# 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, ExprTuple, Lambda, f
from proveit.logic import And, Bijections, Equals, InSet, Injections, Surjections

In [2]:
# build up the expression from sub-expressions
expr = ExprTuple(Lambda(f, 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(f \mapsto \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
0ExprTuple1
1Lambdaparameter: 19
body: 3
2ExprTuple19
3Operationoperator: 4
operands: 5
4Literal
5ExprTuple6, 7
6Operationoperator: 16
operands: 8
7Operationoperator: 9
operands: 10
8ExprTuple19, 11
9Literal
10ExprTuple12, 13
11Operationoperator: 14
operands: 23
12Operationoperator: 16
operands: 15
13Operationoperator: 16
operands: 17
14Literal
15ExprTuple19, 18
16Literal
17ExprTuple19, 20
18Operationoperator: 21
operands: 23
19Variable
20Operationoperator: 22
operands: 23
21Literal
22Literal
23ExprTuple24, 25
24Variable
25Variable