Theorem cnvepima

Description: The image of converse epsilon. (Contributed by Peter Mazsa, 22-Mar-2023)

	Ref	Expression
Assertion	cnvepima	⊢ ( 𝐴 ∈ 𝑉 → ( ◡ E “ 𝐴 ) = ∪ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		cnvepresex	⊢ ( 𝐴 ∈ 𝑉 → ( ◡ E ↾ 𝐴 ) ∈ V )
2		uniqs	⊢ ( ( ◡ E ↾ 𝐴 ) ∈ V → ∪ ( 𝐴 / ◡ E ) = ( ◡ E “ 𝐴 ) )
3	1 2	syl	⊢ ( 𝐴 ∈ 𝑉 → ∪ ( 𝐴 / ◡ E ) = ( ◡ E “ 𝐴 ) )
4		qsid	⊢ ( 𝐴 / ◡ E ) = 𝐴
5	4	unieqi	⊢ ∪ ( 𝐴 / ◡ E ) = ∪ 𝐴
6	3 5	eqtr3di	⊢ ( 𝐴 ∈ 𝑉 → ( ◡ E “ 𝐴 ) = ∪ 𝐴 )