Theorem 2arymaptfv

Description: The value of the mapping of binary (endo)functions. (Contributed by AV, 21-May-2024)

	Ref	Expression
Hypothesis	2arymaptf.h	⊢ 𝐻 = ( ℎ ∈ ( 2 -aryF 𝑋 ) ↦ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑦 ⟩ } ) ) )
Assertion	2arymaptfv	⊢ ( 𝐹 ∈ ( 2 -aryF 𝑋 ) → ( 𝐻 ‘ 𝐹 ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝐹 ‘ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑦 ⟩ } ) ) )

Proof

Step	Hyp	Ref	Expression
1		2arymaptf.h	⊢ 𝐻 = ( ℎ ∈ ( 2 -aryF 𝑋 ) ↦ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑦 ⟩ } ) ) )
2		fveq1	⊢ ( ℎ = 𝐹 → ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑦 ⟩ } ) = ( 𝐹 ‘ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑦 ⟩ } ) )
3	2	mpoeq3dv	⊢ ( ℎ = 𝐹 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑦 ⟩ } ) ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝐹 ‘ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑦 ⟩ } ) ) )
4		eqid	⊢ ( 0 ..^ 2 ) = ( 0 ..^ 2 )
5	4	naryrcl	⊢ ( ℎ ∈ ( 2 -aryF 𝑋 ) → ( 2 ∈ ℕ0 ∧ 𝑋 ∈ V ) )
6		mpoexga	⊢ ( ( 𝑋 ∈ V ∧ 𝑋 ∈ V ) → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑦 ⟩ } ) ) ∈ V )
7	6	anidms	⊢ ( 𝑋 ∈ V → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑦 ⟩ } ) ) ∈ V )
8	5 7	simpl2im	⊢ ( ℎ ∈ ( 2 -aryF 𝑋 ) → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( ℎ ‘ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑦 ⟩ } ) ) ∈ V )
9	3 1 8	fvmpt3	⊢ ( 𝐹 ∈ ( 2 -aryF 𝑋 ) → ( 𝐻 ‘ 𝐹 ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝐹 ‘ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑦 ⟩ } ) ) )