Theorem hashbc 14490

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 6882	. . . . . 6 ⊢ (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅))
2	1	oveq1d 7426	. . . . 5 ⊢ (𝑤 = ∅ → ((♯‘𝑤)C𝑘) = ((♯‘∅)C𝑘))
3		pweq 4579	. . . . . . 7 ⊢ (𝑤 = ∅ → 𝒫 𝑤 = 𝒫 ∅)
4	3	rabeqdv 3437	. . . . . 6 ⊢ (𝑤 = ∅ → {𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})
5	4	fveq2d 6886	. . . . 5 ⊢ (𝑤 = ∅ → (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
6	2, 5	eqeq12d 2785	. . . 4 ⊢ (𝑤 = ∅ → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})))
7	6	ralbidv 3194	. . 3 ⊢ (𝑤 = ∅ → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})))
8		fveq2 6882	. . . . . 6 ⊢ (𝑤 = 𝑦 → (♯‘𝑤) = (♯‘𝑦))
9	8	oveq1d 7426	. . . . 5 ⊢ (𝑤 = 𝑦 → ((♯‘𝑤)C𝑘) = ((♯‘𝑦)C𝑘))
10		pweq 4579	. . . . . . 7 ⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
11	10	rabeqdv 3437	. . . . . 6 ⊢ (𝑤 = 𝑦 → {𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘})
12	11	fveq2d 6886	. . . . 5 ⊢ (𝑤 = 𝑦 → (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘}))
13	9, 12	eqeq12d 2785	. . . 4 ⊢ (𝑤 = 𝑦 → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘})))
14	13	ralbidv 3194	. . 3 ⊢ (𝑤 = 𝑦 → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘})))
15		fveq2 6882	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (♯‘𝑤) = (♯‘(𝑦 ∪ {𝑧})))
16	15	oveq1d 7426	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((♯‘𝑤)C𝑘) = ((♯‘(𝑦 ∪ {𝑧}))C𝑘))
17		pweq 4579	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → 𝒫 𝑤 = 𝒫 (𝑦 ∪ {𝑧}))
18	17	rabeqdv 3437	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘})
19	18	fveq2d 6886	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘}))
20	16, 19	eqeq12d 2785	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘})))
21	20	ralbidv 3194	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘(𝑦 ∪ {𝑧}))C𝑘) = (♯‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝑘})))
22		fveq2 6882	. . . . . 6 ⊢ (𝑤 = 𝐴 → (♯‘𝑤) = (♯‘𝐴))
23	22	oveq1d 7426	. . . . 5 ⊢ (𝑤 = 𝐴 → ((♯‘𝑤)C𝑘) = ((♯‘𝐴)C𝑘))
24		pweq 4579	. . . . . . 7 ⊢ (𝑤 = 𝐴 → 𝒫 𝑤 = 𝒫 𝐴)
25	24	rabeqdv 3437	. . . . . 6 ⊢ (𝑤 = 𝐴 → {𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘})
26	25	fveq2d 6886	. . . . 5 ⊢ (𝑤 = 𝐴 → (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}))
27	23, 26	eqeq12d 2785	. . . 4 ⊢ (𝑤 = 𝐴 → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘})))
28	27	ralbidv 3194	. . 3 ⊢ (𝑤 = 𝐴 → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑤 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘})))
29		hash0 14403	. . . . . . . . . 10 ⊢ (♯‘∅) = 0
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → (♯‘∅) = 0)
31		elfz1eq 13563	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
32	30, 31	oveq12d 7429	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → ((♯‘∅)C𝑘) = (0C0))
33		0nn0 12519	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
34		bcn0 14346	. . . . . . . . 9 ⊢ (0 ∈ ℕ0 → (0C0) = 1)
35	33, 34	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
36	32, 35	eqtrdi 2820	. . . . . . 7 ⊢ (𝑘 ∈ (0...0) → ((♯‘∅)C𝑘) = 1)
37	31	eqcomd 2775	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...0) → 0 = 𝑘)
38		pw0 4780	. . . . . . . . . . . . . 14 ⊢ 𝒫 ∅ = {∅}
39	38	raleqi 3327	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} (♯‘𝑥) = 𝑘)
40		0ex 5272	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
41		fveq2 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
42	41, 29	eqtrdi 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
43	42	eqeq1d 2771	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → ((♯‘𝑥) = 𝑘 ↔ 0 = 𝑘))
44	40, 43	ralsn 4650	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ {∅} (♯‘𝑥) = 𝑘 ↔ 0 = 𝑘)
45	39, 44	bitri 278	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘 ↔ 0 = 𝑘)
46	37, 45	sylibr 237	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...0) → ∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘)
47		rabid2 3455	. . . . . . . . . . 11 ⊢ (𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} ↔ ∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘)
48	46, 47	sylibr 237	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...0) → 𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})
49	48, 38	eqtr3di 2819	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = {∅})
50	49	fveq2d 6886	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘{∅}))
51		hashsng 14405	. . . . . . . . 9 ⊢ (∅ ∈ V → (♯‘{∅}) = 1)
52	40, 51	ax-mp 5	. . . . . . . 8 ⊢ (♯‘{∅}) = 1
53	50, 52	eqtrdi 2820	. . . . . . 7 ⊢ (𝑘 ∈ (0...0) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = 1)
54	36, 53	eqtr4d 2807	. . . . . 6 ⊢ (𝑘 ∈ (0...0) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
55	54	adantl 486	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ 𝑘 ∈ (0...0)) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
56	29	oveq1i 7421	. . . . . 6 ⊢ ((♯‘∅)C𝑘) = (0C𝑘)
57		bcval3 14342	. . . . . . . 8 ⊢ ((0 ∈ ℕ0 ∧ 𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
58	33, 57	mp3an1 1474	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
59		id 23	. . . . . . . . . . . . . 14 ⊢ (0 = 𝑘 → 0 = 𝑘)
60		0z 12602	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
