let x1, x2 be set ; :: thesis: for C being Category
for a being Object of C
for p1, p2 being Morphism of C st dom p1 = a & dom p2 = a holds
(x1,x2) --> (p1,p2) is Projections_family of a,{x1,x2}
let C be Category; :: thesis: for a being Object of C
for p1, p2 being Morphism of C st dom p1 = a & dom p2 = a holds
(x1,x2) --> (p1,p2) is Projections_family of a,{x1,x2}
let a be Object of C; :: thesis: for p1, p2 being Morphism of C st dom p1 = a & dom p2 = a holds
(x1,x2) --> (p1,p2) is Projections_family of a,{x1,x2}
let p1, p2 be Morphism of C; :: thesis: ( dom p1 = a & dom p2 = a implies (x1,x2) --> (p1,p2) is Projections_family of a,{x1,x2} )
assume A1: ( dom p1 = a & dom p2 = a ) ; :: thesis: (x1,x2) --> (p1,p2) is Projections_family of a,{x1,x2}
doms ((x1,x2) --> (p1,p2)) = (x1,x2) --> ((dom p1),(dom p2)) by Th6
.= {x1,x2} --> a by A1, FUNCT_4:65 ;
hence (x1,x2) --> (p1,p2) is Projections_family of a,{x1,x2} by Def13; :: thesis: verum