Definition df-ihash 10965

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 7036), 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 7040).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8697). 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 10964	. 2 class ♯
2		vx	. . . . . 6 setvar 𝑥
3		cz 9414	. . . . . 6 class ℤ
4	2	cv 1374	. . . . . . 7 class 𝑥
5		c1 7968	. . . . . . 7 class 1
6		caddc 7970	. . . . . . 7 class +
7	4, 5, 6	co 5974	. . . . . 6 class (𝑥 + 1)
8	2, 3, 7	cmpt 4124	. . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9		cc0 7967	. . . . 5 class 0
10	8, 9	cfrec 6506	. . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11		com 4659	. . . . . 6 class ω
12		cpnf 8146	. . . . . 6 class +∞
13	11, 12	cop 3649	. . . . 5 class ⟨ω, +∞⟩
14	13	csn 3646	. . . 4 class {⟨ω, +∞⟩}
15	10, 14	cun 3175	. . 3 class (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})
16		cvv 2779	. . . 4 class V
17		vy	. . . . . . . 8 setvar 𝑦
18	17	cv 1374	. . . . . . 7 class 𝑦
19		cdom 6856	. . . . . . 7 class ≼
20	18, 4, 19	wbr 4062	. . . . . 6 wff 𝑦 ≼ 𝑥
21	11	csn 3646	. . . . . . 7 class {ω}
22	11, 21	cun 3175	. . . . . 6 class (ω ∪ {ω})
23	20, 17, 22	crab 2492	. . . . 5 class {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
24	23	cuni 3867	. . . 4 class ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
25	2, 16, 24	cmpt 4124	. . 3 class (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
26	15, 25	ccom 4700	. 2 class ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
27	1, 26	wceq 1375	1 wff ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))

This definition is referenced by:  hashinfom  10967  hashennn  10969