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Definition df-ihash 10534
 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. Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8356). 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}) ∣ 𝑦𝑥}))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-ihash
StepHypRef Expression
1 chash 10533 . 2 class
2 vx . . . . . 6 setvar 𝑥
3 cz 9066 . . . . . 6 class
42cv 1330 . . . . . . 7 class 𝑥
5 c1 7633 . . . . . . 7 class 1
6 caddc 7635 . . . . . . 7 class +
74, 5, 6co 5774 . . . . . 6 class (𝑥 + 1)
82, 3, 7cmpt 3989 . . . . 5 class (𝑥 ∈ ℤ ↦ (𝑥 + 1))
9 cc0 7632 . . . . 5 class 0
108, 9cfrec 6287 . . . 4 class frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
11 com 4504 . . . . . 6 class ω
12 cpnf 7809 . . . . . 6 class +∞
1311, 12cop 3530 . . . . 5 class ⟨ω, +∞⟩
1413csn 3527 . . . 4 class {⟨ω, +∞⟩}
1510, 14cun 3069 . . 3 class (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩})
16 cvv 2686 . . . 4 class V
17 vy . . . . . . . 8 setvar 𝑦
1817cv 1330 . . . . . . 7 class 𝑦
19 cdom 6633 . . . . . . 7 class
2018, 4, 19wbr 3929 . . . . . 6 wff 𝑦𝑥
2111csn 3527 . . . . . . 7 class {ω}
2211, 21cun 3069 . . . . . 6 class (ω ∪ {ω})
2320, 17, 22crab 2420 . . . . 5 class {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}
2423cuni 3736 . . . 4 class {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}
252, 16, 24cmpt 3989 . . 3 class (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥})
2615, 25ccom 4543 . 2 class ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
271, 26wceq 1331 1 wff ♯ = ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ {⟨ω, +∞⟩}) ∘ (𝑥 ∈ V ↦ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦𝑥}))
 Colors of variables: wff set class This definition is referenced by:  hashinfom  10536  hashennn  10538
 Copyright terms: Public domain W3C validator