Expression of type Forall

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, f
from proveit.logic import And, Bijections, Forall, InSet, Injections, Surjections

# build up the expression from sub-expressions
expr = 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())

\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)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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