Theorem eccnvepres3

Description: Condition for a restricted converse epsilon coset of a set to be the set itself. (Contributed by Peter Mazsa, 11-May-2021)

	Ref	Expression
Assertion	eccnvepres3	⊢ ( 𝐵 ∈ dom ( ◡ E ↾ 𝐴 ) → [ 𝐵 ] ( ◡ E ↾ 𝐴 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		resdmres	⊢ ( ◡ E ↾ dom ( ◡ E ↾ 𝐴 ) ) = ( ◡ E ↾ 𝐴 )
2	1	eceq2i	⊢ [ 𝐵 ] ( ◡ E ↾ dom ( ◡ E ↾ 𝐴 ) ) = [ 𝐵 ] ( ◡ E ↾ 𝐴 )
3		eccnvepres2	⊢ ( 𝐵 ∈ dom ( ◡ E ↾ 𝐴 ) → [ 𝐵 ] ( ◡ E ↾ dom ( ◡ E ↾ 𝐴 ) ) = 𝐵 )
4	2 3	eqtr3id	⊢ ( 𝐵 ∈ dom ( ◡ E ↾ 𝐴 ) → [ 𝐵 ] ( ◡ E ↾ 𝐴 ) = 𝐵 )