Definition df-hash 14347

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 (ex-hash 30380). (Contributed by Paul Chapman, 22-Jun-2011.)

Assertion

Ref	Expression
df-hash	⊢ ♯ = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))

Detailed syntax breakdown of Definition df-hash

Step	Hyp	Ref	Expression
1		chash 14346	. 2 class ♯
2		vx	. . . . . . 7 setvar 𝑥
3		cvv 3459	. . . . . . 7 class V
4	2	cv 1539	. . . . . . . 8 class 𝑥
5		c1 11128	. . . . . . . 8 class 1
6		caddc 11130	. . . . . . . 8 class +
7	4, 5, 6	co 7403	. . . . . . 7 class (𝑥 + 1)
8	2, 3, 7	cmpt 5201	. . . . . 6 class (𝑥 ∈ V ↦ (𝑥 + 1))
9		cc0 11127	. . . . . 6 class 0
10	8, 9	crdg 8421	. . . . 5 class rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0)
11		com 7859	. . . . 5 class ω
12	10, 11	cres 5656	. . . 4 class (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
13		ccrd 9947	. . . 4 class card
14	12, 13	ccom 5658	. . 3 class ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card)
15		cfn 8957	. . . . 5 class Fin
16	3, 15	cdif 3923	. . . 4 class (V ∖ Fin)
17		cpnf 11264	. . . . 5 class +∞
18	17	csn 4601	. . . 4 class {+∞}
19	16, 18	cxp 5652	. . 3 class ((V ∖ Fin) × {+∞})
20	14, 19	cun 3924	. 2 class (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))
21	1, 20	wceq 1540	1 wff ♯ = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))

This definition is referenced by:  hashgval  14349  hashinf  14351  hashfxnn0  14353