Theorem relexp0idm

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

	Ref	Expression
Assertion	relexp0idm	⊢ ( 𝑅 ∈ 𝑉 → ( ( 𝑅 ↑𝑟 0 ) ↑𝑟 0 ) = ( 𝑅 ↑𝑟 0 ) )

Proof

Step	Hyp	Ref	Expression
1		ifid	⊢ if ( 0 < 0 , 0 , 0 ) = 0
2	1	eqcomi	⊢ 0 = if ( 0 < 0 , 0 , 0 )
3	2	jctr	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ 𝑉 ∧ 0 = if ( 0 < 0 , 0 , 0 ) ) )
4		c0ex	⊢ 0 ∈ V
5	4	prid1	⊢ 0 ∈ { 0 , 1 }
6	5 5	pm3.2i	⊢ ( 0 ∈ { 0 , 1 } ∧ 0 ∈ { 0 , 1 } )
7		relexp01min	⊢ ( ( ( 𝑅 ∈ 𝑉 ∧ 0 = if ( 0 < 0 , 0 , 0 ) ) ∧ ( 0 ∈ { 0 , 1 } ∧ 0 ∈ { 0 , 1 } ) ) → ( ( 𝑅 ↑𝑟 0 ) ↑𝑟 0 ) = ( 𝑅 ↑𝑟 0 ) )
8	3 6 7	sylancl	⊢ ( 𝑅 ∈ 𝑉 → ( ( 𝑅 ↑𝑟 0 ) ↑𝑟 0 ) = ( 𝑅 ↑𝑟 0 ) )