Definition df-ihash 11147

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 7168), 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 7172).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8861). 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 11146	. 2 class ♯
2		vx	. . . . . 6 setvar 𝑥
3		cz 9582	. . . . . 6 class ℤ
4	2	cv 1397	. . . . . . 7 class 𝑥
5		c1 8133	. . . . . . 7 class 1
6		caddc 8135	. . . . . . 7 class +
7	4, 5, 6	co 6052	. . . . . 6 class (𝑥 + 1)
8	2, 3, 7	cmpt 4173	. . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9		cc0 8132	. . . . 5 class 0
10	8, 9	cfrec 6623	. . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11		com 4714	. . . . . 6 class ω
12		cpnf 8310	. . . . . 6 class +∞
13	11, 12	cop 3694	. . . . 5 class ⟨ω, +∞⟩
14	13	csn 3691	. . . 4 class {⟨ω, +∞⟩}
15	10, 14	cun 3211	. . 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 6976	. . . . . . 7 class ≼
20	18, 4, 19	wbr 4111	. . . . . 6 wff 𝑦 ≼ 𝑥
21	11	csn 3691	. . . . . . 7 class {ω}
22	11, 21	cun 3211	. . . . . 6 class (ω ∪ {ω})
23	20, 17, 22	crab 2526	. . . . 5 class {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
24	23	cuni 3916	. . . 4 class ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
25	2, 16, 24	cmpt 4173	. . 3 class (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
26	15, 25	ccom 4755	. 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  11149  hashennn  11151