Theorem hashfibc 11211

Description: The binomial coefficient counts the number of subsets of a finite set of a given size. This is Metamath 100 proof #58 (formula for the number of combinations). For more on the notation for subsets of a given size, see sseqn 11207. (Contributed by Mario Carneiro, 13-Jul-2014.)

Assertion

Ref	Expression
hashfibc	⊢ ((𝐴 ∈ Fin ∧ 𝐾 ∈ ℤ) → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾}))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐾

Proof of Theorem hashfibc

Dummy variables 𝑘 𝑤 𝑦 𝑧 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5672	. . . . . 6 ⊢ (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅))
2	1	oveq1d 6067	. . . . 5 ⊢ (𝑤 = ∅ → ((♯‘𝑤)C𝑘) = ((♯‘∅)C𝑘))
3		pweq 3674	. . . . . . . 8 ⊢ (𝑤 = ∅ → 𝒫 𝑤 = 𝒫 ∅)
4	3	ineq1d 3423	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝒫 𝑤 ∩ Fin) = (𝒫 ∅ ∩ Fin))
5	4	rabeqdv 2809	. . . . . 6 ⊢ (𝑤 = ∅ → {𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
6	5	fveq2d 5676	. . . . 5 ⊢ (𝑤 = ∅ → (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
7	2, 6	eqeq12d 2249	. . . 4 ⊢ (𝑤 = ∅ → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
8	7	ralbidv 2544	. . 3 ⊢ (𝑤 = ∅ → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
9		fveq2 5672	. . . . . 6 ⊢ (𝑤 = 𝑦 → (♯‘𝑤) = (♯‘𝑦))
10	9	oveq1d 6067	. . . . 5 ⊢ (𝑤 = 𝑦 → ((♯‘𝑤)C𝑘) = ((♯‘𝑦)C𝑘))
11		pweq 3674	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
12	11	ineq1d 3423	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝒫 𝑤 ∩ Fin) = (𝒫 𝑦 ∩ Fin))
13	12	rabeqdv 2809	. . . . . 6 ⊢ (𝑤 = 𝑦 → {𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
14	13	fveq2d 5676	. . . . 5 ⊢ (𝑤 = 𝑦 → (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
15	10, 14	eqeq12d 2249	. . . 4 ⊢ (𝑤 = 𝑦 → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
16	15	ralbidv 2544	. . 3 ⊢ (𝑤 = 𝑦 → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
17		fveq2 5672	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (♯‘𝑤) = (♯‘(𝑦 ∪ {𝑧})))
18	17	oveq1d 6067	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((♯‘𝑤)C𝑘) = ((♯‘(𝑦 ∪ {𝑧}))C𝑘))
19		pweq 3674	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → 𝒫 𝑤 = 𝒫 (𝑦 ∪ {𝑧}))
20	19	ineq1d 3423	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝒫 𝑤 ∩ Fin) = (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin))
21	20	rabeqdv 2809	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
22	21	fveq2d 5676	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
23	18, 22	eqeq12d 2249	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
24	23	ralbidv 2544	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
25		fveq2 5672	. . . . . 6 ⊢ (𝑤 = 𝐴 → (♯‘𝑤) = (♯‘𝐴))
26	25	oveq1d 6067	. . . . 5 ⊢ (𝑤 = 𝐴 → ((♯‘𝑤)C𝑘) = ((♯‘𝐴)C𝑘))
27		pweq 3674	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → 𝒫 𝑤 = 𝒫 𝐴)
28	27	ineq1d 3423	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝒫 𝑤 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
29	28	rabeqdv 2809	. . . . . 6 ⊢ (𝑤 = 𝐴 → {𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
30	29	fveq2d 5676	. . . . 5 ⊢ (𝑤 = 𝐴 → (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
31	26, 30	eqeq12d 2249	. . . 4 ⊢ (𝑤 = 𝐴 → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
32	31	ralbidv 2544	. . 3 ⊢ (𝑤 = 𝐴 → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
33		0nn0 9513	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
34		bcn0 11121	. . . . . . . . 9 ⊢ (0 ∈ ℕ0 → (0C0) = 1)
35	33, 34	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
36		hash0 11163	. . . . . . . . . 10 ⊢ (♯‘∅) = 0
37	36	a1i 9	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → (♯‘∅) = 0)
38		elfz1eq 10372	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
39	37, 38	oveq12d 6070	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → ((♯‘∅)C𝑘) = (0C0))
40	38	eqcomd 2240	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...0) → 0 = 𝑘)
41		pw0 3843	. . . . . . . . . . . . . . 15 ⊢ 𝒫 ∅ = {∅}
42	41	raleqi 2747	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} (♯‘𝑥) = 𝑘)
43		0ex 4239	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ V
44		fveq2 5672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
45	44, 36	eqtrdi 2283	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
