Definition df-ihash 11015

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 7084), 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 7088).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8745). 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 11014	. 2 class ♯
2		vx	. . . . . 6 setvar 𝑥
3		cz 9462	. . . . . 6 class ℤ
4	2	cv 1394	. . . . . . 7 class 𝑥
5		c1 8016	. . . . . . 7 class 1
6		caddc 8018	. . . . . . 7 class +
7	4, 5, 6	co 6010	. . . . . 6 class (𝑥 + 1)
8	2, 3, 7	cmpt 4145	. . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9		cc0 8015	. . . . 5 class 0
10	8, 9	cfrec 6547	. . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11		com 4683	. . . . . 6 class ω
12		cpnf 8194	. . . . . 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 6899	. . . . . . 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 4724	. 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  11017  hashennn  11019