Definition df-ihash 11198

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 7207), 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 7211).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8904). 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 11197	. 2 class ♯
2		vx	. . . . . 6 setvar 𝑥
3		cz 9627	. . . . . 6 class ℤ
4	2	cv 1401	. . . . . . 7 class 𝑥
5		c1 8174	. . . . . . 7 class 1
6		caddc 8176	. . . . . . 7 class +
7	4, 5, 6	co 6079	. . . . . 6 class (𝑥 + 1)
8	2, 3, 7	cmpt 4190	. . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9		cc0 8173	. . . . 5 class 0
10	8, 9	cfrec 6655	. . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11		com 4735	. . . . . 6 class ω
12		cpnf 8351	. . . . . 6 class +∞
13	11, 12	cop 3711	. . . . 5 class ⟨ω, +∞⟩
14	13	csn 3708	. . . 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 7015	. . . . . . 7 class ≼
20	18, 4, 19	wbr 4128	. . . . . 6 wff 𝑦 ≼ 𝑥
21	11	csn 3708	. . . . . . 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 3933	. . . 4 class ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
25	2, 16, 24	cmpt 4190	. . 3 class (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
26	15, 25	ccom 4776	. 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 referenced by:  hashinfom  11200  hashennn  11202