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Definition df-hash 14366
Description: Define the set size function , which gives the cardinality of a finite set as a member of 0, and assigns all infinite sets the value +∞. For example, (♯‘{0, 1, 2}) = 3 (ex-hash 30481). (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
StepHypRef Expression
1 chash 14365 . 2 class
2 vx . . . . . . 7 setvar 𝑥
3 cvv 3477 . . . . . . 7 class V
42cv 1535 . . . . . . . 8 class 𝑥
5 c1 11153 . . . . . . . 8 class 1
6 caddc 11155 . . . . . . . 8 class +
74, 5, 6co 7430 . . . . . . 7 class (𝑥 + 1)
82, 3, 7cmpt 5230 . . . . . 6 class (𝑥 ∈ V ↦ (𝑥 + 1))
9 cc0 11152 . . . . . 6 class 0
108, 9crdg 8447 . . . . 5 class rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0)
11 com 7886 . . . . 5 class ω
1210, 11cres 5690 . . . 4 class (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
13 ccrd 9972 . . . 4 class card
1412, 13ccom 5692 . . 3 class ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card)
15 cfn 8983 . . . . 5 class Fin
163, 15cdif 3959 . . . 4 class (V ∖ Fin)
17 cpnf 11289 . . . . 5 class +∞
1817csn 4630 . . . 4 class {+∞}
1916, 18cxp 5686 . . 3 class ((V ∖ Fin) × {+∞})
2014, 19cun 3960 . 2 class (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))
211, 20wceq 1536 1 wff ♯ = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))
Colors of variables: wff setvar class
This definition is referenced by:  hashgval  14368  hashinf  14370  hashfxnn0  14372
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