Expression of type Implies

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, f, fx, fy, x, y
from proveit.logic import Bijections, Forall, Image, Implies, InSet, NotEquals

# build up the expression from sub-expressions
expr = Implies(Forall(instance_param_or_params = [x, y], instance_expr = NotEquals(fx, fy), domain = A, condition = NotEquals(x, y)), InSet(f, Bijections(A, Image(f, A))))

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[\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)

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 30 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	27, 9
8	Lambda	parameters: 33 body: 10
9	Operation	operator: 11 operands: 12
10	Conditional	value: 13 condition: 14
11	Literal
12	ExprTuple	34, 15
13	Operation	operator: 32 operands: 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 19 operands: 20
16	ExprTuple	21, 22
17	Literal
18	ExprTuple	23, 24, 25
19	Literal
20	ExprTuple	27, 34
21	Operation	operator: 27 operand: 35
22	Operation	operator: 27 operand: 36
23	Operation	operator: 30 operands: 29
24	Operation	operator: 30 operands: 31
25	Operation	operator: 32 operands: 33
26	ExprTuple	35
27	Variable
28	ExprTuple	36
29	ExprTuple	35, 34
30	Literal
31	ExprTuple	36, 34
32	Literal
33	ExprTuple	35, 36
34	Variable
35	Variable
36	Variable