P. 172 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17101-17200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	139prm 17101	139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ ;;139 ∈ ℙ
Theorem	163prm 17102	163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ ;;163 ∈ ℙ
Theorem	317prm 17103	317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ ;;317 ∈ ℙ
Theorem	631prm 17104	631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ ;;631 ∈ ℙ
Theorem	prmo4 17105	The primorial of 4. (Contributed by AV, 28-Aug-2020.)
⊢ (#p‘4) = 6
Theorem	prmo5 17106	The primorial of 5. (Contributed by AV, 28-Aug-2020.)
⊢ (#p‘5) = ;30
Theorem	prmo6 17107	The primorial of 6. (Contributed by AV, 28-Aug-2020.)
⊢ (#p‘6) = ;30
6.2.20  Very large primes
Theorem	1259lem1 17108	Lemma for 1259prm 17113. Calculate a power mod. In decimal, we calculate 2↑16 = 52𝑁 + 68≡68 and 2↑17≡68 · 2 = 136 in this lemma. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;1259    ⇒   ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)
Theorem	1259lem2 17109	Lemma for 1259prm 17113. Calculate a power mod. In decimal, we calculate 2↑34 = (2↑17)↑2≡136↑2≡14𝑁 + 870. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.)
⊢ 𝑁 = ;;;1259    ⇒   ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)
Theorem	1259lem3 17110	Lemma for 1259prm 17113. Calculate a power mod. In decimal, we calculate 2↑38 = 2↑34 · 2↑4≡870 · 16 = 11𝑁 + 71 and 2↑76 = (2↑34)↑2≡71↑2 = 4𝑁 + 5≡5. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;1259    ⇒   ⊢ ((2↑;76) mod 𝑁) = (5 mod 𝑁)
Theorem	1259lem4 17111	Lemma for 1259prm 17113. Calculate a power mod. In decimal, we calculate 2↑306 = (2↑76)↑4 · 4≡5↑4 · 4 = 2𝑁 − 18, 2↑612 = (2↑306)↑2≡18↑2 = 324, 2↑629 = 2↑612 · 2↑17≡324 · 136 = 35𝑁 − 1 and finally 2↑(𝑁 − 1) = (2↑629)↑2≡1↑2 = 1. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;1259    ⇒   ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
Theorem	1259lem5 17112	Lemma for 1259prm 17113. Calculate the GCD of 2↑34 − 1≡869 with 𝑁 = 1259. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
⊢ 𝑁 = ;;;1259    ⇒   ⊢ (((2↑;34) − 1) gcd 𝑁) = 1
Theorem	1259prm 17113	1259 is a prime number. (Contributed by Mario Carneiro, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ 𝑁 = ;;;1259    ⇒   ⊢ 𝑁 ∈ ℙ
Theorem	2503lem1 17114	Lemma for 2503prm 17117. Calculate a power mod. In decimal, we calculate 2↑18 = 512↑2 = 104𝑁 + 1832≡1832. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;2503    ⇒   ⊢ ((2↑;18) mod 𝑁) = (;;;1832 mod 𝑁)
Theorem	2503lem2 17115	Lemma for 2503prm 17117. Calculate a power mod. We calculate 2↑19 = 2↑18 · 2≡1832 · 2 = 𝑁 + 1161, 2↑38 = (2↑19)↑2≡1161↑2 = 538𝑁 + 1307, 2↑39 = 2↑38 · 2≡1307 · 2 = 𝑁 + 111, 2↑78 = (2↑39)↑2≡111↑2 = 5𝑁 − 194, 2↑156 = (2↑78)↑2≡194↑2 = 15𝑁 + 91, 2↑312 = (2↑156)↑2≡91↑2 = 3𝑁 + 772, 2↑624 = (2↑312)↑2≡772↑2 = 238𝑁 + 270, 2↑1248 = (2↑624)↑2≡270↑2 = 29𝑁 + 313, 2↑1251 = 2↑1248 · 8≡313 · 8 = 𝑁 + 1 and finally 2↑(𝑁 − 1) = (2↑1251)↑2≡1↑2 = 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;2503    ⇒   ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
Theorem	2503lem3 17116	Lemma for 2503prm 17117. Calculate the GCD of 2↑18 − 1≡1831 with 𝑁 = 2503. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.)
