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Metamath Proof Explorer

Theorem coepr

Description: Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013)

Ref Expression
Hypotheses coep.1 
|- A e. _V
coep.2 
|- B e. _V
Assertion coepr 
|- ( A ( R o. `' _E ) B <-> E. x e. A x R B )

Proof

Step Hyp Ref Expression
1 coep.1 
 |-  A e. _V
2 coep.2 
 |-  B e. _V
3 vex 
 |-  x e. _V
4 1 3 brcnv 
 |-  ( A `' _E x <-> x _E A )
5 1 epeli 
 |-  ( x _E A <-> x e. A )
6 4 5 bitri 
 |-  ( A `' _E x <-> x e. A )
7 6 anbi1i 
 |-  ( ( A `' _E x /\ x R B ) <-> ( x e. A /\ x R B ) )
8 7 exbii 
 |-  ( E. x ( A `' _E x /\ x R B ) <-> E. x ( x e. A /\ x R B ) )
9 1 2 brco 
 |-  ( A ( R o. `' _E ) B <-> E. x ( A `' _E x /\ x R B ) )
10 df-rex 
 |-  ( E. x e. A x R B <-> E. x ( x e. A /\ x R B ) )
11 8 9 10 3bitr4i 
 |-  ( A ( R o. `' _E ) B <-> E. x e. A x R B )