Definition df-ihash 11238

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 7218), 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 7222).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8919). 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 11237	. 2 class ♯
2		vx	. . . . . 6 setvar 𝑥
3		cz 9656	. . . . . 6 class ℤ
4	2	cv 1401	. . . . . . 7 class 𝑥
5		c1 8187	. . . . . . 7 class 1
6		caddc 8189	. . . . . . 7 class +
7	4, 5, 6	co 6088	. . . . . 6 class (𝑥 + 1)
8	2, 3, 7	cmpt 4193	. . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9		cc0 8186	. . . . 5 class 0
10	8, 9	cfrec 6665	. . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11		com 4739	. . . . . 6 class ω
12		cpnf 8364	. . . . . 6 class +∞
13	11, 12	cop 3712	. . . . 5 class ⟨ω, +∞⟩
14	13	csn 3709	. . . 4 class {⟨ω, +∞⟩}
15	10, 14	cun 3218	. . 3 class (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})
16		cvv 2821	. . . 4 class V
17		vy	. . . . . . . 8 setvar 𝑦
18	17	cv 1401	. . . . . . 7 class 𝑦
19		cdom 7026	. . . . . . 7 class ≼
20	18, 4, 19	wbr 4131	. . . . . 6 wff 𝑦 ≼ 𝑥
21	11	csn 3709	. . . . . . 7 class {ω}
22	11, 21	cun 3218	. . . . . 6 class (ω ∪ {ω})
23	20, 17, 22	crab 2532	. . . . 5 class {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
24	23	cuni 3935	. . . 4 class ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
25	2, 16, 24	cmpt 4193	. . 3 class (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
26	15, 25	ccom 4780	. 2 class ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
27	1, 26	wceq 1402	1 wff ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))

This definition is used by:  hashinfom  11240  hashennn  11242