P. 173 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17201-17300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	structfung 17201	The converse of the converse of a structure is a function. Closed form of structfun 17202. (Contributed by AV, 12-Nov-2021.)
⊢ (𝐹 Struct 𝑋 → Fun ◡◡𝐹)
Theorem	structfun 17202	Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof shortened by AV, 12-Nov-2021.)
⊢ 𝐹 Struct 𝑋    ⇒   ⊢ Fun ◡◡𝐹
Theorem	structfn 17203	Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐹 Struct ⟨𝑀, 𝑁⟩    ⇒   ⊢ (Fun ◡◡𝐹 ∧ dom 𝐹 ⊆ (1...𝑁))
Theorem	strleun 17204	Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐹 Struct ⟨𝐴, 𝐵⟩    &   ⊢ 𝐺 Struct ⟨𝐶, 𝐷⟩    &   ⊢ 𝐵 < 𝐶    ⇒   ⊢ (𝐹 ∪ 𝐺) Struct ⟨𝐴, 𝐷⟩
Theorem	strle1 17205	Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐼 ∈ ℕ    &   ⊢ 𝐴 = 𝐼    ⇒   ⊢ {⟨𝐴, 𝑋⟩} Struct ⟨𝐼, 𝐼⟩
Theorem	strle2 17206	Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐼 ∈ ℕ    &   ⊢ 𝐴 = 𝐼    &   ⊢ 𝐼 < 𝐽    &   ⊢ 𝐽 ∈ ℕ    &   ⊢ 𝐵 = 𝐽    ⇒   ⊢ {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩} Struct ⟨𝐼, 𝐽⟩
Theorem	strle3 17207	Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐼 ∈ ℕ    &   ⊢ 𝐴 = 𝐼    &   ⊢ 𝐼 < 𝐽    &   ⊢ 𝐽 ∈ ℕ    &   ⊢ 𝐵 = 𝐽    &   ⊢ 𝐽 < 𝐾    &   ⊢ 𝐾 ∈ ℕ    &   ⊢ 𝐶 = 𝐾    ⇒   ⊢ {⟨𝐴, 𝑋⟩, ⟨𝐵, 𝑌⟩, ⟨𝐶, 𝑍⟩} Struct ⟨𝐼, 𝐾⟩
Theorem	sbcie2s 17208*	A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.) (Revised by SN, 2-Mar-2025.)
⊢ 𝐴 = (𝐸‘𝑊)    &   ⊢ 𝐵 = (𝐹‘𝑊)    &   ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜑 ↔ 𝜓))
Theorem	sbcie3s 17209*	A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 15-Mar-2019.)
⊢ 𝐴 = (𝐸‘𝑊)    &   ⊢ 𝐵 = (𝐹‘𝑊)    &   ⊢ 𝐶 = (𝐺‘𝑊)    &   ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏][(𝐺‘𝑤) / 𝑐]𝜓 ↔ 𝜑))
7.1.1.2  Substitution of components
Syntax	csts 17210	Set components of a structure.
class sSet
Definition	df-sets 17211*	Set a component of an extensible structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, df-ress 17288 adjusts the base set to match its second argument, which has the effect of making subgroups, subspaces, subrings etc. from the original structures. Or df-mgp 20162, which takes a ring and overrides its addition operation with the multiplicative operation, so that we can consider the "multiplicative group" using group and monoid theorems, which expect the operation to be in the +g slot instead of the .r slot. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
Theorem	reldmsets 17212	The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
⊢ Rel dom sSet
Theorem	setsvalg 17213	Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))
Theorem	setsval 17214	Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
Theorem	fvsetsid 17215	The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ((𝐹 sSet ⟨𝑋, 𝑌⟩)‘𝑋) = 𝑌)
Theorem	fsets 17216	The structure replacement function is a function. (Contributed by SO, 12-Jul-2018.)
⊢ (((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐹 sSet ⟨𝑋, 𝑌⟩):𝐴⟶𝐵)
Theorem	setsdm 17217	The domain of a structure with replacement is the domain of the original structure extended by the index of the replacement. (Contributed by AV, 7-Jun-2021.)
⊢ ((𝐺 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
Theorem	setsfun 17218	A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.)
⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ (𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊)) → Fun (𝐺 sSet ⟨𝐼, 𝐸⟩))
Theorem	setsfun0 17219	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 17218 is useful for proofs based on isstruct2 17196 which requires Fun (𝐹 ∖ {∅}) for 𝐹 to be an extensible structure. (Contributed by AV, 7-Jun-2021.)
⊢ (((𝐺 ∈ 𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
Theorem	setsn0fun 17220	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.)
⊢ (𝜑 → 𝑆 Struct 𝑋)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
Theorem	setsstruct2 17221	An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
⊢ (((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1st ‘𝑋), 𝐼, (1st ‘𝑋)), if(𝐼 ≤ (2nd ‘𝑋), (2nd ‘𝑋), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌)
Theorem	setsexstruct2 17222*	An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → ∃𝑦(𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑦)
Theorem	setsstruct 17223	An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 9-Jun-2021.) (Revised by AV, 14-Nov-2021.)
⊢ ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩)
Theorem	wunsets 17224	Closure of structure replacement in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑆 sSet 𝐴) ∈ 𝑈)
Theorem	setsres 17225	The structure replacement function does not affect the value of 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ (𝑆 ∈ 𝑉 → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴})))
Theorem	setsabs 17226	Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))
Theorem	setscom 17227	Different components can be set in any order. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (((𝑆 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑊 ∧ 𝐷 ∈ 𝑋)) → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))
7.1.1.3  Slots
Syntax	cslot 17228	Extend class notation with the slot function.
class Slot 𝐴
Definition	df-slot 17229*	Define the slot extractor for extensible structures. The class Slot 𝐴 is a function whose argument can be any set, although it is meaningful only if that set is a member of an extensible structure (such as a partially ordered set (df-poset 18383) or a group (df-grp 18976)). Note that Slot 𝐴 is implemented as "evaluation at 𝐴". That is, (Slot 𝐴‘𝑆) is defined to be (𝑆‘𝐴), where 𝐴 will typically be an index (which is implemented as a small natural number) of a component of an extensible structure 𝑆. Each extensible structure is a function defined on specific (natural number) "slots", and the function Slot 𝐴 extracts the structure's component as a function value at a particular slot (with index 𝐴). The special "structure" ndx, defined as the identity function restricted to ℕ, can be used to extract the number 𝐴 from a slot, since (Slot 𝐴‘ndx) = 𝐴 (see ndxarg 17243). This is typically used to refer to the number of a slot when defining structures without having to expose the detail of what that number is (for instance, we use the expression (Base‘ndx) in theorems and proofs instead of its hard-coded, numeric value 1), and discourage using the specific definition of slot extractors like Base = Slot 1 (see df-base 17259). Actually, these definitions are used in two basic theorems named *id (theorems of the form 𝐶 = Slot (𝐶‘ndx)) and *ndx (theorems of the form (𝐶‘ndx) = 𝑁) only (see, for example, baseid 17261 and basendx 17267), except additionally in the discouraged theorem baseval 17260 to demonstrate the representations of the value of the base set extractor. The *id theorems are implementation independent equivalents of the definitions by the means of ndxid 17244, but the *ndx theorems still depend on the hard-coded values of the indices. Therefore, the usage of these *ndx theorems is also discouraged (for more details see the section header comment mmtheorems.html#cnx 17244). Example: The group operation is the second component, i.e., the component in the second slot, of a group-like structure 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩} (see grpstr 17343). The slot extractor +g = Slot 2 (see df-plusg 17324) applied on the structure 𝐺 provides the group operation + = (+g‘𝐺). Expanding the definitions, we get + = (Slot 2‘𝐺) = (𝐺‘2) = (𝐺‘(+g‘ndx)) (for the last equation, see plusgndx 17337). The class Slot cannot be defined as (𝑥 ∈ V ↦ (𝑓 ∈ V ↦ (𝑓‘𝑥))) because each Slot 𝐴 is a function on the proper class V so is itself a proper class, and the values of functions are sets (fvex 6933). It is necessary to allow proper classes as values of Slot 𝐴 since for instance the class of all (base sets of) groups is proper. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥‘𝐴))
Theorem	sloteq 17230	Equality theorem for the Slot construction. The converse holds if 𝐴 (or 𝐵) is a set. (Contributed by BJ, 27-Dec-2021.)
⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)
Theorem	slotfn 17231	A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐸 = Slot 𝑁    ⇒   ⊢ 𝐸 Fn V
Theorem	strfvnd 17232	Deduction version of strfvn 17233. (Contributed by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁))
Theorem	strfvn 17233	Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 17259) and 𝑁 is the index of the component. 𝑆 is a structure, i.e. a specific member of a class of structures such as Poset (df-poset 18383) where 𝑆 ∈ Poset. Hint: Do not substitute 𝑁 by a specific (positive) integer to be independent of a hard-coded index value. Often, (𝐸‘ndx) can be used instead of 𝑁. Alternatively, use strfv 17251 instead of strfvn 17233. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.)
⊢ 𝑆 ∈ V    &   ⊢ 𝐸 = Slot 𝑁    ⇒   ⊢ (𝐸‘𝑆) = (𝑆‘𝑁)
Theorem	strfvss 17234	A structure component extractor produces a value which is contained in a set dependent on 𝑆, but not 𝐸. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝐸 = Slot 𝑁    ⇒   ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆
Theorem	wunstr 17235	Closure of a structure index in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐸‘𝑆) ∈ 𝑈)
Theorem	str0 17236	All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐹 = Slot 𝐼    ⇒   ⊢ ∅ = (𝐹‘∅)
Theorem	strfvi 17237	Structure slot extractors cannot distinguish between proper classes and ∅, so they can be protected using the identity function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑋 = (𝐸‘𝑆)    ⇒   ⊢ 𝑋 = (𝐸‘( I ‘𝑆))
Theorem	fveqprc 17238	Lemma for showing the equality of values for functions like slot extractors 𝐸 at a proper class. Extracted from several former proofs of lemmas like zlmlem 21550. (Contributed by AV, 31-Oct-2024.)
⊢ (𝐸‘∅) = ∅    &   ⊢ 𝑌 = (𝐹‘𝑋)    ⇒   ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑌))
Theorem	oveqprc 17239	Lemma for showing the equality of values for functions like slot extractors 𝐸 at a proper class. Extracted from several former proofs of lemmas like resvlem 33322. (Contributed by AV, 31-Oct-2024.)
⊢ (𝐸‘∅) = ∅    &   ⊢ 𝑍 = (𝑋𝑂𝑌)    &   ⊢ Rel dom 𝑂    ⇒   ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑍))
7.1.1.4  Structure component indices The structure component index extractor ndx, defined in this subsection, is used to get the numeric argument from a defined structure component extractor such as df-base 17259 (see ndxarg 17243). For each defined structure component extractor, there should be a corresponding specific theorem providing its index, like basendx 17267. The usage of these theorems, however, is discouraged since the particular value for the index is an implementation detail. It is generally sufficient to work with (Base‘ndx) instead of the hard-coded index value, and use theorems such as baseid 17261 and basendxnplusgndx 17341. The main circumstance in which it is necessary to look at indices directly is when showing that a set of indices are disjoint (for example in proofs such as cznabel 47983, based on setsnid 17256) or even ordered (in proofs such as lmodstr 17384). The requirement that the indices are distinct is necessary for sets of ordered pairs to be extensible structures, whereas the ordering allows for proofs avoiding the usage of quadradically many inequalities (compare cnfldfun 21401 with cnfldfunALT 21402). As for the inequalities, it is recommended to provide them explicitly as theorems like basendxnplusgndx 17341, whenever they are required. Since these theorems use discouraged slot theorems, they should be placed near the definition of a slot (within the same subsection), so that the range of usages of discouraged theorems is tightly limited. Although there could be quadradically many of them in the total number of indices, much less are actually available (and not much more are expected). As for the ordering, there are some theorems like basendxltplusgndx 17340 providing the less-than relationship between two indices. These theorems are also proved by discouraged theorems, so they should be placed near the definition of a slot (within the same subsection), too. However, since such theorems are rarely used (in structure building theorems *str like rngstr 17357), it is not recommended to provide explicit theorems for all of them, but to use the (discouraged) *ndx theorems as in lmodstr 17384. Therefore, *str theorems generally depend on the hard-coded values of the indices.
