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		qsid	⊢ ( 𝐴 / ◡ E ) = 𝐴
2	1	unieqi	⊢ ∪ ( 𝐴 / ◡ E ) = ∪ 𝐴
3		cnvepresex	⊢ ( 𝐴 ∈ 𝑉 → ( ◡ E ↾ 𝐴 ) ∈ V )
4		uniqsALTV	⊢ ( ( ◡ E ↾ 𝐴 ) ∈ V → ∪ ( 𝐴 / ◡ E ) = ( ◡ E “ 𝐴 ) )
5	3 4	syl	⊢ ( 𝐴 ∈ 𝑉 → ∪ ( 𝐴 / ◡ E ) = ( ◡ E “ 𝐴 ) )
6	2 5	syl5reqr	⊢ ( 𝐴 ∈ 𝑉 → ( ◡ E “ 𝐴 ) = ∪ 𝐴 )