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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | isstruct2r 11501 | The property of being a structure with components in . (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.) |
Struct | ||
Theorem | structex 11502 | A structure is a set. (Contributed by AV, 10-Nov-2021.) |
Struct | ||
Theorem | structn0fun 11503 | A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.) |
Struct | ||
Theorem | isstructim 11504 | The property of being a structure with components in . (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.) |
Struct | ||
Theorem | isstructr 11505 | The property of being a structure with components in . (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.) |
Struct | ||
Theorem | structcnvcnv 11506 | Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
Struct | ||
Theorem | structfung 11507 | The converse of the converse of a structure is a function. Closed form of structfun 11508. (Contributed by AV, 12-Nov-2021.) |
Struct | ||
Theorem | structfun 11508 | Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof shortened by AV, 12-Nov-2021.) |
Struct | ||
Theorem | structfn 11509 | Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
Struct | ||
Theorem | slotfni 11510 | A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by Jim Kingdon, 19-Jan-2023.) |
Slot | ||
Theorem | strnfvnd 11511 | Deduction version of strnfvn 11513. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.) |
Slot | ||
Theorem | basfn 11512 | The base set extractor is a function on . (Contributed by Stefan O'Rear, 8-Jul-2015.) |
Theorem | strnfvn 11513 |
Value of a structure component extractor . Normally, is a
defined constant symbol such as (df-base 11496) and is a
fixed integer such as . is a
structure, i.e. a specific
member of a class of structures.
Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strnfv 11533. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.) |
Slot | ||
Theorem | strfvssn 11514 | A structure component extractor produces a value which is contained in a set dependent on , but not . This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 19-Jan-2023.) |
Slot | ||
Theorem | ndxarg 11515 | Get the numeric argument from a defined structure component extractor such as df-base 11496. (Contributed by Mario Carneiro, 6-Oct-2013.) |
Slot | ||
Theorem | ndxid 11516 |
A structure component extractor is defined by its own index. This
theorem, together with strnfv 11533 below, is useful for avoiding direct
reference to the hard-coded numeric index in component extractor
definitions, such as the in df-base 11496, making it easier to change
should the need arise.
(Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.) |
Slot Slot | ||
Theorem | strndxid 11517 | The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.) |
Slot | ||
Theorem | reldmsets 11518 | The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
sSet | ||
Theorem | setsvalg 11519 | Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.) |
sSet | ||
Theorem | setsvala 11520 | Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 20-Jan-2023.) |
sSet | ||
Theorem | setsex 11521 | Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.) |
sSet | ||
Theorem | setsssvald 11522 | Value of the structure replacement function, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 21-Jan-2023.) |
Slot sSet | ||
Theorem | fvsetsid 11523 | The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.) |
sSet | ||
Theorem | setsfun 11524 | A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.) |
sSet | ||
Theorem | setsfun0 11525 | A structure with replacement without the empty set is a function if the original structure without the empty set is a function. This variant of setsfun 11524 is useful for proofs based on isstruct2r 11501 which requires for to be an extensible structure. (Contributed by AV, 7-Jun-2021.) |
sSet | ||
Theorem | setsn0fun 11526 | The value of the structure replacement function (without the empty set) is a function if the structure (without the empty set)is a function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.) |
Struct sSet | ||
Theorem | setsresg 11527 | The structure replacement function does not affect the value of away from . (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.) |
sSet | ||
Theorem | setsabsd 11528 | Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.) |
sSet sSet sSet | ||
Theorem | setscom 11529 | Component-setting is commutative when the x-values are different. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
sSet sSet sSet sSet | ||
Theorem | strnfvd 11530 | Deduction version of strnfv 11533. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 22-Jan-2023.) |
Slot | ||
Theorem | strnfv2d 11531 | Deduction version of strnfv 11533. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 23-Jan-2023.) |
Slot | ||
Theorem | strnfv2 11532 | A variation on strnfv 11533 to avoid asserting that itself is a function, which involves sethood of all the ordered pair components of . (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 23-Jan-2023.) |
Slot | ||
Theorem | strnfv 11533 | Extract a structure component (such as the base set) from a structure with a component extractor (such as the base set extractor df-base 11496). By virtue of ndxid 11516, this can be done without having to refer to the hard-coded numeric index of . (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 23-Jan-2023.) |
Struct Slot | ||
Theorem | strnfv3 11534 | Variant on strnfv 11533 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 23-Jan-2023.) |
Struct Slot | ||
Theorem | strnssd 11535 | Deduction version of strnss 11536. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 23-Jan-2023.) |
Slot | ||
Theorem | strnss 11536 | Propagate component extraction to a structure from a subset structure . (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Jim Kingdon, 23-Jan-2023.) |
Slot | ||
Theorem | strn0 11537 | All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 23-Jan-2023.) |
Slot | ||
Theorem | base0 11538 | The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | setsidn 11539 | Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.) |
Slot sSet | ||
Theorem | setsnidn 11540 | Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.) |
Slot sSet | ||
Theorem | baseval 11541 | Value of the base set extractor. (Normally it is preferred to work with rather than the hard-coded in order to make structure theorems portable. This is an example of how to obtain it when needed.) (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.) |