Syntax	cnx 17240	Extend class notation with the structure component index extractor.
class ndx
Definition	df-ndx 17241	Define the structure component index extractor. See Theorem ndxarg 17243 to understand its purpose. The restriction to ℕ ensures that ndx is a set. The restriction to some set is necessary since I is a proper class. In principle, we could have chosen ℂ or (if we revise all structure component definitions such as df-base 17259) another set such as the set of finite ordinals ω (df-om 7904). (Contributed by NM, 4-Sep-2011.)
⊢ ndx = ( I ↾ ℕ)
Theorem	wunndx 17242	Closure of the index extractor in an infinite weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ (𝜑 → ndx ∈ 𝑈)
Theorem	ndxarg 17243	Get the numeric argument from a defined structure component extractor such as df-base 17259. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝐸‘ndx) = 𝑁
Theorem	ndxid 17244	A structure component extractor is defined by its own index. This theorem, together with strfv 17251 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 17259 and the ;10 in df-ple 17331, making it easier to change should the need arise. For example, we can refer to a specific poset with base set 𝐵 and order relation 𝐿 using {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), 𝐿⟩} rather than {⟨1, 𝐵⟩, ⟨;10, 𝐿⟩}. The latter, while shorter to state, requires revision if we later change ;10 to some other number, and it may also be harder to remember. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝐸 = Slot (𝐸‘ndx)
Theorem	strndxid 17245	The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.) (New usage is discouraged.) Use strfvnd 17232 directly with 𝑁 set to (𝐸‘ndx) if possible.
⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸‘𝑆))
Theorem	setsidvald 17246	Value of the structure replacement function, deduction version. Hint: Do not substitute 𝑁 by a specific (positive) integer to be independent of a hard-coded index value. Often, (𝐸‘ndx) can be used instead of 𝑁. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 17-Oct-2024.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → Fun 𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ dom 𝑆)    ⇒   ⊢ (𝜑 → 𝑆 = (𝑆 sSet ⟨𝑁, (𝐸‘𝑆)⟩))
Theorem	setsidvaldOLD 17247	Obsolete version of setsidvald 17246 as of 17-Oct-2024. (Contributed by AV, 14-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → Fun 𝑆)    &   ⊢ (𝜑 → (𝐸‘ndx) ∈ dom 𝑆)    ⇒   ⊢ (𝜑 → 𝑆 = (𝑆 sSet ⟨(𝐸‘ndx), (𝐸‘𝑆)⟩))
Theorem	strfvd 17248	Deduction version of strfv 17251. (Contributed by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → Fun 𝑆)    &   ⊢ (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆))
Theorem	strfv2d 17249	Deduction version of strfv2 17250. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → Fun ◡◡𝑆)    &   ⊢ (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆))
Theorem	strfv2 17250	A variation on strfv 17251 to avoid asserting that 𝑆 itself is a function, which involves sethood of all the ordered pair components of 𝑆. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ 𝑆 ∈ V    &   ⊢ Fun ◡◡𝑆    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆    ⇒   ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))
Theorem	strfv 17251	Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 (such as a member of Poset, df-poset 18383) with a component extractor 𝐸 (such as the base set extractor df-base 17259). By virtue of ndxid 17244, this can be done without having to refer to the hard-coded numeric index of 𝐸. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
⊢ 𝑆 Struct 𝑋    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆    ⇒   ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆))
Theorem	strfv3 17252	Variant on strfv 17251 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.)
⊢ (𝜑 → 𝑈 = 𝑆)    &   ⊢ 𝑆 Struct 𝑋    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ 𝐴 = (𝐸‘𝑈)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	strssd 17253	Deduction version of strss 17254. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    &   ⊢ (𝜑 → Fun 𝑇)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑇)    &   ⊢ (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐸‘𝑇) = (𝐸‘𝑆))
Theorem	strss 17254	Propagate component extraction to a structure 𝑇 from a subset structure 𝑆. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 15-Jan-2014.)
