# 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, Forall, InSet, Injections, Surjections

In [2]:
# build up the expression from sub-expressions
expr = ExprTuple(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(\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]\right)

In [5]:
stored_expr.style_options()

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

core typesub-expressionsexpression
0ExprTuple1
1Lambdaparameters: 27
body: 2
2Operationoperator: 3
operand: 5
3Literal
4ExprTuple5
5Lambdaparameter: 23
body: 7
6ExprTuple23
7Operationoperator: 8
operands: 9
8Literal
9ExprTuple10, 11
10Operationoperator: 20
operands: 12
11Operationoperator: 13
operands: 14
12ExprTuple23, 15
13Literal
14ExprTuple16, 17
15Operationoperator: 18
operands: 27
16Operationoperator: 20
operands: 19
17Operationoperator: 20
operands: 21
18Literal
19ExprTuple23, 22
20Literal
21ExprTuple23, 24
22Operationoperator: 25
operands: 27
23Variable
24Operationoperator: 26
operands: 27
25Literal
26Literal
27ExprTuple28, 29
28Variable
29Variable