46	45	eqeq1d 2243	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → ((♯‘𝑥) = 𝑘 ↔ 0 = 𝑘))
47	43, 46	ralsn 3734	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ {∅} (♯‘𝑥) = 𝑘 ↔ 0 = 𝑘)
48	42, 47	bitri 184	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘 ↔ 0 = 𝑘)
49	40, 48	sylibr 134	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...0) → ∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘)
50		rabid2 2723	. . . . . . . . . . . 12 ⊢ (𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} ↔ ∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘)
51	49, 50	sylibr 134	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...0) → 𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})
52	51, 41	eqtr3di 2282	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...0) → {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = {∅})
53	52	fveq2d 5676	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘{∅}))
54		hashsng 11165	. . . . . . . . . 10 ⊢ (∅ ∈ V → (♯‘{∅}) = 1)
55	43, 54	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘{∅}) = 1
56	53, 55	eqtrdi 2283	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = 1)
57	35, 39, 56	3eqtr4a 2293	. . . . . . 7 ⊢ (𝑘 ∈ (0...0) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
58	57	adantl 277	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ 𝑘 ∈ (0...0)) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
59	36	oveq1i 6062	. . . . . . 7 ⊢ ((♯‘∅)C𝑘) = (0C𝑘)
60		bcval3 11117	. . . . . . . . 9 ⊢ ((0 ∈ ℕ0 ∧ 𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
61	33, 60	mp3an1 1361	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
62		id 19	. . . . . . . . . . . . . . 15 ⊢ (0 = 𝑘 → 0 = 𝑘)
63		0z 9590	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
64		elfz3 10371	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ ℤ → 0 ∈ (0...0))
65	63, 64	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ (0...0)
66	62, 65	eqeltrrdi 2326	. . . . . . . . . . . . . 14 ⊢ (0 = 𝑘 → 𝑘 ∈ (0...0))
67	66	con3i 637	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 ∈ (0...0) → ¬ 0 = 𝑘)
68	67	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ¬ 0 = 𝑘)
69	41	raleqi 2747	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} ¬ (♯‘𝑥) = 𝑘)
70	46	notbid 673	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → (¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘))
71	43, 70	ralsn 3734	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {∅} ¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
72	69, 71	bitri 184	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
73	68, 72	sylibr 134	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘)
74		rabeq0 3540	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = ∅ ↔ ∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘)
75	73, 74	sylibr 134	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = ∅)
76	75	fveq2d 5676	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘∅))
77	76, 36	eqtrdi 2283	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = 0)
78	61, 77	eqtr4d 2270	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
79	59, 78	eqtrid 2279	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
80		0zd 9591	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ → 0 ∈ ℤ)
81		fzdcel 10377	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ 0 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑘 ∈ (0...0))
82	80, 80, 81	mpd3an23 1376	. . . . . . 7 ⊢ (𝑘 ∈ ℤ → DECID 𝑘 ∈ (0...0))
83		exmiddc 844	. . . . . . 7 ⊢ (DECID 𝑘 ∈ (0...0) → (𝑘 ∈ (0...0) ∨ ¬ 𝑘 ∈ (0...0)))
84	82, 83	syl 14	. . . . . 6 ⊢ (𝑘 ∈ ℤ → (𝑘 ∈ (0...0) ∨ ¬ 𝑘 ∈ (0...0)))
85	58, 79, 84	mpjaodan 806	. . . . 5 ⊢ (𝑘 ∈ ℤ → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
86		velsn 3708	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
87		0fi 7143	. . . . . . . . . . . 12 ⊢ ∅ ∈ Fin
88		eleq1 2297	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑥 ∈ Fin ↔ ∅ ∈ Fin))
89	87, 88	mpbiri 168	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → 𝑥 ∈ Fin)
90	86, 89	sylbi 121	. . . . . . . . . 10 ⊢ (𝑥 ∈ {∅} → 𝑥 ∈ Fin)
91	90, 41	eleq2s 2329	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ∅ → 𝑥 ∈ Fin)
92	91	ssriv 3244	. . . . . . . 8 ⊢ 𝒫 ∅ ⊆ Fin
93		dfss 3227	. . . . . . . 8 ⊢ (𝒫 ∅ ⊆ Fin ↔ 𝒫 ∅ = (𝒫 ∅ ∩ Fin))