⊢ 𝑇 ∈ V    &   ⊢ Fun 𝑇    &   ⊢ 𝑆 ⊆ 𝑇    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆    ⇒   ⊢ (𝐸‘𝑇) = (𝐸‘𝑆)
Theorem	setsid 17255	Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐸 = Slot (𝐸‘ndx)    ⇒   ⊢ ((𝑊 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 = (𝐸‘(𝑊 sSet ⟨(𝐸‘ndx), 𝐶⟩)))
Theorem	setsnid 17256	Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 7-Nov-2024.)
⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ 𝐷    ⇒   ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩))
Theorem	setsnidOLD 17257	Obsolete proof of setsnid 17256 as of 7-Nov-2024. Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ 𝐷    ⇒   ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩))
7.1.1.5  Base sets
Syntax	cbs 17258	Extend class notation with the class of all base set extractors.
class Base
Definition	df-base 17259	Define the base set (also called underlying set, ground set, carrier set, or carrier) extractor for extensible structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) Use its index-independent form baseid 17261 instead. (New usage is discouraged.)
⊢ Base = Slot 1
Theorem	baseval 17260	Value of the base set extractor. (Normally it is preferred to work with (Base‘ndx) rather than the hard-coded 1 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.)
⊢ 𝐾 ∈ V    ⇒   ⊢ (Base‘𝐾) = (𝐾‘1)
Theorem	baseid 17261	Utility theorem: index-independent form of df-base 17259. (Contributed by NM, 20-Oct-2012.)
⊢ Base = Slot (Base‘ndx)
Theorem	basfn 17262	The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
⊢ Base Fn V
Theorem	base0 17263	The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ ∅ = (Base‘∅)
Theorem	elbasfv 17264	Utility theorem: reverse closure for any structure defined as a function. (Contributed by Stefan O'Rear, 24-Aug-2015.)
⊢ 𝑆 = (𝐹‘𝑍)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ (𝑋 ∈ 𝐵 → 𝑍 ∈ V)
Theorem	elbasov 17265	Utility theorem: reverse closure for any structure defined as a two-argument function. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ Rel dom 𝑂    &   ⊢ 𝑆 = (𝑋𝑂𝑌)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Theorem	strov2rcl 17266	Partial reverse closure for any structure defined as a two-argument function. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 2-Dec-2019.)
⊢ 𝑆 = (𝐼𝐹𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ Rel dom 𝐹    ⇒   ⊢ (𝑋 ∈ 𝐵 → 𝐼 ∈ V)
Theorem	basendx 17267	Index value of the base set extractor. (Contributed by Mario Carneiro, 2-Aug-2013.) Use of this theorem is discouraged since the particular value 1 for the index is an implementation detail, see section header comment mmtheorems.html#cnx for more information. (New usage is discouraged.)
⊢ (Base‘ndx) = 1
Theorem	basendxnn 17268	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.) (Proof shortened by AV, 13-Oct-2024.)
⊢ (Base‘ndx) ∈ ℕ
Theorem	basendxnnOLD 17269	Obsolete proof of basendxnn 17268 as of 13-Oct-2024. (Contributed by AV, 23-Sep-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (Base‘ndx) ∈ ℕ
Theorem	basndxelwund 17270	The index of the base set is an element in a weak universe containing the natural numbers. Formerly part of proof for 1strwun 17278. (Contributed by AV, 27-Mar-2020.) (Revised by AV, 17-Oct-2024.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
Theorem	basprssdmsets 17271	The pair of the base index and another index is a subset of the domain of the structure obtained by replacing/adding a slot at the other index in a structure having a base slot. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
⊢ (𝜑 → 𝑆 Struct 𝑋)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)    ⇒   ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))
Theorem	opelstrbas 17272	The base set of a structure with a base set. (Contributed by AV, 10-Nov-2021.)
