Theorem relexp1idm

Description: Repeated raising a relation to the first power is idempotent. (Contributed by RP, 12-Jun-2020)

	Ref	Expression
Assertion	relexp1idm	⊢ ( 𝑅 ∈ 𝑉 → ( ( 𝑅 ↑𝑟 1 ) ↑𝑟 1 ) = ( 𝑅 ↑𝑟 1 ) )

Proof

Step	Hyp	Ref	Expression
1		ifid	⊢ if ( 1 < 1 , 1 , 1 ) = 1
2	1	eqcomi	⊢ 1 = if ( 1 < 1 , 1 , 1 )
3	2	jctr	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ 𝑉 ∧ 1 = if ( 1 < 1 , 1 , 1 ) ) )
4		1ex	⊢ 1 ∈ V
5	4	prid2	⊢ 1 ∈ { 0 , 1 }
6	5 5	pm3.2i	⊢ ( 1 ∈ { 0 , 1 } ∧ 1 ∈ { 0 , 1 } )
7		relexp01min	⊢ ( ( ( 𝑅 ∈ 𝑉 ∧ 1 = if ( 1 < 1 , 1 , 1 ) ) ∧ ( 1 ∈ { 0 , 1 } ∧ 1 ∈ { 0 , 1 } ) ) → ( ( 𝑅 ↑𝑟 1 ) ↑𝑟 1 ) = ( 𝑅 ↑𝑟 1 ) )
8	3 6 7	sylancl	⊢ ( 𝑅 ∈ 𝑉 → ( ( 𝑅 ↑𝑟 1 ) ↑𝑟 1 ) = ( 𝑅 ↑𝑟 1 ) )