Theorem hashfibc 11200

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 11196. (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 5669	. . . . . 6 ⊢ (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅))
2	1	oveq1d 6064	. . . . 5 ⊢ (𝑤 = ∅ → ((♯‘𝑤)C𝑘) = ((♯‘∅)C𝑘))
3		pweq 3671	. . . . . . . 8 ⊢ (𝑤 = ∅ → 𝒫 𝑤 = 𝒫 ∅)
4	3	ineq1d 3420	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝒫 𝑤 ∩ Fin) = (𝒫 ∅ ∩ Fin))
5	4	rabeqdv 2806	. . . . . 6 ⊢ (𝑤 = ∅ → {𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
6	5	fveq2d 5673	. . . . 5 ⊢ (𝑤 = ∅ → (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
7	2, 6	eqeq12d 2247	. . . 4 ⊢ (𝑤 = ∅ → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
8	7	ralbidv 2542	. . 3 ⊢ (𝑤 = ∅ → (∀𝑘 ∈ ℤ ((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
9		fveq2 5669	. . . . . 6 ⊢ (𝑤 = 𝑦 → (♯‘𝑤) = (♯‘𝑦))
10	9	oveq1d 6064	. . . . 5 ⊢ (𝑤 = 𝑦 → ((♯‘𝑤)C𝑘) = ((♯‘𝑦)C𝑘))
11		pweq 3671	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
12	11	ineq1d 3420	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝒫 𝑤 ∩ Fin) = (𝒫 𝑦 ∩ Fin))
13	12	rabeqdv 2806	. . . . . 6 ⊢ (𝑤 = 𝑦 → {𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
14	13	fveq2d 5673	. . . . 5 ⊢ (𝑤 = 𝑦 → (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
15	10, 14	eqeq12d 2247	. . . 4 ⊢ (𝑤 = 𝑦 → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
16	15	ralbidv 2542	. . 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 5669	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (♯‘𝑤) = (♯‘(𝑦 ∪ {𝑧})))
18	17	oveq1d 6064	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((♯‘𝑤)C𝑘) = ((♯‘(𝑦 ∪ {𝑧}))C𝑘))
19		pweq 3671	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → 𝒫 𝑤 = 𝒫 (𝑦 ∪ {𝑧}))
20	19	ineq1d 3420	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝒫 𝑤 ∩ Fin) = (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin))
21	20	rabeqdv 2806	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
22	21	fveq2d 5673	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 (𝑦 ∪ {𝑧}) ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
23	18, 22	eqeq12d 2247	. . . 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 2542	. . 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 5669	. . . . . 6 ⊢ (𝑤 = 𝐴 → (♯‘𝑤) = (♯‘𝐴))
26	25	oveq1d 6064	. . . . 5 ⊢ (𝑤 = 𝐴 → ((♯‘𝑤)C𝑘) = ((♯‘𝐴)C𝑘))
27		pweq 3671	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → 𝒫 𝑤 = 𝒫 𝐴)
28	27	ineq1d 3420	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝒫 𝑤 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
29	28	rabeqdv 2806	. . . . . 6 ⊢ (𝑤 = 𝐴 → {𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
30	29	fveq2d 5673	. . . . 5 ⊢ (𝑤 = 𝐴 → (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
31	26, 30	eqeq12d 2247	. . . 4 ⊢ (𝑤 = 𝐴 → (((♯‘𝑤)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑤 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘})))
32	31	ralbidv 2542	. . 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 9507	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
34		bcn0 11113	. . . . . . . . 9 ⊢ (0 ∈ ℕ0 → (0C0) = 1)
35	33, 34	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
36		hash0 11154	. . . . . . . . . 10 ⊢ (♯‘∅) = 0
37	36	a1i 9	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → (♯‘∅) = 0)
38		elfz1eq 10365	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
39	37, 38	oveq12d 6067	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → ((♯‘∅)C𝑘) = (0C0))
40	38	eqcomd 2238	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...0) → 0 = 𝑘)
41		pw0 3840	. . . . . . . . . . . . . . 15 ⊢ 𝒫 ∅ = {∅}
42	41	raleqi 2744	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝒫 ∅(♯‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} (♯‘𝑥) = 𝑘)
43		0ex 4236	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ V
44		fveq2 5669	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
45	44, 36	eqtrdi 2281	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
46	45	eqeq1d 2241	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → ((♯‘𝑥) = 𝑘 ↔ 0 = 𝑘))
47	43, 46	ralsn 3731	. . . . . . . . . . . . . 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 2720	. . . . . . . . . . . 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 2280	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...0) → {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = {∅})
53	52	fveq2d 5673	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘{∅}))
54		hashsng 11156	. . . . . . . . . 10 ⊢ (∅ ∈ V → (♯‘{∅}) = 1)
55	43, 54	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘{∅}) = 1
56	53, 55	eqtrdi 2281	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = 1)
57	35, 39, 56	3eqtr4a 2291	. . . . . . 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 6059	. . . . . . 7 ⊢ ((♯‘∅)C𝑘) = (0C𝑘)
