Theorem isclintop

Description: The predicate "is a closed (internal binary) operations for a set". (Contributed by FL, 2-Nov-2009) (Revised by AV, 20-Jan-2020)

	Ref	Expression
Assertion	isclintop	⊢ ( 𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘ 𝑀 ) ↔ ⚬ : ( 𝑀 × 𝑀 ) ⟶ 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		clintopval	⊢ ( 𝑀 ∈ 𝑉 → ( clIntOp ‘ 𝑀 ) = ( 𝑀 ↑m ( 𝑀 × 𝑀 ) ) )
2	1	eleq2d	⊢ ( 𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘ 𝑀 ) ↔ ⚬ ∈ ( 𝑀 ↑m ( 𝑀 × 𝑀 ) ) ) )
3		sqxpexg	⊢ ( 𝑀 ∈ 𝑉 → ( 𝑀 × 𝑀 ) ∈ V )
4		elmapg	⊢ ( ( 𝑀 ∈ 𝑉 ∧ ( 𝑀 × 𝑀 ) ∈ V ) → ( ⚬ ∈ ( 𝑀 ↑m ( 𝑀 × 𝑀 ) ) ↔ ⚬ : ( 𝑀 × 𝑀 ) ⟶ 𝑀 ) )
5	3 4	mpdan	⊢ ( 𝑀 ∈ 𝑉 → ( ⚬ ∈ ( 𝑀 ↑m ( 𝑀 × 𝑀 ) ) ↔ ⚬ : ( 𝑀 × 𝑀 ) ⟶ 𝑀 ) )
6	2 5	bitrd	⊢ ( 𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘ 𝑀 ) ↔ ⚬ : ( 𝑀 × 𝑀 ) ⟶ 𝑀 ) )