Theorem hashbc 13237

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). (Contributed by Mario Carneiro, 13-Jul-2014.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐾

Proof of Theorem hashbc

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

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 ⊢ (𝑤 = ∅ → (#‘𝑤) = (#‘∅))
2	1	oveq1d 6665	. . . . 5 ⊢ (𝑤 = ∅ → ((#‘𝑤)C𝑘) = ((#‘∅)C𝑘))
3		pweq 4161	. . . . . . 7 ⊢ (𝑤 = ∅ → 𝒫 𝑤 = 𝒫 ∅)
4		rabeq 3192	. . . . . . 7 ⊢ (𝒫 𝑤 = 𝒫 ∅ → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})
5	3, 4	syl 17	. . . . . 6 ⊢ (𝑤 = ∅ → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})
6	5	fveq2d 6195	. . . . 5 ⊢ (𝑤 = ∅ → (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
7	2, 6	eqeq12d 2637	. . . 4 ⊢ (𝑤 = ∅ → (((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})))
8	7	ralbidv 2986	. . 3 ⊢ (𝑤 = ∅ → (∀𝑘 ∈ ℤ ((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})))
9		fveq2 6191	. . . . . 6 ⊢ (𝑤 = 𝑦 → (#‘𝑤) = (#‘𝑦))
10	9	oveq1d 6665	. . . . 5 ⊢ (𝑤 = 𝑦 → ((#‘𝑤)C𝑘) = ((#‘𝑦)C𝑘))
11		pweq 4161	. . . . . . 7 ⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
12		rabeq 3192	. . . . . . 7 ⊢ (𝒫 𝑤 = 𝒫 𝑦 → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘})
13	11, 12	syl 17	. . . . . 6 ⊢ (𝑤 = 𝑦 → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘})
14	13	fveq2d 6195	. . . . 5 ⊢ (𝑤 = 𝑦 → (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}))
15	10, 14	eqeq12d 2637	. . . 4 ⊢ (𝑤 = 𝑦 → (((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘})))
16	15	ralbidv 2986	. . 3 ⊢ (𝑤 = 𝑦 → (∀𝑘 ∈ ℤ ((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘})))
17		fveq2 6191	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (#‘𝑤) = (#‘(𝑦 ∪ {𝑧})))
18	17	oveq1d 6665	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((#‘𝑤)C𝑘) = ((#‘(𝑦 ∪ {𝑧}))C𝑘))
19		pweq 4161	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → 𝒫 𝑤 = 𝒫 (𝑦 ∪ {𝑧}))
20		rabeq 3192	. . . . . . 7 ⊢ (𝒫 𝑤 = 𝒫 (𝑦 ∪ {𝑧}) → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})
21	19, 20	syl 17	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})
22	21	fveq2d 6195	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘}))
23	18, 22	eqeq12d 2637	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
24	23	ralbidv 2986	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ ℤ ((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
25		fveq2 6191	. . . . . 6 ⊢ (𝑤 = 𝐴 → (#‘𝑤) = (#‘𝐴))
26	25	oveq1d 6665	. . . . 5 ⊢ (𝑤 = 𝐴 → ((#‘𝑤)C𝑘) = ((#‘𝐴)C𝑘))
27		pweq 4161	. . . . . . 7 ⊢ (𝑤 = 𝐴 → 𝒫 𝑤 = 𝒫 𝐴)
28		rabeq 3192	. . . . . . 7 ⊢ (𝒫 𝑤 = 𝒫 𝐴 → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘})
29	27, 28	syl 17	. . . . . 6 ⊢ (𝑤 = 𝐴 → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘})
30	29	fveq2d 6195	. . . . 5 ⊢ (𝑤 = 𝐴 → (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}))
31	26, 30	eqeq12d 2637	. . . 4 ⊢ (𝑤 = 𝐴 → (((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘})))
32	31	ralbidv 2986	. . 3 ⊢ (𝑤 = 𝐴 → (∀𝑘 ∈ ℤ ((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘})))
33		hash0 13158	. . . . . . . . . 10 ⊢ (#‘∅) = 0
34	33	a1i 11	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → (#‘∅) = 0)
35		elfz1eq 12352	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
36	34, 35	oveq12d 6668	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → ((#‘∅)C𝑘) = (0C0))
37		0nn0 11307	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
38		bcn0 13097	. . . . . . . . 9 ⊢ (0 ∈ ℕ0 → (0C0) = 1)
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
40	36, 39	syl6eq 2672	. . . . . . 7 ⊢ (𝑘 ∈ (0...0) → ((#‘∅)C𝑘) = 1)
41		pw0 4343	. . . . . . . . . 10 ⊢ 𝒫 ∅ = {∅}
42	35	eqcomd 2628	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...0) → 0 = 𝑘)
