Definition df-ihash 11031

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 7093), 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 7097).

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8755). 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 11030	. 2 class ♯
2		vx	. . . . . 6 setvar 𝑥
3		cz 9472	. . . . . 6 class ℤ
4	2	cv 1394	. . . . . . 7 class 𝑥
5		c1 8026	. . . . . . 7 class 1
6		caddc 8028	. . . . . . 7 class +
7	4, 5, 6	co 6013	. . . . . 6 class (𝑥 + 1)
8	2, 3, 7	cmpt 4148	. . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9		cc0 8025	. . . . 5 class 0
10	8, 9	cfrec 6551	. . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11		com 4686	. . . . . 6 class ω
12		cpnf 8204	. . . . . 6 class +∞
13	11, 12	cop 3670	. . . . 5 class ⟨ω, +∞⟩
14	13	csn 3667	. . . 4 class {⟨ω, +∞⟩}
15	10, 14	cun 3196	. . 3 class (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})
16		cvv 2800	. . . 4 class V
17		vy	. . . . . . . 8 setvar 𝑦
18	17	cv 1394	. . . . . . 7 class 𝑦
19		cdom 6903	. . . . . . 7 class ≼
20	18, 4, 19	wbr 4086	. . . . . 6 wff 𝑦 ≼ 𝑥
21	11	csn 3667	. . . . . . 7 class {ω}
22	11, 21	cun 3196	. . . . . 6 class (ω ∪ {ω})
23	20, 17, 22	crab 2512	. . . . 5 class {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
24	23	cuni 3891	. . . 4 class ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}
25	2, 16, 24	cmpt 4148	. . 3 class (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥})
26	15, 25	ccom 4727	. 2 class ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))
27	1, 26	wceq 1395	1 wff ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝑥}))

This definition is referenced by:  hashinfom  11033  hashennn  11035