# from the theory of proveit.logic.sets.functions.bijections¶

In [1]:
import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
# import Expression classes needed to build the expression
from proveit import A, B, Lambda, f
from proveit.logic import And, Bijections, Equals, Forall, InSet, Injections, Surjections

In [2]:
# build up the expression from sub-expressions
expr = Lambda([A, B], Forall(instance_param_or_params = [f], instance_expr = Equals(InSet(f, Bijections(A, B)), And(InSet(f, Injections(A, B)), InSet(f, Surjections(A, B)))).with_wrapping_at(2)))

expr:
In [3]:
# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

In [4]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left(A, B\right) \mapsto \left[\forall_{f}~\left(\begin{array}{c} \begin{array}{l} \left(f \in \left[A \xrightarrow[\text{onto}]{\text{1-to-1}} B\right]\right) =  \\ \left(\left(f \in \left[A \xrightarrow[]{\text{1-to-1}} B\right]\right) \land \left(f \in \left[A \xrightarrow[\text{onto}]{} B\right]\right)\right) \end{array} \end{array}\right)\right]

In [5]:
stored_expr.style_options()

no style options
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Lambdaparameters: 26
body: 1
1Operationoperator: 2
operand: 4
2Literal
3ExprTuple4
4Lambdaparameter: 22
body: 6
5ExprTuple22
6Operationoperator: 7
operands: 8
7Literal
8ExprTuple9, 10
9Operationoperator: 19
operands: 11
10Operationoperator: 12
operands: 13
11ExprTuple22, 14
12Literal
13ExprTuple15, 16
14Operationoperator: 17
operands: 26
15Operationoperator: 19
operands: 18
16Operationoperator: 19
operands: 20
17Literal
18ExprTuple22, 21
19Literal
20ExprTuple22, 23
21Operationoperator: 24
operands: 26
22Variable
23Operationoperator: 25
operands: 26
24Literal
25Literal
26ExprTuple27, 28
27Variable
28Variable