Definition df-ihash 11044

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 7103), 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 7107).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8767). 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 11043	. 2 class ♯
2		vx	. . . . . 6 setvar 𝑥
3		cz 9484	. . . . . 6 class ℤ
4	2	cv 1396	. . . . . . 7 class 𝑥
5		c1 8038	. . . . . . 7 class 1
6		caddc 8040	. . . . . . 7 class +
7	4, 5, 6	co 6023	. . . . . 6 class (𝑥 + 1)
8	2, 3, 7	cmpt 4151	. . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9		cc0 8037	. . . . 5 class 0
10	8, 9	cfrec 6561	. . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11		com 4690	. . . . . 6 class ω
12		cpnf 8216	. . . . . 6 class +∞
13	11, 12	cop 3673	. . . . 5 class ⟨ω, +∞⟩
14	13	csn 3670	. . . 4 class {⟨ω, +∞⟩}
15	10, 14	cun 3197	. . 3 class (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})
16		cvv 2801	. . . 4 class V
17		vy	. . . . . . . 8 setvar 𝑦
18	17	cv 1396	. . . . . . 7 class 𝑦
19		cdom 6913	. . . . . . 7 class ≼
20	18, 4, 19	wbr 4089	. . . . . 6 wff 𝑦 ≼ 𝑥
21	11	csn 3670	. . . . . . 7 class {ω}
22	11, 21	cun 3197	. . . . . 6 class (ω ∪ {ω})
23	20, 17, 22	crab 2513	. . . . 5 class {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
24	23	cuni 3894	. . . 4 class ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
25	2, 16, 24	cmpt 4151	. . 3 class (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
26	15, 25	ccom 4731	. 2 class ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
27	1, 26	wceq 1397	1 wff ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))

This definition is referenced by:  hashinfom  11046  hashennn  11048