# 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, Conditional, Lambda, f, fx, x
from proveit.logic import Bijections, Forall, InSet

In [2]:
# build up the expression from sub-expressions
sub_expr1 = Bijections(A, B)
expr = Lambda([A, B], Forall(instance_param_or_params = [f], instance_expr = InSet(f, sub_expr1), condition = InSet(Lambda(x, Conditional(fx, InSet(x, A))), sub_expr1)))

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[x \mapsto \left\{f\left(x\right) \textrm{ if } x \in A\right..\right] \in \left[A \xrightarrow[\text{onto}]{\text{1-to-1}} B\right]}~\left(f \in \left[A \xrightarrow[\text{onto}]{\text{1-to-1}} B\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
0Lambdaparameters: 15
body: 1
1Operationoperator: 2
operand: 4
2Literal
3ExprTuple4
4Lambdaparameter: 19
body: 6
5ExprTuple19
6Conditionalvalue: 7
condition: 8
7Operationoperator: 21
operands: 9
8Operationoperator: 21
operands: 10
9ExprTuple19, 12
10ExprTuple11, 12
11Lambdaparameter: 23
body: 13
12Operationoperator: 14
operands: 15
13Conditionalvalue: 16
condition: 17
14Literal
15ExprTuple24, 18
16Operationoperator: 19
operand: 23
17Operationoperator: 21
operands: 22
18Variable
19Variable
20ExprTuple23
21Literal
22ExprTuple23, 24
23Variable
24Variable