⊢ 𝑁 = ;;;2503    ⇒   ⊢ (((2↑;18) − 1) gcd 𝑁) = 1
Theorem	2503prm 17117	2503 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
⊢ 𝑁 = ;;;2503    ⇒   ⊢ 𝑁 ∈ ℙ
Theorem	4001lem1 17118	Lemma for 4001prm 17122. Calculate a power mod. In decimal, we calculate 2↑12 = 4096 = 𝑁 + 95, 2↑24 = (2↑12)↑2≡95↑2 = 2𝑁 + 1023, 2↑25 = 2↑24 · 2≡1023 · 2 = 2046, 2↑50 = (2↑25)↑2≡2046↑2 = 1046𝑁 + 1070, 2↑100 = (2↑50)↑2≡1070↑2 = 286𝑁 + 614 and 2↑200 = (2↑100)↑2≡614↑2 = 94𝑁 + 902 ≡902. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;4001    ⇒   ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁)
Theorem	4001lem2 17119	Lemma for 4001prm 17122. Calculate a power mod. In decimal, we calculate 2↑400 = (2↑200)↑2≡902↑2 = 203𝑁 + 1401 and 2↑800 = (2↑400)↑2≡1401↑2 = 490𝑁 + 2311 ≡2311. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;4001    ⇒   ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁)
Theorem	4001lem3 17120	Lemma for 4001prm 17122. Calculate a power mod. In decimal, we calculate 2↑1000 = 2↑800 · 2↑200≡2311 · 902 = 521𝑁 + 1 and finally 2↑(𝑁 − 1) = (2↑1000)↑4≡1↑4 = 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;4001    ⇒   ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
Theorem	4001lem4 17121	Lemma for 4001prm 17122. Calculate the GCD of 2↑800 − 1≡2310 with 𝑁 = 4001. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;4001    ⇒   ⊢ (((2↑;;800) − 1) gcd 𝑁) = 1
Theorem	4001prm 17122	4001 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
⊢ 𝑁 = ;;;4001    ⇒   ⊢ 𝑁 ∈ ℙ
PART 7  BASIC STRUCTURES
7.1  Extensible structures
7.1.1  Basic definitions An "extensible structure" (or "structure" in short, at least in this section) is used to define a specific group, ring, poset, and so on. An extensible structure can contain many components. For example, a group will have at least two components (base set and operation), although it can be further specialized by adding other components such as a multiplicative operation for rings (and still remain a group per our definition). Thus, every ring is also a group. This extensible structure approach allows theorems from more general structures (such as groups) to be reused for more specialized structures (such as rings) without having to reprove anything. Structures are common in mathematics, but in informal (natural language) proofs the details are assumed in ways that we must make explicit. An extensible structure is implemented as a function (a set of ordered pairs) on a finite (and not necessarily sequential) subset of ℕ. The function's argument is the index of a structure component (such as 1 for the base set of a group), and its value is the component (such as the base set). By convention, we normally avoid direct reference to the hard-coded numeric index and instead use structure component extractors such as ndxid 17174 and strfv 17180. Using extractors makes it easier to change numeric indices and also makes the components' purpose clearer. For example, as noted in ndxid 17174, we can refer to a specific poset with base set 𝐵 and order relation 𝐿 using the extensible structure {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), 𝐿⟩} rather than {⟨1, 𝐵⟩, ⟨;10, 𝐿⟩}. See section header comment mmtheorems.html#cnx 17174 for more details on numeric indices versus the structure component extractors. There are many other possible ways to handle structures. We chose this extensible structure approach because this approach (1) results in simpler notation than other approaches we are aware of, and (2) is easier to do proofs with. We cannot use an approach that uses "hidden" arguments; Metamath does not support hidden arguments, and in any case we want nothing hidden. It would be possible to use a categorical approach (e.g., something vaguely similar to Lean's mathlib). However, instances (the chain of proofs that an 𝑋 is a 𝑌 via a bunch of forgetful functors) can cause serious performance problems for automated tooling, and the resulting proofs would be painful to look at directly (in the case of Lean, they are long past the level where people would find it acceptable to look at them directly). Metamath is working under much stricter conditions than this, and it has still managed to achieve about the same level of flexibility through this "extensible structure" approach. To create a substructure of a given extensible structure, you can simply use the multifunction restriction operator for extensible structures ↾s as defined in df-ress 17208. This 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). For example, the unital ring of integers ℤring is defined in df-zring 21364 as simply ℤring = (ℂfld ↾s ℤ). This can be similarly done for all other subsets of ℂ, which has all the structure we can show applies to it, and this all comes "for free". Should we come up with some new structure in the future that we wish ℂ to inherit, then we change the definition of ℂfld, reprove all the slot extraction theorems, add a new one, and that's it. None of the other downstream theorems have to change. Note that the construct of df-prds 17417 addresses a different situation. It is not possible to have SubGrp and SubRing be the same thing because they produce different outputs on the same input. The subgroups of an extensible structure treated as a group are not the same as the subrings of that same structure. With df-prds 17417 it can actually reasonably perform the task, that is, being the product group given a family of groups, while also being the product ring given a family of rings. There is no contradiction here because the group part of a product ring is a product group. There is also a general theory of "substructure algebras", in the form of df-mre 17554 and df-acs 17557. SubGrp is a Moore collection, as is SubRing, SubRng and many other substructure collections. But it is not useful for picking out a particular collection of interest; SubRing and SubGrp still need to be defined and they are distinct --- nothing is going to select these definitions for us. Extensible structures only work well when they represent concrete categories, where there is a "base set", morphisms are functions, and subobjects are subsets with induced operations. In short, they primarily work well for "sets with (some) extra structure". Extensible structures may not suffice for more complicated situations. For example, in manifolds, ↾s would not work. That said, extensible structures are sufficient for many of the structures that set.mm currently considers, and offer a good compromise for a goal-oriented formalization.
