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Metamath Proof Explorer

Theorem eccnvepres2

Description: The restricted converse epsilon coset of an element of the restriction is the element itself. (Contributed by Peter Mazsa, 16-Jul-2019)

		Ref	Expression
	Assertion	eccnvepres2	⊢ ( 𝐵 ∈ 𝐴 → [ 𝐵 ] ( ^◡ E ↾ 𝐴 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elecreseq	⊢ ( 𝐵 ∈ 𝐴 → [ 𝐵 ] ( ^◡ E ↾ 𝐴 ) = [ 𝐵 ] ^◡ E )
2		eccnvep	⊢ ( 𝐵 ∈ 𝐴 → [ 𝐵 ] ^◡ E = 𝐵 )
3	1 2	eqtrd	⊢ ( 𝐵 ∈ 𝐴 → [ 𝐵 ] ( ^◡ E ↾ 𝐴 ) = 𝐵 )