let C be Category; :: thesis: for a, b being Object of C st Hom (a,b) <> {} holds

for f being Morphism of a,b holds

( dom (f opp) = cod f & cod (f opp) = dom f )

let a, b be Object of C; :: thesis: ( Hom (a,b) <> {} implies for f being Morphism of a,b holds

( dom (f opp) = cod f & cod (f opp) = dom f ) )

assume A1: Hom (a,b) <> {} ; :: thesis: for f being Morphism of a,b holds

( dom (f opp) = cod f & cod (f opp) = dom f )

then A2: Hom ((b opp),(a opp)) <> {} by Th4;

let f be Morphism of a,b; :: thesis: ( dom (f opp) = cod f & cod (f opp) = dom f )

thus dom (f opp) = b by A2, CAT_1:5

.= cod f by A1, CAT_1:5 ; :: thesis: cod (f opp) = dom f

thus cod (f opp) = a by A2, CAT_1:5

.= dom f by A1, CAT_1:5 ; :: thesis: verum

for f being Morphism of a,b holds

( dom (f opp) = cod f & cod (f opp) = dom f )

let a, b be Object of C; :: thesis: ( Hom (a,b) <> {} implies for f being Morphism of a,b holds

( dom (f opp) = cod f & cod (f opp) = dom f ) )

assume A1: Hom (a,b) <> {} ; :: thesis: for f being Morphism of a,b holds

( dom (f opp) = cod f & cod (f opp) = dom f )

then A2: Hom ((b opp),(a opp)) <> {} by Th4;

let f be Morphism of a,b; :: thesis: ( dom (f opp) = cod f & cod (f opp) = dom f )

thus dom (f opp) = b by A2, CAT_1:5

.= cod f by A1, CAT_1:5 ; :: thesis: cod (f opp) = dom f

thus cod (f opp) = a by A2, CAT_1:5

.= dom f by A1, CAT_1:5 ; :: thesis: verum