94	92, 93	mpbi 145	. . . . . . 7 ⊢ 𝒫 ∅ = (𝒫 ∅ ∩ Fin)
95	94	rabeqi 2808	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘}
96	95	fveq2i 5675	. . . . 5 ⊢ (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
97	85, 96	eqtrdi 2283	. . . 4 ⊢ (𝑘 ∈ ℤ → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
98	97	rgen 2597	. . 3 ⊢ ∀𝑘 ∈ ℤ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
99		oveq2 6060	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((♯‘𝑦)C𝑘) = ((♯‘𝑦)C𝑗))
100		eqeq2 2244	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((♯‘𝑥) = 𝑘 ↔ (♯‘𝑥) = 𝑗))
101	100	rabbidv 2804	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗})
102		fveqeq2 5681	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑧) = 𝑗))
103	102	cbvrabv 2814	. . . . . . . 8 ⊢ {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗} = {𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}
104	101, 103	eqtrdi 2283	. . . . . . 7 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗})
105	104	fveq2d 5676	. . . . . 6 ⊢ (𝑘 = 𝑗 → (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))
106	99, 105	eqeq12d 2249	. . . . 5 ⊢ (𝑘 = 𝑗 → (((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗})))
107	106	cbvralvw 2784	. . . 4 ⊢ (∀𝑘 ∈ ℤ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))
108		simpll 527	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))) → 𝑦 ∈ Fin)
109		simplr 529	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))) → ¬ 𝑧 ∈ 𝑦)
110		simprr 533	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))) → ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))
111	103	fveq2i 5675	. . . . . . . . . 10 ⊢ (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗})
112	111	eqeq2i 2245	. . . . . . . . 9 ⊢ (((♯‘𝑦)C𝑗) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))
113	112	ralbii 2550	. . . . . . . 8 ⊢ (∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}) ↔ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))
114	110, 113	sylibr 134	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))) → ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}))
115		simprl 531	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))) → 𝑘 ∈ ℤ)
116	108, 109, 114, 115	hashfibclem 11210	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))) → ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
117	116	expr 375	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ 𝑘 ∈ ℤ) → (∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}) → ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
118	117	ralrimdva 2624	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}) → ∀𝑘 ∈ ℤ ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
119	107, 118	biimtrid 152	. . 3 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑘 ∈ ℤ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) → ∀𝑘 ∈ ℤ ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
120	8, 16, 24, 32, 98, 119	findcard2s 7149	. 2 ⊢ (𝐴 ∈ Fin → ∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
121		oveq2 6060	. . . 4 ⊢ (𝑘 = 𝐾 → ((♯‘𝐴)C𝑘) = ((♯‘𝐴)C𝐾))
122		eqeq2 2244	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((♯‘𝑥) = 𝑘 ↔ (♯‘𝑥) = 𝐾))
123	122	rabbidv 2804	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾})
124	123	fveq2d 5676	. . . 4 ⊢ (𝑘 = 𝐾 → (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾}))
125	121, 124	eqeq12d 2249	. . 3 ⊢ (𝑘 = 𝐾 → (((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾})))
126	125	rspccva 2922	. 2 ⊢ ((∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ∧ 𝐾 ∈ ℤ) → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾}))
127	120, 126	sylan 283	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐾 ∈ ℤ) → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716  DECID wdc 842   = wceq 1398   ∈ wcel 2205  ∀wral 2522  {crab 2526  Vcvv 2815   ∪ cun 3211   ∩ cin 3212   ⊆ wss 3213  ∅c0 3510  𝒫 cpw 3671  {csn 3691  ‘cfv 5354  (class class class)co 6052  Fincfn 6977  0cc0 8129  1c1 8130  ℕ₀cn0 9498  ℤcz 9579  ...cfz 10345  Ccbc 11113  ♯chash 11142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-2o 6650  df-oadd 6653  df-er 6769  df-map 6886  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-fz 10346  df-seqfrec 10814  df-fac 11092  df-bc 11114  df-ihash 11143

This theorem is referenced by:  ballotfilem1  13143  ballotfilem2  13149