RationalMap || Ideal -- restriction of a rational map

i1 : P5 = ZZ/190181[x_0..x_5]

o1 = P5

o1 : PolynomialRing

i2 : Phi = rationalMap {x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3}

o2 = -- rational map --
                    ZZ
     source: Proj(------[x , x , x , x , x , x ])
                  190181  0   1   2   3   4   5
                    ZZ
     target: Proj(------[x , x , x , x , x , x ])
                  190181  0   1   2   3   4   5
     defining forms: {
                       2
                      x  - x x ,
                       4    3 5
                      
                      x x  - x x ,
                       2 4    1 5
                      
                      x x  - x x ,
                       2 3    1 4
                      
                       2
                      x  - x x ,
                       2    0 5
                      
                      x x  - x x ,
                       1 2    0 4
                      
                       2
                      x  - x x
                       1    0 3
                     }

o2 : RationalMap (quadratic rational map from PP^5 to PP^5)

i3 : J = ideal random(1,P5);

o3 : Ideal of P5

i4 : Phi' = Phi||J

o4 = -- rational map --
                                  ZZ
     source: subvariety of Proj(------[x , x , x , x , x , x ]) defined by
                                190181  0   1   2   3   4   5
             {
               2                    2                                                               2
              x  + 9702x x  - 94294x  - x x  + 68094x x  - 9702x x  - 68094x x  + 93593x x  - 53251x  + 94294x x  - 93593x x  + 53251x x
               1        1 2         2    0 3         2 3        0 4         1 4         2 4         4         0 5         1 5         3 5
             }
                                  ZZ
     target: subvariety of Proj(------[x , x , x , x , x , x ]) defined by
                                190181  0   1   2   3   4   5
             {
              x  + 16566x  - 70158x  - 38148x  - 77864x  - 71321x
               0         1         2         3         4         5
             }
     defining forms: {
                       2
                      x  - x x ,
                       4    3 5
                      
                      x x  - x x ,
                       2 4    1 5
                      
                      x x  - x x ,
                       2 3    1 4
                      
                       2
                      x  - x x ,
                       2    0 5
                      
                      x x  - x x ,
                       1 2    0 4
                      
                                         2                                                        2
                      - 9702x x  + 94294x  - 68094x x  + 9702x x  + 68094x x  - 93593x x  + 53251x  - 94294x x  + 93593x x  - 53251x x
                             1 2         2         2 3        0 4         1 4         2 4         4         0 5         1 5         3 5
                     }

o4 : RationalMap (quadratic rational map from hypersurface in PP^5 to hypersurface in PP^5)

i5 : describe Phi

o5 = rational map defined by forms of degree 2
     source variety: PP^5
     target variety: PP^5
     coefficient ring: ZZ/190181

i6 : describe Phi'

o6 = rational map defined by forms of degree 2
     source variety: smooth quadric hypersurface in PP^5
     target variety: hyperplane in PP^5
     coefficient ring: ZZ/190181