Metamath Proof Explorer

Theorem cnvepima

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

Ref Expression
Assertion cnvepima 
|- ( A e. V -> ( `' _E " A ) = U. A )

Proof

Step Hyp Ref Expression
1 cnvepresex 
 |-  ( A e. V -> ( `' _E |` A ) e. _V )
2 uniqsALTV 
 |-  ( ( `' _E |` A ) e. _V -> U. ( A /. `' _E ) = ( `' _E " A ) )
3 1 2 syl 
 |-  ( A e. V -> U. ( A /. `' _E ) = ( `' _E " A ) )
4 qsid 
 |-  ( A /. `' _E ) = A
5 4 unieqi 
 |-  U. ( A /. `' _E ) = U. A
6 3 5 eqtr3di 
 |-  ( A e. V -> ( `' _E " A ) = U. A )