Theorem ackbij1lem14 10320

Description: Lemma for ackbij1 10325. (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 10314	. 2 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
3		pweq 4573	. . . . 5 ⊢ (𝑎 = ∅ → 𝒫 𝑎 = 𝒫 ∅)
4	3	fveq2d 6897	. . . 4 ⊢ (𝑎 = ∅ → (card‘𝒫 𝑎) = (card‘𝒫 ∅))
5		fveq2 6893	. . . . 5 ⊢ (𝑎 = ∅ → (𝐹‘𝑎) = (𝐹‘∅))
6		suceq 6437	. . . . 5 ⊢ ((𝐹‘𝑎) = (𝐹‘∅) → suc (𝐹‘𝑎) = suc (𝐹‘∅))
7	5, 6	syl 18	. . . 4 ⊢ (𝑎 = ∅ → suc (𝐹‘𝑎) = suc (𝐹‘∅))
8	4, 7	eqeq12d 2779	. . 3 ⊢ (𝑎 = ∅ → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 ∅) = suc (𝐹‘∅)))
9		pweq 4573	. . . . 5 ⊢ (𝑎 = 𝑏 → 𝒫 𝑎 = 𝒫 𝑏)
10	9	fveq2d 6897	. . . 4 ⊢ (𝑎 = 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 𝑏))
11		fveq2 6893	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
12		suceq 6437	. . . . 5 ⊢ ((𝐹‘𝑎) = (𝐹‘𝑏) → suc (𝐹‘𝑎) = suc (𝐹‘𝑏))
13	11, 12	syl 18	. . . 4 ⊢ (𝑎 = 𝑏 → suc (𝐹‘𝑎) = suc (𝐹‘𝑏))
14	10, 13	eqeq12d 2779	. . 3 ⊢ (𝑎 = 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)))
15		pweq 4573	. . . . 5 ⊢ (𝑎 = suc 𝑏 → 𝒫 𝑎 = 𝒫 suc 𝑏)
16	15	fveq2d 6897	. . . 4 ⊢ (𝑎 = suc 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 suc 𝑏))
17		fveq2 6893	. . . . 5 ⊢ (𝑎 = suc 𝑏 → (𝐹‘𝑎) = (𝐹‘suc 𝑏))
18		suceq 6437	. . . . 5 ⊢ ((𝐹‘𝑎) = (𝐹‘suc 𝑏) → suc (𝐹‘𝑎) = suc (𝐹‘suc 𝑏))
19	17, 18	syl 18	. . . 4 ⊢ (𝑎 = suc 𝑏 → suc (𝐹‘𝑎) = suc (𝐹‘suc 𝑏))
20	16, 19	eqeq12d 2779	. . 3 ⊢ (𝑎 = suc 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
21		pweq 4573	. . . . 5 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
22	21	fveq2d 6897	. . . 4 ⊢ (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
23		fveq2 6893	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
24		suceq 6437	. . . . 5 ⊢ ((𝐹‘𝑎) = (𝐹‘𝐴) → suc (𝐹‘𝑎) = suc (𝐹‘𝐴))
25	23, 24	syl 18	. . . 4 ⊢ (𝑎 = 𝐴 → suc (𝐹‘𝑎) = suc (𝐹‘𝐴))
26	22, 25	eqeq12d 2779	. . 3 ⊢ (𝑎 = 𝐴 → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 𝐴) = suc (𝐹‘𝐴)))
27		df-1o 8487	. . . 4 ⊢ 1o = suc ∅
28		pw0 4832	. . . . . 6 ⊢ 𝒫 ∅ = {∅}
29	28	fveq2i 6896	. . . . 5 ⊢ (card‘𝒫 ∅) = (card‘{∅})
30		0ex 5263	. . . . . 6 ⊢ ∅ ∈ V
31		cardsn 10060	. . . . . 6 ⊢ (∅ ∈ V → (card‘{∅}) = 1o)
32	30, 31	ax-mp 5	. . . . 5 ⊢ (card‘{∅}) = 1o
33	29, 32	eqtri 2786	. . . 4 ⊢ (card‘𝒫 ∅) = 1o
34	1	ackbij1lem13 10319	. . . . 5 ⊢ (𝐹‘∅) = ∅
35		suceq 6437	. . . . 5 ⊢ ((𝐹‘∅) = ∅ → suc (𝐹‘∅) = suc ∅)
36	34, 35	ax-mp 5	. . . 4 ⊢ suc (𝐹‘∅) = suc ∅
37	27, 33, 36	3eqtr4i 2796	. . 3 ⊢ (card‘𝒫 ∅) = suc (𝐹‘∅)
38		oveq2 7435	. . . . . 6 ⊢ ((card‘𝒫 𝑏) = suc (𝐹‘𝑏) → ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)) = ((card‘𝒫 𝑏) +o suc (𝐹‘𝑏)))
39	38	adantl 487	. . . . 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 10311	. . . . . 6 ⊢ (𝑏 ∈ ω → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
41	40	adantr 486	. . . . 5 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
42		df-suc 6374	. . . . . . . . . 10 ⊢ suc 𝑏 = (𝑏 ∪ {𝑏})
43	42	equncomi 4161	. . . . . . . . 9 ⊢ suc 𝑏 = ({𝑏} ∪ 𝑏)
44	43	fveq2i 6896	. . . . . . . 8 ⊢ (𝐹‘suc 𝑏) = (𝐹‘({𝑏} ∪ 𝑏))
45		ackbij1lem4 10310	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → {𝑏} ∈ (𝒫 ω ∩ Fin))
46	45	adantr 486	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → {𝑏} ∈ (𝒫 ω ∩ Fin))
47		ackbij1lem3 10309	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → 𝑏 ∈ (𝒫 ω ∩ Fin))