⊢ (𝜑 → 𝑆 Struct 𝑋)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑌)    &   ⊢ (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝑉 = (Base‘𝑆))
Theorem	1strstr 17273	A constructed one-slot structure. Depending on hard-coded index. Use 1strstr1 17274 instead. (Contributed by AV, 27-Mar-2020.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ 𝐺 Struct ⟨1, 1⟩
Theorem	1strstr1 17274	A constructed one-slot structure. (Contributed by AV, 15-Nov-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ 𝐺 Struct ⟨(Base‘ndx), (Base‘ndx)⟩
Theorem	1strbas 17275	The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Proof shortened by AV, 15-Nov-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	1strbasOLD 17276	Obsolete proof of 1strbas 17275 as of 15-Nov-2024. The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	1strwunbndx 17277	A constructed one-slot structure in a weak universe containing the index of the base set extractor. (Contributed by AV, 27-Mar-2020.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)
Theorem	1strwun 17278	A constructed one-slot structure in a weak universe. (Contributed by AV, 27-Mar-2020.) (Proof shortened by AV, 17-Oct-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)
Theorem	1strwunOLD 17279	Obsolete version of 1strwun 17278 as of 17-Oct-2024. A constructed one-slot structure in a weak universe. (Contributed by AV, 27-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑈) → 𝐺 ∈ 𝑈)
Theorem	2strstr 17280	A constructed two-slot structure. Depending on hard-coded indices. Use 2strstr1 17283 instead. (Contributed by Mario Carneiro, 29-Aug-2015.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 1 < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝐺 Struct ⟨1, 𝑁⟩
Theorem	2strbas 17281	The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) Use the index-independent version 2strbas1 17285 instead. (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 1 < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	2strop 17282	The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) Use the index-independent version 2strop1 17286 instead. (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(𝐸‘ndx), + ⟩}    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 1 < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ ( + ∈ 𝑉 → + = (𝐸‘𝐺))
Theorem	2strstr1 17283	A constructed two-slot structure. Version of 2strstr 17280 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Proof shortened by AV, 17-Oct-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩
Theorem	2strstr1OLD 17284	Obsolete version of 2strstr1 17283 as of 27-Oct-2024. A constructed two-slot structure. Version of 2strstr 17280 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩
Theorem	2strbas1 17285	The base set of a constructed two-slot structure. Version of 2strbas 17281 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
Theorem	2strop1 17286	The other slot of a constructed two-slot structure. Version of 2strop 17282 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩}    &   ⊢ (Base‘ndx) < 𝑁    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐸 = Slot 𝑁    ⇒   ⊢ ( + ∈ 𝑉 → + = (𝐸‘𝐺))
7.1.1.6  Base set restrictions
Syntax	cress 17287	Extend class notation with the extensible structure builder restriction operator.
class ↾s
Definition	df-ress 17288*	Define a multifunction restriction operator for extensible structures, which can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the Base set while leaving operators alone; individual kinds of structures will need to handle this behavior, by ignoring operators' values outside the range (like Ring), defining a function using the base set and applying that (like TopGrp), or explicitly truncating the slot before use (like MetSp). (Credit for this operator goes to Mario Carneiro.) See ressbas 17293 for the altered base set, and resseqnbas 17300 (subrg0 20607, ressplusg 17349, subrg1 20610, ressmulr 17366) for the (un)altered other operations. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
Theorem	reldmress 17289	The structure restriction is a proper operator, so it can be used with ovprc1 7487. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ Rel dom ↾s
Theorem	ressval 17290	Value of structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
Theorem	ressid2 17291	General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊)
Theorem	ressval2 17292	Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
Theorem	ressbas 17293	Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅))
Theorem	ressbasOLD 17294	Obsolete proof of ressbas 17293 as of 7-Nov-2024. Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅))
Theorem	ressbasssg 17295	The base set of a restriction to 𝐴 is a subset of 𝐴 and the base set 𝐵 of the original structure. (Contributed by SN, 10-Jan-2025.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (Base‘𝑅) ⊆ (𝐴 ∩ 𝐵)
Theorem	ressbas2 17296	Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝑅))
Theorem	ressbasss 17297	The base set of a restriction is a subset of the base set of the original structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof shortened by SN, 25-Feb-2025.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (Base‘𝑅) ⊆ 𝐵
Theorem	ressbasssOLD 17298	Obsolete proof of ressbas 17293 as of 25-Feb-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (Base‘𝑅) ⊆ 𝐵
Theorem	ressbasss2 17299	The base set of a restriction to 𝐴 is a subset of 𝐴. (Contributed by SN, 10-Jan-2025.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    ⇒   ⊢ (Base‘𝑅) ⊆ 𝐴
Theorem	resseqnbas 17300	The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)
⊢ 𝑅 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐶 = (𝐸‘𝑊)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (Base‘ndx)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅))