Theorem 2arympt

Description: A binary (endo)function in maps-to notation. (Contributed by AV, 20-May-2024)

	Ref	Expression
Hypothesis	2arympt.f	⊢ 𝐹 = ( 𝑥 ∈ ( 𝑋 ↑m { 0 , 1 } ) ↦ ( ( 𝑥 ‘ 0 ) 𝑂 ( 𝑥 ‘ 1 ) ) )
Assertion	2arympt	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑂 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ) → 𝐹 ∈ ( 2 -aryF 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		2arympt.f	⊢ 𝐹 = ( 𝑥 ∈ ( 𝑋 ↑m { 0 , 1 } ) ↦ ( ( 𝑥 ‘ 0 ) 𝑂 ( 𝑥 ‘ 1 ) ) )
2		simplr	⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑂 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ) ∧ 𝑥 ∈ ( 𝑋 ↑m { 0 , 1 } ) ) → 𝑂 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 )
3		elmapi	⊢ ( 𝑥 ∈ ( 𝑋 ↑m { 0 , 1 } ) → 𝑥 : { 0 , 1 } ⟶ 𝑋 )
4		0elpr01	⊢ 0 ∈ { 0 , 1 }
5	4	a1i	⊢ ( 𝑥 ∈ ( 𝑋 ↑m { 0 , 1 } ) → 0 ∈ { 0 , 1 } )
6	3 5	ffvelcdmd	⊢ ( 𝑥 ∈ ( 𝑋 ↑m { 0 , 1 } ) → ( 𝑥 ‘ 0 ) ∈ 𝑋 )
7	6	adantl	⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑂 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ) ∧ 𝑥 ∈ ( 𝑋 ↑m { 0 , 1 } ) ) → ( 𝑥 ‘ 0 ) ∈ 𝑋 )
8		1elpr01	⊢ 1 ∈ { 0 , 1 }
9	8	a1i	⊢ ( 𝑥 ∈ ( 𝑋 ↑m { 0 , 1 } ) → 1 ∈ { 0 , 1 } )
10	3 9	ffvelcdmd	⊢ ( 𝑥 ∈ ( 𝑋 ↑m { 0 , 1 } ) → ( 𝑥 ‘ 1 ) ∈ 𝑋 )
11	10	adantl	⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑂 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ) ∧ 𝑥 ∈ ( 𝑋 ↑m { 0 , 1 } ) ) → ( 𝑥 ‘ 1 ) ∈ 𝑋 )
12	2 7 11	fovcdmd	⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑂 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ) ∧ 𝑥 ∈ ( 𝑋 ↑m { 0 , 1 } ) ) → ( ( 𝑥 ‘ 0 ) 𝑂 ( 𝑥 ‘ 1 ) ) ∈ 𝑋 )
13	12 1	fmptd	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑂 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ) → 𝐹 : ( 𝑋 ↑m { 0 , 1 } ) ⟶ 𝑋 )
14		2aryfvalel	⊢ ( 𝑋 ∈ 𝑉 → ( 𝐹 ∈ ( 2 -aryF 𝑋 ) ↔ 𝐹 : ( 𝑋 ↑m { 0 , 1 } ) ⟶ 𝑋 ) )
15	14	adantr	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑂 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ) → ( 𝐹 ∈ ( 2 -aryF 𝑋 ) ↔ 𝐹 : ( 𝑋 ↑m { 0 , 1 } ) ⟶ 𝑋 ) )
16	13 15	mpbird	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑂 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ) → 𝐹 ∈ ( 2 -aryF 𝑋 ) )