60		bcval3 11109	. . . . . . . . 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 9584	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
64		elfz3 10364	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ ℤ → 0 ∈ (0...0))
65	63, 64	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ (0...0)
66	62, 65	eqeltrrdi 2324	. . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝒫 ∅ ¬ (♯‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} ¬ (♯‘𝑥) = 𝑘)
70	46	notbid 673	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → (¬ (♯‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘))
71	43, 70	ralsn 3731	. . . . . . . . . . . . 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 3537	. . . . . . . . . . 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 5673	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘∅))
77	76, 36	eqtrdi 2281	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = 0)
78	61, 77	eqtr4d 2268	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
79	59, 78	eqtrid 2277	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}))
80		0zd 9585	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ → 0 ∈ ℤ)
81		fzdcel 10370	. . . . . . . 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 3705	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
87		0fi 7140	. . . . . . . . . . . 12 ⊢ ∅ ∈ Fin
88		eleq1 2295	. . . . . . . . . . . 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 2327	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ∅ → 𝑥 ∈ Fin)
92	91	ssriv 3241	. . . . . . . 8 ⊢ 𝒫 ∅ ⊆ Fin
93		dfss 3224	. . . . . . . 8 ⊢ (𝒫 ∅ ⊆ Fin ↔ 𝒫 ∅ = (𝒫 ∅ ∩ Fin))
94	92, 93	mpbi 145	. . . . . . 7 ⊢ 𝒫 ∅ = (𝒫 ∅ ∩ Fin)
95	94	rabeqi 2805	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘}
96	95	fveq2i 5672	. . . . 5 ⊢ (♯‘{𝑥 ∈ 𝒫 ∅ ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
97	85, 96	eqtrdi 2281	. . . 4 ⊢ (𝑘 ∈ ℤ → ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
98	97	rgen 2595	. . 3 ⊢ ∀𝑘 ∈ ℤ ((♯‘∅)C𝑘) = (♯‘{𝑥 ∈ (𝒫 ∅ ∩ Fin) ∣ (♯‘𝑥) = 𝑘})
99		oveq2 6057	. . . . . 6 ⊢ (𝑘 = 𝑗 → ((♯‘𝑦)C𝑘) = ((♯‘𝑦)C𝑗))
100		eqeq2 2242	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((♯‘𝑥) = 𝑘 ↔ (♯‘𝑥) = 𝑗))
101	100	rabbidv 2801	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗})
102		fveqeq2 5678	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑧) = 𝑗))
103	102	cbvrabv 2811	. . . . . . . 8 ⊢ {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗} = {𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}
104	101, 103	eqtrdi 2281	. . . . . . 7 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗})
105	104	fveq2d 5673	. . . . . 6 ⊢ (𝑘 = 𝑗 → (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))
106	99, 105	eqeq12d 2247	. . . . 5 ⊢ (𝑘 = 𝑗 → (((♯‘𝑦)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗})))
107	106	cbvralvw 2781	. . . 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 5672	. . . . . . . . . 10 ⊢ (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗})
112	111	eqeq2i 2243	. . . . . . . . 9 ⊢ (((♯‘𝑦)C𝑗) = (♯‘{𝑥 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝑦)C𝑗) = (♯‘{𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∣ (♯‘𝑧) = 𝑗}))
113	112	ralbii 2548	. . . . . . . 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 11199	. . . . . 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 2622	. . . 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 7146	. 2 ⊢ (𝐴 ∈ Fin → ∀𝑘 ∈ ℤ ((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}))
121		oveq2 6057	. . . 4 ⊢ (𝑘 = 𝐾 → ((♯‘𝐴)C𝑘) = ((♯‘𝐴)C𝐾))
122		eqeq2 2242	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((♯‘𝑥) = 𝑘 ↔ (♯‘𝑥) = 𝐾))
123	122	rabbidv 2801	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘} = {𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾})
124	123	fveq2d 5673	. . . 4 ⊢ (𝑘 = 𝐾 → (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾}))
125	121, 124	eqeq12d 2247	. . 3 ⊢ (𝑘 = 𝐾 → (((♯‘𝐴)C𝑘) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝑘}) ↔ ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∣ (♯‘𝑥) = 𝐾})))
126	125	rspccva 2919	. 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 2203  ∀wral 2520  {crab 2524  Vcvv 2812   ∪ cun 3208   ∩ cin 3209   ⊆ wss 3210  ∅c0 3507  𝒫 cpw 3668  {csn 3688  ‘cfv 5351  (class class class)co 6049  Fincfn 6974  0cc0 8123  1c1 8124  ℕ₀cn0 9492  ℤcz 9573  ...cfz 10338  Ccbc 11105  ♯chash 11133

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-map 6883  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-n0 9493  df-z 9574  df-uz 9850  df-q 9948  df-rp 9983  df-fz 10339  df-seqfrec 10806  df-fac 11084  df-bc 11106  df-ihash 11134

This theorem is referenced by: (None)