Theorem | baseid 11542 | Utility theorem: index-independent form of df-base 11496. (Contributed by NM, 20-Oct-2012.) |
Slot | ||
Theorem | basendx 11543 | Index value of the base set extractor. (Normally it is preferred to work with rather than the hard-coded in order to make structure theorems portable. This is an example of how to obtain it when needed.) (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Aug-2013.) |
Theorem | basendxnn 11544 | The index value of the base set extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 23-Sep-2020.) |
Theorem | reldmress 11545 | The structure restriction is a proper operator, so it can be used with ovprc1 5685. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
↾_{s} | ||
Theorem | ressid2 11546 | General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 26-Jan-2023.) |
↾_{s} | ||
Theorem | ressval2 11547 | Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
↾_{s} sSet | ||
Theorem | ressid 11548 | Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
↾_{s} | ||
A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union), and it may sometimes be more convenient to consider topologies without reference to the underlying set. | ||
Syntax | ctop 11549 | Syntax for the class of topologies. |
Definition | df-top 11550* |
Define the class of topologies. It is a proper class. See istopg 11551 and
istopfin 11552 for the corresponding characterizations,
using respectively
binary intersections like in this definition and nonempty finite
intersections.
The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241. (Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.) |
Theorem | istopg 11551* |
Express the predicate " is a topology". See istopfin 11552 for
another characterization using nonempty finite intersections instead of
binary intersections.
Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Theorem | istopfin 11552* | Express the predicate " is a topology" using nonempty finite intersections instead of binary intersections as in istopg 11551. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.) |
Theorem | uniopn 11553 | The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.) |
Theorem | iunopn 11554* | The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.) |
Theorem | inopn 11555 | The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
Theorem | fiinopn 11556 | The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.) |
Theorem | unopn 11557 | The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | 0opn 11558 | The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.) |
Theorem | 0ntop 11559 | The empty set is not a topology. (Contributed by FL, 1-Jun-2008.) |
Theorem | topopn 11560 | The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
Theorem | eltopss 11561 | A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.) |
Syntax | ctopon 11562 | Syntax for the function of topologies on sets. |
TopOn | ||
Definition | df-topon 11563* | Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
TopOn | ||
Theorem | funtopon 11564 | The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.) |
TopOn | ||
Theorem | istopon 11565 | Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | topontop 11566 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | toponuni 11567 | The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | topontopi 11568 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | toponunii 11569 | The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | toptopon 11570 | Alternative definition of in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | toptopon2 11571 | A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
TopOn | ||
Theorem | topontopon 11572 | A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
TopOn TopOn | ||
Theorem | toponsspwpwg 11573 | The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.) |
TopOn | ||
Theorem | dmtopon 11574 | The domain of TopOn is . (Contributed by BJ, 29-Apr-2021.) |
TopOn | ||
Theorem | fntopon 11575 | The class TopOn is a function with domain . (Contributed by BJ, 29-Apr-2021.) |
TopOn | ||
Theorem | toponmax 11576 | The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | toponss 11577 | A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.) |
TopOn | ||
Theorem | toponcom 11578 | If is a topology on the base set of topology , then is a topology on the base of . (Contributed by Mario Carneiro, 22-Aug-2015.) |
TopOn TopOn | ||
Theorem | toponcomb 11579 | Biconditional form of toponcom 11578. (Contributed by BJ, 5-Dec-2021.) |
TopOn TopOn | ||
Theorem | topgele 11580 | The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.) |
TopOn | ||
Syntax | ccncf 11581 | Extend class notation to include the operation which returns a class of continuous complex functions. |
Definition | df-cncf 11582* | Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.) |
Theorem | cncfval 11583* | The value of the continuous complex function operation is the set of continuous functions from to . (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.) |
Theorem | elcncf 11584* | Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.) |
Theorem | elcncf2 11585* | Version of elcncf 11584 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.) |
Theorem | cncfrss 11586 | Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.) |
Theorem | cncfrss2 11587 | Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.) |
Theorem | cncff 11588 | A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.) |
Theorem | cncfi 11589* | Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.) |
Theorem | elcncf1di 11590* | Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 26-Nov-2007.) |
Theorem | elcncf1ii 11591* | Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 26-Nov-2007.) |
Theorem | rescncf 11592 | A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.) |
Theorem | cncffvrn 11593 | Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.) |
Theorem | cncfss 11594 | The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.) |
Theorem | climcncf 11595 | Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.) |
Theorem | abscncf 11596 | Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
Theorem | recncf 11597 | Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
Theorem | imcncf 11598 | Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
Theorem | cjcncf 11599 | Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
Theorem | mulc1cncf 11600* | Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.) |
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