Theorem 1arymaptf

Description: The mapping of unary (endo)functions is a function into the set of endofunctions. (Contributed by AV, 18-May-2024)

	Ref	Expression
Hypothesis	1arymaptfv.h	⊢ 𝐻 = ( ℎ ∈ ( 1 -aryF 𝑋 ) ↦ ( 𝑥 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) )
Assertion	1arymaptf	⊢ ( 𝑋 ∈ 𝑉 → 𝐻 : ( 1 -aryF 𝑋 ) ⟶ ( 𝑋 ↑m 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		1arymaptfv.h	⊢ 𝐻 = ( ℎ ∈ ( 1 -aryF 𝑋 ) ↦ ( 𝑥 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) )
2		fv1arycl	⊢ ( ( ℎ ∈ ( 1 -aryF 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) → ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) ∈ 𝑋 )
3	2	adantll	⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ ℎ ∈ ( 1 -aryF 𝑋 ) ) ∧ 𝑥 ∈ 𝑋 ) → ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) ∈ 𝑋 )
4	3	fmpttd	⊢ ( ( 𝑋 ∈ 𝑉 ∧ ℎ ∈ ( 1 -aryF 𝑋 ) ) → ( 𝑥 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) : 𝑋 ⟶ 𝑋 )
5		simpl	⊢ ( ( 𝑋 ∈ 𝑉 ∧ ℎ ∈ ( 1 -aryF 𝑋 ) ) → 𝑋 ∈ 𝑉 )
6	5 5	elmapd	⊢ ( ( 𝑋 ∈ 𝑉 ∧ ℎ ∈ ( 1 -aryF 𝑋 ) ) → ( ( 𝑥 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) ∈ ( 𝑋 ↑m 𝑋 ) ↔ ( 𝑥 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) : 𝑋 ⟶ 𝑋 ) )
7	4 6	mpbird	⊢ ( ( 𝑋 ∈ 𝑉 ∧ ℎ ∈ ( 1 -aryF 𝑋 ) ) → ( 𝑥 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) ∈ ( 𝑋 ↑m 𝑋 ) )
8	7 1	fmptd	⊢ ( 𝑋 ∈ 𝑉 → 𝐻 : ( 1 -aryF 𝑋 ) ⟶ ( 𝑋 ↑m 𝑋 ) )