Definition df-ihash 11217

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 7213), 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 7217).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8911). 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 11216	. 2 class ♯
2		vx	. . . . . 6 setvar 𝑥
3		cz 9646	. . . . . 6 class ℤ
4	2	cv 1401	. . . . . . 7 class 𝑥
5		c1 8180	. . . . . . 7 class 1
6		caddc 8182	. . . . . . 7 class +
7	4, 5, 6	co 6085	. . . . . 6 class (𝑥 + 1)
8	2, 3, 7	cmpt 4192	. . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9		cc0 8179	. . . . 5 class 0
10	8, 9	cfrec 6661	. . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11		com 4737	. . . . . 6 class ω
12		cpnf 8357	. . . . . 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 7021	. . . . . . 7 class ≼
20	18, 4, 19	wbr 4130	. . . . . 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 4192	. . 3 class (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
26	15, 25	ccom 4778	. 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  11219  hashennn  11221