Theorem bijection_transitivity 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 bijection_transitivity

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

Passed sanity check: bijection_transitivity matches stored_expr

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

\forall_{f, g, A, B, C}~\left(\begin{array}{c} \begin{array}{l} \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)\right) \\  \Rightarrow \left(\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) \end{array} \end{array}\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	parameters: 4 body: 5
4	ExprTuple	36, 28, 35, 32, 33
5	Operation	operator: 6 operands: 7
6	Literal
7	ExprTuple	8, 9
8	Operation	operator: 10 operands: 11
9	Operation	operator: 30 operands: 12
10	Literal
11	ExprTuple	13, 14
12	ExprTuple	15, 16
13	Operation	operator: 30 operands: 17
14	Operation	operator: 30 operands: 18
15	Lambda	parameter: 38 body: 19
16	Operation	operator: 26 operands: 20
17	ExprTuple	36, 21
18	ExprTuple	28, 22
19	Conditional	value: 23 condition: 24
20	ExprTuple	35, 33
21	Operation	operator: 26 operands: 25
22	Operation	operator: 26 operands: 27
23	Operation	operator: 28 operand: 34
24	Operation	operator: 30 operands: 31
25	ExprTuple	35, 32
26	Literal
27	ExprTuple	32, 33
28	Variable
29	ExprTuple	34
30	Literal
31	ExprTuple	38, 35
32	Variable
33	Variable
34	Operation	operator: 36 operand: 38
35	Variable
36	Variable
37	ExprTuple	38
38	Variable