Expression of type Equals

from the theory of proveit.logic.sets.intersection

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 x
from proveit.core_expr_types import Q__y_1_to_n, R__y_1_to_n, S_1_to_n, y_1_to_n
from proveit.logic import Equals, Forall, InSet
from proveit.logic.sets import general_intersectall_Ryn

# build up the expression from sub-expressions
expr = Equals(InSet(x, general_intersectall_Ryn), Forall(instance_param_or_params = [y_1_to_n], instance_expr = InSet(x, R__y_1_to_n), domains = [S_1_to_n], condition = Q__y_1_to_n)).with_wrapping_at(1)

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())

\begin{array}{c} \begin{array}{l} \left(x \in \left[\bigcap_{\left(y_{1} \in S_{1}\right), \left(y_{2} \in S_{2}\right), \ldots, \left(y_{n} \in S_{n}\right)~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~R\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right]\right) \\  = \left[\forall_{\left(y_{1} \in S_{1}\right), \left(y_{2} \in S_{2}\right), \ldots, \left(y_{n} \in S_{n}\right)~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\left(x \in R\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right)\right] \end{array} \end{array}

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.	()	(1)	('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: 30 operands: 5
4	Operation	operator: 6 operand: 9
5	ExprTuple	18, 8
6	Literal
7	ExprTuple	9
8	Operation	operator: 10 operand: 13
9	Lambda	parameters: 27 body: 12
10	Literal
11	ExprTuple	13
12	Conditional	value: 14 condition: 17
13	Lambda	parameters: 27 body: 15
14	Operation	operator: 30 operands: 16
15	Conditional	value: 19 condition: 17
16	ExprTuple	18, 19
17	Operation	operator: 20 operands: 21
18	Variable
19	Operation	operator: 22 operands: 27
20	Literal
21	ExprTuple	23, 24
22	Variable
23	ExprRange	lambda_map: 25 start_index: 33 end_index: 34
24	Operation	operator: 26 operands: 27
25	Lambda	parameter: 40 body: 28
26	Variable
27	ExprTuple	29
28	Operation	operator: 30 operands: 31
29	ExprRange	lambda_map: 32 start_index: 33 end_index: 34
30	Literal
31	ExprTuple	36, 35
32	Lambda	parameter: 40 body: 36
33	Literal
34	Variable
35	IndexedVar	variable: 37 index: 40
36	IndexedVar	variable: 38 index: 40
37	Variable
38	Variable
39	ExprTuple	40
40	Variable