61		elfz3 13562	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℤ → 0 ∈ (0...0))
62	60, 61	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 0 ∈ (0...0)
63	59, 62	eqeltrrdi 2878	. . . . . . . . . . . . 13 ⊢ (0 = 𝑘 → 𝑘 ∈ (0...0))
64	63	con3i 155	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 ∈ (0...0) → ¬ 0 = 𝑘)
65	64	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ¬ 0 = 𝑘)
66	38	raleqi 3327	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} ¬ (♯‘𝑥) = 𝑘)
67	43	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘))
68	40, 67	ralsn 4650	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ {∅} ¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
69	66, 68	bitri 278	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
70	65, 69	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘)
71		rabeq0 4350	. . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = ∅ ↔ ∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘)
72	70, 71	sylibr 237	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = ∅)
73	72	fveq2d 6886	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘∅))
74	73, 29	eqtrdi 2820	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = 0)
75	58, 74	eqtr4d 2807	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
76	56, 75	eqtrid 2816	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
77	55, 76	pm2.61dan 824	. . . 4 ⊢ (𝑘 ∈ ℤ → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
78	77	rgen 3087	. . 3 ⊢ ∀𝑘 ∈ ℤ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘})
79		oveq2 7419	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((♯‘𝑦)C𝑘) = ((♯‘𝑦)C𝑗))
80		eqeq2 2781	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((♯‘𝑥) = 𝑘 ↔ (♯‘𝑥) = 𝑗))
81	80	rabbidv 3429	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗})
82		fveqeq2 6891	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑧) = 𝑗))
83	82	cbvrabv 3432	. . . . . . . 8 ⊢ {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗} = {𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}
84	81, 83	eqtrdi 2820	. . . . . . 7 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘} = {𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗})
85	84	fveq2d 6886	. . . . . 6 ⊢ (𝑘 = 𝑗 → (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))
86	79, 85	eqeq12d 2785	. . . . 5 ⊢ (𝑘 = 𝑗 → (((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗})))
87	86	cbvralvw 3249	. . . 4 ⊢ (∀𝑘 ∈ ℤ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))
88		simpll 778	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))) → 𝑦 ∈ Fin)
89		simplr 780	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))) → ¬ 𝑧 ∈ 𝑦)
90		simprr 784	. . . . . . . 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}))
91	83	fveq2i 6885	. . . . . . . . . 10 ⊢ (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗})
92	91	eqeq2i 2782	. . . . . . . . 9 ⊢ (((♯‘𝑦)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))
93	92	ralbii 3117	. . . . . . . 8 ⊢ (∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝑦 ∣ (♯‘𝑥) = 𝑗}) ↔ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))
94	90, 93	sylibr 237	. . . . . . 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}))
95		simprl 782	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ 𝒫 𝑦 ∣ (♯‘𝑧) = 𝑗}))) → 𝑘 ∈ ℤ)
96	88, 89, 94, 95	hashbclem 14489	. . . . . 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}))
97	96	expr 461	. . . . 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})))
98	97	ralrimdva 3171	. . . 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})))
99	87, 98	biimtrid 245	. . 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})))
100	7, 14, 21, 28, 78, 99	findcard2s 9150	. 2 ⊢ (𝐴 ∈ Fin → ∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}))
101		oveq2 7419	. . . 4 ⊢ (𝑘 = 𝐾 → ((♯‘𝐴)C𝑘) = ((♯‘𝐴)C𝐾))
102		eqeq2 2781	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((♯‘𝑥) = 𝑘 ↔ (♯‘𝑥) = 𝐾))
103	102	rabbidv 3429	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})
104	103	fveq2d 6886	. . . 4 ⊢ (𝑘 = 𝐾 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
105	101, 104	eqeq12d 2785	. . 3 ⊢ (𝑘 = 𝐾 → (((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})))
106	105	rspccva 3587	. 2 ⊢ ((∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑘}) ∧ 𝐾 ∈ ℤ) → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
107	100, 106	sylan 591	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐾 ∈ ℤ) → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3422  Vcvv 3461   ∪ cun 3909  ∅c0 4292  𝒫 cpw 4565  {csn 4592  ‘cfv 6537  (class class class)co 7411  Fincfn 8943  0cc0 11100  1c1 11101  ℕ₀cn0 12504  ℤcz 12591  ...cfz 13535  Ccbc 14338  ♯chash 14366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-oadd 8457  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9887  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-n0 12505  df-z 12592  df-uz 12863  df-rp 13017  df-fz 13536  df-seq 14038  df-fac 14310  df-bc 14339  df-hash 14367

This theorem is referenced by:  hashbc2  17066  sylow1lem1  19668  musum  27321  esplympl  33902  ballotlem1  34822  ballotlem2  34824  sticksstones5  42842