Theorem derangf

Description: The derangement number is a function from finite sets to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015)

	Ref	Expression
Hypothesis	derang.d	⊢ 𝐷 = ( 𝑥 ∈ Fin ↦ ( ♯ ‘ { 𝑓 ∣ ( 𝑓 : 𝑥 –1-1-onto→ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } ) )
Assertion	derangf	⊢ 𝐷 : Fin ⟶ ℕ0

Proof

Step	Hyp	Ref	Expression
1		derang.d	⊢ 𝐷 = ( 𝑥 ∈ Fin ↦ ( ♯ ‘ { 𝑓 ∣ ( 𝑓 : 𝑥 –1-1-onto→ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } ) )
2		deranglem	⊢ ( 𝑥 ∈ Fin → { 𝑓 ∣ ( 𝑓 : 𝑥 –1-1-onto→ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } ∈ Fin )
3		hashcl	⊢ ( { 𝑓 ∣ ( 𝑓 : 𝑥 –1-1-onto→ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } ∈ Fin → ( ♯ ‘ { 𝑓 ∣ ( 𝑓 : 𝑥 –1-1-onto→ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } ) ∈ ℕ0 )
4	2 3	syl	⊢ ( 𝑥 ∈ Fin → ( ♯ ‘ { 𝑓 ∣ ( 𝑓 : 𝑥 –1-1-onto→ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } ) ∈ ℕ0 )
5	1 4	fmpti	⊢ 𝐷 : Fin ⟶ ℕ0