Theorem 1arymaptfv

Description: The value of the mapping of unary (endo)functions. (Contributed by AV, 18-May-2024)

	Ref	Expression
Hypothesis	1arymaptfv.h	⊢ 𝐻 = ( ℎ ∈ ( 1 -aryF 𝑋 ) ↦ ( 𝑥 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) )
Assertion	1arymaptfv	⊢ ( 𝐹 ∈ ( 1 -aryF 𝑋 ) → ( 𝐻 ‘ 𝐹 ) = ( 𝑥 ∈ 𝑋 ↦ ( 𝐹 ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) )

Proof

Step	Hyp	Ref	Expression
1		1arymaptfv.h	⊢ 𝐻 = ( ℎ ∈ ( 1 -aryF 𝑋 ) ↦ ( 𝑥 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) )
2		fveq1	⊢ ( ℎ = 𝐹 → ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) = ( 𝐹 ‘ { ⟨ 0 , 𝑥 ⟩ } ) )
3	2	mpteq2dv	⊢ ( ℎ = 𝐹 → ( 𝑥 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) = ( 𝑥 ∈ 𝑋 ↦ ( 𝐹 ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) )
4		eqid	⊢ ( 0 ..^ 1 ) = ( 0 ..^ 1 )
5	4	naryrcl	⊢ ( ℎ ∈ ( 1 -aryF 𝑋 ) → ( 1 ∈ ℕ0 ∧ 𝑋 ∈ V ) )
6	5	simprd	⊢ ( ℎ ∈ ( 1 -aryF 𝑋 ) → 𝑋 ∈ V )
7	6	mptexd	⊢ ( ℎ ∈ ( 1 -aryF 𝑋 ) → ( 𝑥 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) ∈ V )
8	3 1 7	fvmpt3	⊢ ( 𝐹 ∈ ( 1 -aryF 𝑋 ) → ( 𝐻 ‘ 𝐹 ) = ( 𝑥 ∈ 𝑋 ↦ ( 𝐹 ‘ { ⟨ 0 , 𝑥 ⟩ } ) ) )