43	41	raleqi 3142	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝒫 ∅(#‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} (#‘𝑥) = 𝑘)
44		0ex 4790	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
45		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (#‘𝑥) = (#‘∅))
46	45, 33	syl6eq 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → (#‘𝑥) = 0)
47	46	eqeq1d 2624	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → ((#‘𝑥) = 𝑘 ↔ 0 = 𝑘))
48	44, 47	ralsn 4222	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {∅} (#‘𝑥) = 𝑘 ↔ 0 = 𝑘)
49	43, 48	bitri 264	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝒫 ∅(#‘𝑥) = 𝑘 ↔ 0 = 𝑘)
50	42, 49	sylibr 224	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...0) → ∀𝑥 ∈ 𝒫 ∅(#‘𝑥) = 𝑘)
51		rabid2 3118	. . . . . . . . . . 11 ⊢ (𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘} ↔ ∀𝑥 ∈ 𝒫 ∅(#‘𝑥) = 𝑘)
52	50, 51	sylibr 224	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...0) → 𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})
53	41, 52	syl5reqr 2671	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘} = {∅})
54	53	fveq2d 6195	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}) = (#‘{∅}))
55		hashsng 13159	. . . . . . . . 9 ⊢ (∅ ∈ V → (#‘{∅}) = 1)
56	44, 55	ax-mp 5	. . . . . . . 8 ⊢ (#‘{∅}) = 1
57	54, 56	syl6eq 2672	. . . . . . 7 ⊢ (𝑘 ∈ (0...0) → (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}) = 1)
58	40, 57	eqtr4d 2659	. . . . . 6 ⊢ (𝑘 ∈ (0...0) → ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
59	58	adantl 482	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ 𝑘 ∈ (0...0)) → ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
60	33	oveq1i 6660	. . . . . 6 ⊢ ((#‘∅)C𝑘) = (0C𝑘)
61		bcval3 13093	. . . . . . . 8 ⊢ ((0 ∈ ℕ0 ∧ 𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
62	37, 61	mp3an1 1411	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
63		id 22	. . . . . . . . . . . . . 14 ⊢ (0 = 𝑘 → 0 = 𝑘)
64		0z 11388	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
65		elfz3 12351	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℤ → 0 ∈ (0...0))
66	64, 65	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 0 ∈ (0...0)
67	63, 66	syl6eqelr 2710	. . . . . . . . . . . . 13 ⊢ (0 = 𝑘 → 𝑘 ∈ (0...0))
68	67	con3i 150	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 ∈ (0...0) → ¬ 0 = 𝑘)
69	68	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ¬ 0 = 𝑘)
70	41	raleqi 3142	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝒫 ∅ ¬ (#‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} ¬ (#‘𝑥) = 𝑘)
71	47	notbid 308	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (¬ (#‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘))
72	44, 71	ralsn 4222	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ {∅} ¬ (#‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
73	70, 72	bitri 264	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝒫 ∅ ¬ (#‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
74	69, 73	sylibr 224	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ∀𝑥 ∈ 𝒫 ∅ ¬ (#‘𝑥) = 𝑘)
75		rabeq0 3957	. . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘} = ∅ ↔ ∀𝑥 ∈ 𝒫 ∅ ¬ (#‘𝑥) = 𝑘)
76	74, 75	sylibr 224	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘} = ∅)
77	76	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}) = (#‘∅))
78	77, 33	syl6eq 2672	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}) = 0)
79	62, 78	eqtr4d 2659	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
80	60, 79	syl5eq 2668	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
81	59, 80	pm2.61dan 832	. . . 4 ⊢ (𝑘 ∈ ℤ → ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
82	81	rgen 2922	. . 3 ⊢ ∀𝑘 ∈ ℤ ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})
