Definition df-ihash 11134

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 7165), 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 7169).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8852). 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 11133	. 2 class ♯
2		vx	. . . . . 6 setvar 𝑥
3		cz 9573	. . . . . 6 class ℤ
4	2	cv 1397	. . . . . . 7 class 𝑥
5		c1 8124	. . . . . . 7 class 1
6		caddc 8126	. . . . . . 7 class +
7	4, 5, 6	co 6049	. . . . . 6 class (𝑥 + 1)
8	2, 3, 7	cmpt 4170	. . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9		cc0 8123	. . . . 5 class 0
10	8, 9	cfrec 6620	. . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11		com 4711	. . . . . 6 class ω
12		cpnf 8301	. . . . . 6 class +∞
13	11, 12	cop 3691	. . . . 5 class ⟨ω, +∞⟩
14	13	csn 3688	. . . 4 class {⟨ω, +∞⟩}
15	10, 14	cun 3208	. . 3 class (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})
16		cvv 2812	. . . 4 class V
17		vy	. . . . . . . 8 setvar 𝑦
18	17	cv 1397	. . . . . . 7 class 𝑦
19		cdom 6973	. . . . . . 7 class ≼
20	18, 4, 19	wbr 4108	. . . . . 6 wff 𝑦 ≼ 𝑥
21	11	csn 3688	. . . . . . 7 class {ω}
22	11, 21	cun 3208	. . . . . 6 class (ω ∪ {ω})
23	20, 17, 22	crab 2524	. . . . 5 class {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
24	23	cuni 3913	. . . 4 class ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
25	2, 16, 24	cmpt 4170	. . 3 class (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
26	15, 25	ccom 4752	. 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  11136  hashennn  11138