Definition df-ihash 11167

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 7179), 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 7183).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8874). 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 11166	. 2 class ♯
2		vx	. . . . . 6 setvar 𝑥
3		cz 9597	. . . . . 6 class ℤ
4	2	cv 1397	. . . . . . 7 class 𝑥
5		c1 8144	. . . . . . 7 class 1
6		caddc 8146	. . . . . . 7 class +
7	4, 5, 6	co 6058	. . . . . 6 class (𝑥 + 1)
8	2, 3, 7	cmpt 4176	. . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9		cc0 8143	. . . . 5 class 0
10	8, 9	cfrec 6634	. . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11		com 4717	. . . . . 6 class ω
12		cpnf 8321	. . . . . 6 class +∞
13	11, 12	cop 3697	. . . . 5 class ⟨ω, +∞⟩
14	13	csn 3694	. . . 4 class {⟨ω, +∞⟩}
15	10, 14	cun 3212	. . 3 class (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})
16		cvv 2815	. . . 4 class V
17		vy	. . . . . . . 8 setvar 𝑦
18	17	cv 1397	. . . . . . 7 class 𝑦
19		cdom 6987	. . . . . . 7 class ≼
20	18, 4, 19	wbr 4114	. . . . . 6 wff 𝑦 ≼ 𝑥
21	11	csn 3694	. . . . . . 7 class {ω}
22	11, 21	cun 3212	. . . . . 6 class (ω ∪ {ω})
23	20, 17, 22	crab 2526	. . . . 5 class {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
24	23	cuni 3919	. . . 4 class ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
25	2, 16, 24	cmpt 4176	. . 3 class (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
26	15, 25	ccom 4758	. 2 class ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
27	1, 26	wceq 1398	1 wff ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))

This definition is referenced by:  hashinfom  11169  hashennn  11171