Theorem ackbij1lem14 9644

Description: Lemma for ackbij1 9649. (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 9638	. 2 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
3		pweq 4513	. . . . 5 ⊢ (𝑎 = ∅ → 𝒫 𝑎 = 𝒫 ∅)
4	3	fveq2d 6649	. . . 4 ⊢ (𝑎 = ∅ → (card‘𝒫 𝑎) = (card‘𝒫 ∅))
5		fveq2 6645	. . . . 5 ⊢ (𝑎 = ∅ → (𝐹‘𝑎) = (𝐹‘∅))
6		suceq 6224	. . . . 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 2814	. . 3 ⊢ (𝑎 = ∅ → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 ∅) = suc (𝐹‘∅)))
9		pweq 4513	. . . . 5 ⊢ (𝑎 = 𝑏 → 𝒫 𝑎 = 𝒫 𝑏)
10	9	fveq2d 6649	. . . 4 ⊢ (𝑎 = 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 𝑏))
11		fveq2 6645	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
12		suceq 6224	. . . . 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 2814	. . 3 ⊢ (𝑎 = 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)))
15		pweq 4513	. . . . 5 ⊢ (𝑎 = suc 𝑏 → 𝒫 𝑎 = 𝒫 suc 𝑏)
16	15	fveq2d 6649	. . . 4 ⊢ (𝑎 = suc 𝑏 → (card‘𝒫 𝑎) = (card‘𝒫 suc 𝑏))
17		fveq2 6645	. . . . 5 ⊢ (𝑎 = suc 𝑏 → (𝐹‘𝑎) = (𝐹‘suc 𝑏))
18		suceq 6224	. . . . 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 2814	. . 3 ⊢ (𝑎 = suc 𝑏 → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏)))
21		pweq 4513	. . . . 5 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
22	21	fveq2d 6649	. . . 4 ⊢ (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
23		fveq2 6645	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
24		suceq 6224	. . . . 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 2814	. . 3 ⊢ (𝑎 = 𝐴 → ((card‘𝒫 𝑎) = suc (𝐹‘𝑎) ↔ (card‘𝒫 𝐴) = suc (𝐹‘𝐴)))
27		df-1o 8085	. . . 4 ⊢ 1o = suc ∅
28		pw0 4705	. . . . . 6 ⊢ 𝒫 ∅ = {∅}
29	28	fveq2i 6648	. . . . 5 ⊢ (card‘𝒫 ∅) = (card‘{∅})
30		0ex 5175	. . . . . 6 ⊢ ∅ ∈ V
31		cardsn 9382	. . . . . 6 ⊢ (∅ ∈ V → (card‘{∅}) = 1o)
32	30, 31	ax-mp 5	. . . . 5 ⊢ (card‘{∅}) = 1o
33	29, 32	eqtri 2821	. . . 4 ⊢ (card‘𝒫 ∅) = 1o
34	1	ackbij1lem13 9643	. . . . 5 ⊢ (𝐹‘∅) = ∅
35		suceq 6224	. . . . 5 ⊢ ((𝐹‘∅) = ∅ → suc (𝐹‘∅) = suc ∅)
36	34, 35	ax-mp 5	. . . 4 ⊢ suc (𝐹‘∅) = suc ∅
37	27, 33, 36	3eqtr4i 2831	. . 3 ⊢ (card‘𝒫 ∅) = suc (𝐹‘∅)
38		oveq2 7143	. . . . . 6 ⊢ ((card‘𝒫 𝑏) = suc (𝐹‘𝑏) → ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)) = ((card‘𝒫 𝑏) +o suc (𝐹‘𝑏)))
39	38	adantl 485	. . . . 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 9635	. . . . . 6 ⊢ (𝑏 ∈ ω → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
41	40	adantr 484	. . . . 5 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (card‘𝒫 suc 𝑏) = ((card‘𝒫 𝑏) +o (card‘𝒫 𝑏)))
42		df-suc 6165	. . . . . . . . . 10 ⊢ suc 𝑏 = (𝑏 ∪ {𝑏})
43	42	equncomi 4082	. . . . . . . . 9 ⊢ suc 𝑏 = ({𝑏} ∪ 𝑏)
44	43	fveq2i 6648	. . . . . . . 8 ⊢ (𝐹‘suc 𝑏) = (𝐹‘({𝑏} ∪ 𝑏))
45		ackbij1lem4 9634	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → {𝑏} ∈ (𝒫 ω ∩ Fin))
46	45	adantr 484	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → {𝑏} ∈ (𝒫 ω ∩ Fin))
47		ackbij1lem3 9633	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → 𝑏 ∈ (𝒫 ω ∩ Fin))
48	47	adantr 484	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → 𝑏 ∈ (𝒫 ω ∩ Fin))
49		incom 4128	. . . . . . . . . . . 12 ⊢ ({𝑏} ∩ 𝑏) = (𝑏 ∩ {𝑏})
50		nnord 7568	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ω → Ord 𝑏)
51		orddisj 6197	. . . . . . . . . . . . 13 ⊢ (Ord 𝑏 → (𝑏 ∩ {𝑏}) = ∅)
52	50, 51	syl 17	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ω → (𝑏 ∩ {𝑏}) = ∅)
53	49, 52	syl5eq 2845	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → ({𝑏} ∩ 𝑏) = ∅)
54	53	adantr 484	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → ({𝑏} ∩ 𝑏) = ∅)
55	1	ackbij1lem9 9639	. . . . . . . . . 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 1368	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((𝐹‘{𝑏}) +o (𝐹‘𝑏)))
57	1	ackbij1lem8 9638	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
58	57	adantr 484	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘{𝑏}) = (card‘𝒫 𝑏))
59	58	oveq1d 7150	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → ((𝐹‘{𝑏}) +o (𝐹‘𝑏)) = ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
60	56, 59	eqtrd 2833	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘({𝑏} ∪ 𝑏)) = ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
61	44, 60	syl5eq 2845	. . . . . . 7 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (𝐹‘suc 𝑏) = ((card‘𝒫 𝑏) +o (𝐹‘𝑏)))
62		suceq 6224	. . . . . . 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 8696	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → 𝑏 ∈ Fin)
65		pwfi 8803	. . . . . . . . . 10 ⊢ (𝑏 ∈ Fin ↔ 𝒫 𝑏 ∈ Fin)
66	64, 65	sylib 221	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → 𝒫 𝑏 ∈ Fin)
67	66	adantr 484	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → 𝒫 𝑏 ∈ Fin)
68		ficardom 9374	. . . . . . . 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 9640	. . . . . . . . 9 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
71	70	ffvelrni 6827	. . . . . . . 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 8215	. . . . . . 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 587	. . . . . 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 2836	. . . . 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 2843	. . . 4 ⊢ ((𝑏 ∈ ω ∧ (card‘𝒫 𝑏) = suc (𝐹‘𝑏)) → (card‘𝒫 suc 𝑏) = suc (𝐹‘suc 𝑏))
77	76	ex 416	. . 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 7589	. 2 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) = suc (𝐹‘𝐴))
79	2, 78	eqtrd 2833	1 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹‘𝐴))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∪ cun 3879   ∩ cin 3880  ∅c0 4243  𝒫 cpw 4497  {csn 4525  ∪ ciun 4881   ↦ cmpt 5110   × cxp 5517  Ord word 6158  suc csuc 6161  ‘cfv 6324  (class class class)co 7135  ωcom 7560  1_oc1o 8078   +_o coa 8082  Fincfn 8492  cardccrd 9348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352

This theorem is referenced by:  ackbij1lem15  9645  ackbij1lem18  9648  ackbij1b  9650