Theorem bijective_by_uniqueness of type Forall

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

see dependencies

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.logic.sets.functions.bijections import bijective_by_uniqueness

# check that the built expression is the same as the stored expression
assert bijective_by_uniqueness.expr == stored_expr
assert bijective_by_uniqueness.expr._style_id == stored_expr._style_id
print("Passed sanity check: bijective_by_uniqueness matches stored_expr")

Passed sanity check: bijective_by_uniqueness matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{f, A}~\left(\left[\forall_{x, y \in A~|~x \neq y}~\left(f\left(x\right) \neq f\left(y\right)\right)\right] \Rightarrow \left(f \in \left[A \xrightarrow[\text{onto}]{\text{1-to-1}} f^{\rightarrow}(A)\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: 8 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 23 body: 3
3	Operation	operator: 4 operands: 5
4	Literal
5	ExprTuple	6, 7
6	Operation	operator: 8 operand: 11
7	Operation	operator: 33 operands: 10
8	Literal
9	ExprTuple	11
10	ExprTuple	30, 12
11	Lambda	parameters: 36 body: 13
12	Operation	operator: 14 operands: 15
13	Conditional	value: 16 condition: 17
14	Literal
15	ExprTuple	37, 18
16	Operation	operator: 35 operands: 19
17	Operation	operator: 20 operands: 21
18	Operation	operator: 22 operands: 23
19	ExprTuple	24, 25
20	Literal
21	ExprTuple	26, 27, 28
22	Literal
23	ExprTuple	30, 37
24	Operation	operator: 30 operand: 38
25	Operation	operator: 30 operand: 39
26	Operation	operator: 33 operands: 32
27	Operation	operator: 33 operands: 34
28	Operation	operator: 35 operands: 36
29	ExprTuple	38
30	Variable
31	ExprTuple	39
32	ExprTuple	38, 37
33	Literal
34	ExprTuple	39, 37
35	Literal
36	ExprTuple	38, 39
37	Variable
38	Variable
39	Variable