83		oveq2 6658	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((#‘𝑦)C𝑘) = ((#‘𝑦)C𝑗))
84		eqeq2 2633	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((#‘𝑥) = 𝑘 ↔ (#‘𝑥) = 𝑗))
85	84	rabbidv 3189	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗})
86		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (#‘𝑥) = (#‘𝑧))
87	86	eqeq1d 2624	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((#‘𝑥) = 𝑗 ↔ (#‘𝑧) = 𝑗))
88	87	cbvrabv 3199	. . . . . . . 8 ⊢ {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗} = {𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}
89	85, 88	syl6eq 2672	. . . . . . 7 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘} = {𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗})
90	89	fveq2d 6195	. . . . . 6 ⊢ (𝑘 = 𝑗 → (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
91	83, 90	eqeq12d 2637	. . . . 5 ⊢ (𝑘 = 𝑗 → (((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗})))
92	91	cbvralv 3171	. . . 4 ⊢ (∀𝑘 ∈ ℤ ((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
93		simpll 790	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → 𝑦 ∈ Fin)
94		simplr 792	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → ¬ 𝑧 ∈ 𝑦)
95		simprr 796	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
96	88	fveq2i 6194	. . . . . . . . . 10 ⊢ (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗}) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗})
97	96	eqeq2i 2634	. . . . . . . . 9 ⊢ (((#‘𝑦)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗}) ↔ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
98	97	ralbii 2980	. . . . . . . 8 ⊢ (∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗}) ↔ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
99	95, 98	sylibr 224	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗}))
100		simprl 794	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → 𝑘 ∈ ℤ)
101	93, 94, 99, 100	hashbclem 13236	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘}))
102	101	expr 643	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ 𝑘 ∈ ℤ) → (∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}) → ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
103	102	ralrimdva 2969	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}) → ∀𝑘 ∈ ℤ ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
104	92, 103	syl5bi 232	. . 3 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑘 ∈ ℤ ((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}) → ∀𝑘 ∈ ℤ ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
105	8, 16, 24, 32, 82, 104	findcard2s 8201	. 2 ⊢ (𝐴 ∈ Fin → ∀𝑘 ∈ ℤ ((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}))
106		oveq2 6658	. . . 4 ⊢ (𝑘 = 𝐾 → ((#‘𝐴)C𝑘) = ((#‘𝐴)C𝐾))
107		eqeq2 2633	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((#‘𝑥) = 𝑘 ↔ (#‘𝑥) = 𝐾))
108	107	rabbidv 3189	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾})
109	108	fveq2d 6195	. . . 4 ⊢ (𝑘 = 𝐾 → (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))
110	106, 109	eqeq12d 2637	. . 3 ⊢ (𝑘 = 𝐾 → (((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾})))
111	110	rspccva 3308	. 2 ⊢ ((∀𝑘 ∈ ℤ ((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}) ∧ 𝐾 ∈ ℤ) → ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))
112	105, 111	sylan 488	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐾 ∈ ℤ) → ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ∪ cun 3572  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  0cc0 9936  1c1 9937  ℕ₀cn0 11292  ℤcz 11377  ...cfz 12326  Ccbc 13089  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-fac 13061  df-bc 13090  df-hash 13118

This theorem is referenced by:  hashbc2  15710  sylow1lem1  18013  musum  24917  ballotlem1  30548  ballotlem2  30550