 5 ⊢ (card‘𝒫 ∅) = (card‘{∅})
30		0ex 5313	. . . . . 6 ⊢ ∅ ∈ V
31		cardsn 10007	. . . . . 6 ⊢ (∅ ∈ V → (card‘{∅}) = 1o)
32	30, 31	ax-mp 5	. . . . 5 ⊢ (card‘{∅}) = 1o
33	29, 32	eqtri 2763	. . . 4 ⊢ (card‘𝒫 ∅) = 1o
34	1	ackbij1lem13 10269	. . . . 5 ⊢ (𝐹‘∅) = ∅
35		suceq 6452	. . . . 5 ⊢ ((𝐹‘∅) = ∅ → suc (𝐹‘∅) = suc ∅)
36	34, 35	ax-mp 5	. . . 4 ⊢ suc (𝐹‘∅) = suc ∅
37	27, 33, 36	3eqtr4i 2773	. . 3 ⊢ (card‘𝒫 ∅) = suc (𝐹‘∅)
38		oveq2 7439	. . . . . 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 10261	. . . . . 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 6392	. . . . . . . . . 10 ⊢ suc 𝑏 = (𝑏 ∪ {𝑏})
43	42	equncomi 4170	. . . . . . . . 9 ⊢ suc 𝑏 = ({𝑏} ∪ 𝑏)
44	43	fveq2i 6910	. . . . . . . 8 ⊢ (𝐹‘suc 𝑏) = (𝐹‘({𝑏} ∪ 𝑏))
45		ackbij1lem4 10260	. . . . . . . . . . 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 10259	. . . . . . . . . . 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 4217	. . . . . . . . . . . 12 ⊢ ({𝑏} ∩ 𝑏) = (𝑏 ∩ {𝑏})
50		nnord 7895	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ω → Ord 𝑏)
51		orddisj 6424	. . . . . . . . . . . . 13 ⊢ (Ord 𝑏 → (𝑏 ∩ {𝑏}) = ∅)
52	50, 51	syl 17	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ω → (𝑏 ∩ {𝑏}) = ∅)
53	49, 52	eqtrid 2787	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → ({𝑏} ∩ 𝑏) = ∅)
54	53	adantr 480	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → ({𝑏} ∩ 𝑏) = ∅)
55	1	ackbij1lem9 10265	. . . . . . . . . 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 1370	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +o (𝐹‘𝑏)))
57	1	ackbij1lem8 10264	. . . . . . . . . . 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 7446	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → ((𝐹‘{𝑏}) +o (𝐹‘𝑏)) = ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
60	56, 59	eqtrd 2775	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
61	44, 60	eqtrid 2787	. . . . . . 7 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
62		suceq 6452	. . . . . . 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 9206	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → 𝑏 ∈ Fin)
65		pwfi 9355	. . . . . . . . . 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 9999	. . . . . . . 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 10266	. . . . . . . . 9 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
71	70	ffvelcdmi 7103	. . . . . . . 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 8643	. . . . . . 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 2778	. . . . 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 2785	. . . 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 2775	1 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∪ cun 3961   ∩ cin 3962  ∅c0 4339  𝒫 cpw 4605  {csn 4631  ∪ ciun 4996   ↦ cmpt 5231   × cxp 5687  Ord word 6385  suc csuc 6388  ‘cfv 6563  (class class class)co 7431  ωcom 7887  1_oc1o 8498   +_o coa 8502  Fincfn 8984  cardccrd 9973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977

This theorem is referenced by:  ackbij1lem15  10271  ackbij1lem18  10274  ackbij1b  10276