Theorem hashfibc 11266

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 11262. (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 5693	. . . . . 6 ⊢ (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅))
2	1	oveq1d 6094	. . . . 5 ⊢ (𝑤 = ∅ → ((♯‘𝑤)C𝑘) = ((♯‘∅)C𝑘))
3		pweq 3691	. . . . . . . 8 ⊢ (𝑤 = ∅ → 𝒫 𝑤 = 𝒫 ∅)
4	3	ineq1d 3431	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝒫 𝑤 ∩ Fin) = (𝒫 ∅ ∩ Fin))
5	4	rabeqdv 2815	. . . . . 6 ⊢ (𝑤 = ∅ → {𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
6	5	fveq2d 5697	. . . . 5 ⊢ (𝑤 = ∅ → (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
7	2, 6	eqeq12d 2253	. . . 4 ⊢ (𝑤 = ∅ → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
8	7	ralbidv 2550	. . 3 ⊢ (𝑤 = ∅ → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
9		fveq2 5693	. . . . . 6 ⊢ (𝑤 = 𝑦 → (♯‘𝑤) = (♯‘𝑦))
10	9	oveq1d 6094	. . . . 5 ⊢ (𝑤 = 𝑦 → ((♯‘𝑤)C𝑘) = ((♯‘𝑦)C𝑘))
11		pweq 3691	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
12	11	ineq1d 3431	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝒫 𝑤 ∩ Fin) = (𝒫 𝑦 ∩ Fin))
13	12	rabeqdv 2815	. . . . . 6 ⊢ (𝑤 = 𝑦 → {𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
14	13	fveq2d 5697	. . . . 5 ⊢ (𝑤 = 𝑦 → (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
15	10, 14	eqeq12d 2253	. . . 4 ⊢ (𝑤 = 𝑦 → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
16	15	ralbidv 2550	. . 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 5693	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (♯‘𝑤) = (♯‘(𝑦 ∪ {𝑧})))
18	17	oveq1d 6094	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((♯‘𝑤)C𝑘) = ((♯‘(𝑦 ∪ {𝑧}))C𝑘))
19		pweq 3691	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → 𝒫 𝑤 = 𝒫 (𝑦 ∪ {𝑧}))
20	19	ineq1d 3431	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝒫 𝑤 ∩ Fin) = (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin))
21	20	rabeqdv 2815	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
22	21	fveq2d 5697	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
23	18, 22	eqeq12d 2253	. . . 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 2550	. . 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 5693	. . . . . 6 ⊢ (𝑤 = 𝐴 → (♯‘𝑤) = (♯‘𝐴))
26	25	oveq1d 6094	. . . . 5 ⊢ (𝑤 = 𝐴 → ((♯‘𝑤)C𝑘) = ((♯‘𝐴)C𝑘))
27		pweq 3691	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → 𝒫 𝑤 = 𝒫 𝐴)
28	27	ineq1d 3431	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝒫 𝑤 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
29	28	rabeqdv 2815	. . . . . 6 ⊢ (𝑤 = 𝐴 → {𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
30	29	fveq2d 5697	. . . . 5 ⊢ (𝑤 = 𝐴 → (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
31	26, 30	eqeq12d 2253	. . . 4 ⊢ (𝑤 = 𝐴 → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
32	31	ralbidv 2550	. . 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 9561	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
34		bcn0 11176	. . . . . . . . 9 ⊢ (0 ∈ ℕ0 → (0C0) = 1)
35	33, 34	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
36		hash0 11218	. . . . . . . . . 10 ⊢ (♯‘∅) = 0
37	36	a1i 9	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → (♯‘∅) = 0)
38		elfz1eq 10422	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
39	37, 38	oveq12d 6097	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → ((♯‘∅)C𝑘) = (0C0))
40	38	eqcomd 2244	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...0) → 0 = 𝑘)
41		pw0 3860	. . . . . . . . . . . . . . 15 ⊢ 𝒫 ∅ = {∅}
42	41	raleqi 2753	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} (♯‘𝑥) = 𝑘)
43		0ex 4258	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ V
44		fveq2 5693	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
45	44, 36	eqtrdi 2287	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
46	45	eqeq1d 2247	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → ((♯‘𝑥) = 𝑘 ↔ 0 = 𝑘))
47	43, 46	ralsn 3751	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . 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 2286	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...0) → {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = {∅})
53	52	fveq2d 5697	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘{∅}))
54		hashsng 11220	. . . . . . . . . 10 ⊢ (∅ ∈ V → (♯‘{∅}) = 1)
55	43, 54	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘{∅}) = 1
56	53, 55	eqtrdi 2287	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = 1)
57	35, 39, 56	3eqtr4a 2297	. . . . . . 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 6089	. . . . . . 7 ⊢ ((♯‘∅)C𝑘) = (0C𝑘)
60		bcval3 11172	. . . . . . . . 9 ⊢ ((0 ∈ ℕ0 ∧ 𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
61	33, 60	mp3an1 1365	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
62		id 19	. . . . . . . . . . . . . . 15 ⊢ (0 = 𝑘 → 0 = 𝑘)
63		0z 9638	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
64		elfz3 10421	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ ℤ → 0 ∈ (0...0))
65	63, 64	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ (0...0)
66	62, 65	eqeltrrdi 2330	. . . . . . . . . . . . . 14 ⊢ (0 = 𝑘 → 𝑘 ∈ (0...0))
67	66	con3i 641	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 ∈ (0...0) → ¬ 0 = 𝑘)
68	67	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ¬ 0 = 𝑘)
69	41	raleqi 2753	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} ¬ (♯‘𝑥) = 𝑘)
70	46	notbid 677	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → (¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘))
71	43, 70	ralsn 3751	. . . . . . . . . . . . 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 3552	. . . . . . . . . . 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 5697	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘∅))
77	76, 36	eqtrdi 2287	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = 0)
78	61, 77	eqtr4d 2274	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
79	59, 78	eqtrid 2283	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
80		0zd 9639	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ → 0 ∈ ℤ)
81		fzdcel 10427	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ 0 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑘 ∈ (0...0))
82	80, 80, 81	mpd3an23 1380	. . . . . . 7 ⊢ (𝑘 ∈ ℤ → DECID 𝑘 ∈ (0...0))
83		exmiddc 848	. . . . . . 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 810	. . . . 5 ⊢ (𝑘 ∈ ℤ → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
86		velsn 3725	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
87		0fi 7182	. . . . . . . . . . . 12 ⊢ ∅ ∈ Fin
88		eleq1 2301	. . . . . . . . . . . 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 2333	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ∅ → 𝑥 ∈ Fin)
92	91	ssriv 3252	. . . . . . . 8 ⊢ 𝒫 ∅ ⊆ Fin
93		dfss 3234	. . . . . . . 8 ⊢ (𝒫 ∅ ⊆ Fin ↔ 𝒫 ∅ = (𝒫 ∅ ∩ Fin))
94	92, 93	mpbi 145	. . . . . . 7 ⊢ 𝒫 ∅ = (𝒫 ∅ ∩ Fin)
95	94	rabeqi 2814	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘}
96	95	fveq2i 5696	. . . . 5 ⊢ (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
97	85, 96	eqtrdi 2287	. . . 4 ⊢ (𝑘 ∈ ℤ → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
98	97	rgen 2603	. . 3 ⊢ ∀𝑘 ∈ ℤ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
99		oveq2 6087	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((♯‘𝑦)C𝑘) = ((♯‘𝑦)C𝑗))
100		eqeq2 2248	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((♯‘𝑥) = 𝑘 ↔ (♯‘𝑥) = 𝑗))
101	100	rabbidv 2810	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗})
102		fveqeq2 5702	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑧) = 𝑗))
103	102	cbvrabv 2820	. . . . . . . 8 ⊢ {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗} = {𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}
104	101, 103	eqtrdi 2287	. . . . . . 7 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗})
105	104	fveq2d 5697	. . . . . 6 ⊢ (𝑘 = 𝑗 → (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))
106	99, 105	eqeq12d 2253	. . . . 5 ⊢ (𝑘 = 𝑗 → (((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗})))
107	106	cbvralvw 2790	. . . 4 ⊢ (∀𝑘 ∈ ℤ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))
108		simpll 531	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))) → 𝑦 ∈ Fin)
109		simplr 533	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))) → ¬ 𝑧 ∈ 𝑦)
110		simprr 537	. . . . . . . 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 5696	. . . . . . . . . 10 ⊢ (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗})
112	111	eqeq2i 2249	. . . . . . . . 9 ⊢ (((♯‘𝑦)C𝑗) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))
113	112	ralbii 2556	. . . . . . . 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 535	. . . . . . 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 11265	. . . . . 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 2630	. . . 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 7188	. 2 ⊢ (𝐴 ∈ Fin → ∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
121		oveq2 6087	. . . 4 ⊢ (𝑘 = 𝐾 → ((♯‘𝐴)C𝑘) = ((♯‘𝐴)C𝐾))
122		eqeq2 2248	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((♯‘𝑥) = 𝑘 ↔ (♯‘𝑥) = 𝐾))
123	122	rabbidv 2810	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾})
124	123	fveq2d 5697	. . . 4 ⊢ (𝑘 = 𝐾 → (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾}))
125	121, 124	eqeq12d 2253	. . 3 ⊢ (𝑘 = 𝐾 → (((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾})))
126	125	rspccva 2928	. 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 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  𝒫 cpw 3688  {csn 3708  ‘cfv 5375  (class class class)co 6079  Fincfn 7016  0cc0 8173  1c1 8174  ℕ₀cn0 9546  ℤcz 9627  ...cfz 10394  Ccbc 11168  ♯chash 11197

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-2o 6682  df-oadd 6685  df-er 6801  df-map 6918  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-seqfrec 10868  df-fac 11147  df-bc 11169  df-ihash 11198

This theorem is referenced by:  ballotfilem1  13203  ballotfilem2  13211