# 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, C, Conditional, ExprTuple, Function, Lambda, f, fx, g, x
from proveit.logic import And, Bijections, InSet

In [2]:
# build up the expression from sub-expressions
expr = ExprTuple(And(InSet(f, Bijections(A, B)), InSet(g, Bijections(B, C))), InSet(Lambda(x, Conditional(Function(g, [fx]), InSet(x, A))), Bijections(A, C)))

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(f \in \left[A \xrightarrow[\text{onto}]{\text{1-to-1}} B\right]\right) \land \left(g \in \left[B \xrightarrow[\text{onto}]{\text{1-to-1}} C\right]\right), \left[x \mapsto \left\{g\left(f\left(x\right)\right) \textrm{ if } x \in A\right..\right] \in \left[A \xrightarrow[\text{onto}]{\text{1-to-1}} C\right]\right)

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
wrap_positionsposition(s) at which wrapping is to occur; 'n' is after the nth comma.()()('with_wrapping_at',)
justificationif any wrap positions are set, justify to the 'left', 'center', or 'right'leftleft('with_justification',)
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0ExprTuple1, 2
1Operationoperator: 3
operands: 4
2Operationoperator: 23
operands: 5
3Literal
4ExprTuple6, 7
5ExprTuple8, 9
6Operationoperator: 23
operands: 10
7Operationoperator: 23
operands: 11
8Lambdaparameter: 31
body: 12
9Operationoperator: 19
operands: 13
10ExprTuple29, 14
11ExprTuple21, 15
12Conditionalvalue: 16
condition: 17
13ExprTuple28, 26
14Operationoperator: 19
operands: 18
15Operationoperator: 19
operands: 20
16Operationoperator: 21
operand: 27
17Operationoperator: 23
operands: 24
18ExprTuple28, 25
19Literal
20ExprTuple25, 26
21Variable
22ExprTuple27
23Literal
24ExprTuple31, 28
25Variable
26Variable
27Operationoperator: 29
operand: 31
28Variable
29Variable
30ExprTuple31
31Variable