Definition df-ihash 11006

Description: Define the set size function ♯, which gives the cardinality of a finite set as a member of ℕ₀, and assigns all infinite sets the value +∞. For example, (♯‘{0, 1, 2}) = 3.

Since we don't know that an arbitrary set is either finite or infinite (by inffiexmid 7076), the behavior beyond finite sets is not as useful as it might appear. For example, we wouldn't expect to be able to define this function in a meaningful way on 𝒫 1_o, which cannot be shown to be finite (per pw1fin 7080).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8737). We adopt the former notation from Corollary 8.2.4 of [AczelRathjen], p. 80 (although that work only defines it for finite sets).

This definition (in terms of ∪ and ≼) is not taken directly from the literature, but for finite sets should be equivalent to the conventional definition that the size of a finite set is the unique natural number which is equinumerous to the given set. (Contributed by Jim Kingdon, 19-Feb-2022.)

Assertion

Ref	Expression
df-ihash	⊢ ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-ihash

Step	Hyp	Ref	Expression
1		chash 11005	. 2 class ♯
2		vx	. . . . . 6 setvar 𝑥
3		cz 9454	. . . . . 6 class ℤ
4	2	cv 1394	. . . . . . 7 class 𝑥
5		c1 8008	. . . . . . 7 class 1
6		caddc 8010	. . . . . . 7 class +
7	4, 5, 6	co 6007	. . . . . 6 class (𝑥 + 1)
8	2, 3, 7	cmpt 4145	. . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9		cc0 8007	. . . . 5 class 0
10	8, 9	cfrec 6542	. . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11		com 4682	. . . . . 6 class ω
12		cpnf 8186	. . . . . 6 class +∞
13	11, 12	cop 3669	. . . . 5 class ⟨ω, +∞⟩
14	13	csn 3666	. . . 4 class {⟨ω, +∞⟩}
15	10, 14	cun 3195	. . 3 class (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})
16		cvv 2799	. . . 4 class V
17		vy	. . . . . . . 8 setvar 𝑦
18	17	cv 1394	. . . . . . 7 class 𝑦
19		cdom 6894	. . . . . . 7 class ≼
20	18, 4, 19	wbr 4083	. . . . . 6 wff 𝑦 ≼ 𝑥
21	11	csn 3666	. . . . . . 7 class {ω}
22	11, 21	cun 3195	. . . . . 6 class (ω ∪ {ω})
23	20, 17, 22	crab 2512	. . . . 5 class {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
24	23	cuni 3888	. . . 4 class ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
25	2, 16, 24	cmpt 4145	. . 3 class (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
26	15, 25	ccom 4723	. 2 class ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
27	1, 26	wceq 1395	1 wff ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))

This definition is referenced by:  hashinfom  11008  hashennn  11010