let A1, A2 be a_partition of Y; :: thesis: ( ( for x being set holds

( x in A1 iff ex h being Function ex F being Subset-Family of Y st

( dom h = G & rng h = F & ( for d being set st d in G holds

h . d in d ) & x = Intersect F & x <> {} ) ) ) & ( for x being set holds

( x in A2 iff ex h being Function ex F being Subset-Family of Y st

( dom h = G & rng h = F & ( for d being set st d in G holds

h . d in d ) & x = Intersect F & x <> {} ) ) ) implies A1 = A2 )

assume that

A27: for x being set holds

( x in A1 iff ex h being Function ex F being Subset-Family of Y st

( dom h = G & rng h = F & ( for d being set st d in G holds

h . d in d ) & x = Intersect F & x <> {} ) ) and

A28: for x being set holds

( x in A2 iff ex h being Function ex F being Subset-Family of Y st

( dom h = G & rng h = F & ( for d being set st d in G holds

h . d in d ) & x = Intersect F & x <> {} ) ) ; :: thesis: A1 = A2

( x in A1 iff ex h being Function ex F being Subset-Family of Y st

( dom h = G & rng h = F & ( for d being set st d in G holds

h . d in d ) & x = Intersect F & x <> {} ) ) ) & ( for x being set holds

( x in A2 iff ex h being Function ex F being Subset-Family of Y st

( dom h = G & rng h = F & ( for d being set st d in G holds

h . d in d ) & x = Intersect F & x <> {} ) ) ) implies A1 = A2 )

assume that

A27: for x being set holds

( x in A1 iff ex h being Function ex F being Subset-Family of Y st

( dom h = G & rng h = F & ( for d being set st d in G holds

h . d in d ) & x = Intersect F & x <> {} ) ) and

A28: for x being set holds

( x in A2 iff ex h being Function ex F being Subset-Family of Y st

( dom h = G & rng h = F & ( for d being set st d in G holds

h . d in d ) & x = Intersect F & x <> {} ) ) ; :: thesis: A1 = A2

now :: thesis: for y being object holds

( y in A1 iff y in A2 )

hence
A1 = A2
by TARSKI:2; :: thesis: verum( y in A1 iff y in A2 )

let y be object ; :: thesis: ( y in A1 iff y in A2 )

( y in A1 iff ex h being Function ex F being Subset-Family of Y st

( dom h = G & rng h = F & ( for d being set st d in G holds

h . d in d ) & y = Intersect F & y <> {} ) ) by A27;

hence ( y in A1 iff y in A2 ) by A28; :: thesis: verum

end;( y in A1 iff ex h being Function ex F being Subset-Family of Y st

( dom h = G & rng h = F & ( for d being set st d in G holds

h . d in d ) & y = Intersect F & y <> {} ) ) by A27;

hence ( y in A1 iff y in A2 ) by A28; :: thesis: verum