To solve Bob's animal feed mixing problem using linear programming and Gurobi, we first need to define the decision variables, objective function, and constraints based on the given information.

Let's denote:
- \(x_1\) as the amount of oats (in kilograms) to be included in the mixture.
- \(x_2\) as the amount of sunflower seeds (in kilograms) to be included in the mixture.

The objective is to minimize the total cost of the mixture. The cost function can be represented as:
\[ \text{Minimize:} \quad 50x_1 + 70x_2 \]

Given constraints are:
1. The mixture must contain a minimum of 250 units of protein.
2. The mixture must contain a minimum of 400 units of fat.

From the problem statement, we know:
- Oats contain 5 units of protein and 16 units of fat per kilogram.
- Sunflower seeds contain 10 units of protein and 22 units of fat per kilogram.

Thus, the constraints can be formulated as follows:
1. Protein constraint: \(5x_1 + 10x_2 \geq 250\)
2. Fat constraint: \(16x_1 + 22x_2 \geq 400\)

Additionally, since we cannot have negative amounts of oats or sunflower seeds, we also include non-negativity constraints:
\[ x_1 \geq 0, x_2 \geq 0 \]

With this formulation, we can now write the Gurobi code in Python to solve this linear programming problem.

```python
from gurobipy import *

# Create a new model
m = Model("Animal_Feed_Mixing")

# Define variables
x1 = m.addVar(lb=0, name="oats")  # Amount of oats in kg
x2 = m.addVar(lb=0, name="sunflower_seeds")  # Amount of sunflower seeds in kg

# Set the objective function: Minimize cost
m.setObjective(50*x1 + 70*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(5*x1 + 10*x2 >= 250, name="protein_constraint")
m.addConstr(16*x1 + 22*x2 >= 400, name="fat_constraint")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Oats: {x1.x} kg")
    print(f"Sunflower seeds: {x2.x} kg")
    print(f"Total cost: ${50*x1.x + 70*x2.x}")
else:
    print("No optimal solution found")
```