48	47	adantr 486	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → 𝑏 ∈ (𝒫 ω ∩ Fin))
49		incom 4108	. . . . . . . . . . . 12 ⊢ ({𝑏} ∩ 𝑏) = (𝑏 ∩ {𝑏})
50		nnord 7892	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ω → Ord 𝑏)
51		orddisj 6407	. . . . . . . . . . . . 13 ⊢ (Ord 𝑏 → (𝑏 ∩ {𝑏}) = ∅)
52	50, 51	syl 18	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ω → (𝑏 ∩ {𝑏}) = ∅)
53	49, 52	eqtrid 2810	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → ({𝑏} ∩ 𝑏) = ∅)
54	53	adantr 486	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → ({𝑏} ∩ 𝑏) = ∅)
55	1	ackbij1lem9 10315	. . . . . . . . . 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 1398	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +o (𝐹‘𝑏)))
57	1	ackbij1lem8 10314	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
58	57	adantr 486	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
59	58	oveq1d 7442	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → ((𝐹‘{𝑏}) +o (𝐹‘𝑏)) = ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
60	56, 59	eqtrd 2798	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
61	44, 60	eqtrid 2810	. . . . . . 7 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
62		suceq 6437	. . . . . . 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 18	. . . . . 6 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → suc (𝐹‘suc 𝑏) = suc ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
64		nnfi 9194	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → 𝑏 ∈ Fin)
65		pwfi 9321	. . . . . . . . . 10 ⊢ (𝑏 ∈ Fin ↔ 𝒫 𝑏 ∈ Fin)
66	64, 65	sylib 221	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → 𝒫 𝑏 ∈ Fin)
67	66	adantr 486	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → 𝒫 𝑏 ∈ Fin)
68		ficardom 10052	. . . . . . . 8 ⊢ (𝒫 𝑏 ∈ Fin → (card‘𝒫 𝑏) ∈ ω)
69	67, 68	syl 18	. . . . . . 7 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (card‘𝒫 𝑏) ∈ ω)
70	1	ackbij1lem10 10316	. . . . . . . . 9 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
71	70	ffvelcdmi 7091	. . . . . . . 8 ⊢ (𝑏 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝑏) ∈ ω)
72	48, 71	syl 18	. . . . . . 7 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘𝑏) ∈ ω)
73		nnasuc 8626	. . . . . . 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 596	. . . . . 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 2801	. . . . 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 2808	. . . 4 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏))
77	76	ex 418	. . 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 7915	. 2 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) = suc (𝐹‘𝐴))
79	2, 78	eqtrd 2798	1 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  Vcvv 3453   ∩ cin 4091   ∪ cun 4151  ∅c0 4280  𝒫 cpw 4559  {csn 4587  ∪ ciun 4953   ↦ cmpt 5188   × cxp 5652  Ord word 6367  suc csuc 6370  ‘cfv 6546  (class class class)co 7427  ωcom 7884  1_oc1o 8480   +_o coa 8484  Fincfn 8984  cardccrd 10026

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-2o 8488  df-oadd 8491  df-er 8728  df-map 8860  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9992  df-card 10030

This theorem is used by:  ackbij1lem15  10321  ackbij1lem18  10324  ackbij1b  10326