Definition df-ihash 11039

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 7098), 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 7102).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8762). 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 11038	. 2 class ♯
2		vx	. . . . . 6 setvar 𝑥
3		cz 9479	. . . . . 6 class ℤ
4	2	cv 1396	. . . . . . 7 class 𝑥
5		c1 8033	. . . . . . 7 class 1
6		caddc 8035	. . . . . . 7 class +
7	4, 5, 6	co 6018	. . . . . 6 class (𝑥 + 1)
8	2, 3, 7	cmpt 4150	. . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9		cc0 8032	. . . . 5 class 0
10	8, 9	cfrec 6556	. . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11		com 4688	. . . . . 6 class ω
12		cpnf 8211	. . . . . 6 class +∞
13	11, 12	cop 3672	. . . . 5 class ⟨ω, +∞⟩
14	13	csn 3669	. . . 4 class {⟨ω, +∞⟩}
15	10, 14	cun 3198	. . 3 class (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})
16		cvv 2802	. . . 4 class V
17		vy	. . . . . . . 8 setvar 𝑦
18	17	cv 1396	. . . . . . 7 class 𝑦
19		cdom 6908	. . . . . . 7 class ≼
20	18, 4, 19	wbr 4088	. . . . . 6 wff 𝑦 ≼ 𝑥
21	11	csn 3669	. . . . . . 7 class {ω}
22	11, 21	cun 3198	. . . . . 6 class (ω ∪ {ω})
23	20, 17, 22	crab 2514	. . . . 5 class {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
24	23	cuni 3893	. . . 4 class ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
25	2, 16, 24	cmpt 4150	. . 3 class (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
26	15, 25	ccom 4729	. 2 class ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
27	1, 26	wceq 1397	1 wff ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))

This definition is referenced by:  hashinfom  11041  hashennn  11043