Expression of type Lambda

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

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.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import A, B, Lambda, f
from proveit.logic import And, Bijections, Forall, InSet, Injections, Surjections

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

expr:

# 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

# 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 \in \left[A \xrightarrow[\text{onto}]{\text{1-to-1}} B\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)\right]

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 24 body: 1
1	Operation	operator: 2 operand: 4
2	Literal
3	ExprTuple	4
4	Lambda	parameter: 20 body: 6
5	ExprTuple	20
6	Conditional	value: 7 condition: 8
7	Operation	operator: 9 operands: 10
8	Operation	operator: 16 operands: 11
9	Literal
10	ExprTuple	12, 13
11	ExprTuple	20, 14
12	Operation	operator: 16 operands: 15
13	Operation	operator: 16 operands: 17
14	Operation	operator: 18 operands: 24
15	ExprTuple	20, 19
16	Literal
17	ExprTuple	20, 21
18	Literal
19	Operation	operator: 22 operands: 24
20	Variable
21	Operation	operator: 23 operands: 24
22	Literal
23	Literal
24	ExprTuple	25, 26
25	Variable
26	Variable