7.1.1.1  Extensible structures as structures with components
Syntax	cstr 17123	Extend class notation with the class of structures with components numbered below 𝐴.
class Struct
Definition	df-struct 17124*	Define a structure with components in 𝑀...𝑁. This is not a requirement for groups, posets, etc., but it is a useful assumption for component extraction theorems. As mentioned in the section header, an "extensible structure should be implemented as a function (a set of ordered pairs)". The current definition, however, is less restrictive: it allows for classes which contain the empty set ∅ to be extensible structures. Because of 0nelfun 6537, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17128: 𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅}). Allowing an extensible structure to contain the empty set ensures that expressions like {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} are structures without asserting or implying that 𝐴, 𝐵, 𝐶 and 𝐷 are sets (if 𝐴 or 𝐵 is a proper class, then ⟨𝐴, 𝐵⟩ = ∅, see opprc 4863). This is used critically in strle1 17135, strle2 17136, strle3 17137 and strleun 17134 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17306 and theorems like it will have many sethood assumptions, and may not even be usable in the empty context. Instead, the sethood assumption is deferred until it is actually needed, e.g., ipsbase 17307, which requires that the base set be a set but not any of the other components. Usually, a concrete structure like ℂfld does not contain the empty set, and therefore is a function, see cnfldfun 21285. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
Theorem	brstruct 17125	The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ Rel Struct
Theorem	isstruct2 17126	The property of being a structure with components in (1st ‘𝑋)...(2nd ‘𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))
Theorem	structex 17127	A structure is a set. (Contributed by AV, 10-Nov-2021.)
⊢ (𝐺 Struct 𝑋 → 𝐺 ∈ V)
Theorem	structn0fun 17128	A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.)
⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅}))
Theorem	isstruct 17129	The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ (𝐹 Struct ⟨𝑀, 𝑁⟩ ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁)))
Theorem	structcnvcnv 17130	Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ (𝐹 Struct 𝑋 → ◡◡𝐹 = (𝐹 ∖ {∅}))
Theorem	structfung 17131	The converse of the converse of a structure is a function. Closed form of structfun 17132. (Contributed by AV, 12-Nov-2021.)
⊢ (𝐹 Struct 𝑋 → Fun ◡◡𝐹)
Theorem	structfun 17132	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 17133	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 17134	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 17135	Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
⊢ 𝐼 ∈ ℕ    &   ⊢ 𝐴 = 𝐼    ⇒   ⊢ {⟨𝐴, 𝑋⟩} Struct ⟨𝐼, 𝐼⟩
Theorem	strle2 17136	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 17137	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 17138*	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 17139*	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 17140	Set components of a structure.
class sSet
Definition	df-sets 17141*	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 17208 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 20057, 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 17142	The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
⊢ Rel dom sSet
Theorem	setsvalg 17143	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 17144	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 17145	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 17146	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 17147	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 17148	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 17149	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 17148 is useful for proofs based on isstruct2 17126 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 17150	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 17151	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 17152*	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 17153	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 17154	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 17155	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 17156	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 17157	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 17158	Extend class notation with the slot function.
class Slot 𝐴
Definition	df-slot 17159*	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 18281) or a group (df-grp 18875)). 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 17173). 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 17187). 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 17189 and basendx 17195), except additionally in the discouraged theorem baseval 17188 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 17174, 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 17174). 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), + ⟩}. The slot extractor +g = Slot 2 (see df-plusg 17240) 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 17253). 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 6874). 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 17160	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 17161	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 17162	Deduction version of strfvn 17163. (Contributed by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁))
Theorem	strfvn 17163	Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 17187) 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 18281) 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 17180 instead of strfvn 17163. (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 17164	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 17165	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 17166	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 17167	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 17168	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 21433. (Contributed by AV, 31-Oct-2024.)
⊢ (𝐸‘∅) = ∅    &   ⊢ 𝑌 = (𝐹‘𝑋)    ⇒   ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑌))
Theorem	oveqprc 17169	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 33312. (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 17187 (see ndxarg 17173). For each defined structure component extractor, there should be a corresponding specific theorem providing its index, like basendx 17195. 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 17189 and basendxnplusgndx 17257. 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 48252, based on setsnid 17185) or even ordered (in proofs such as lmodstr 17295). 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 21285 with cnfldfunALT 21286). As for the inequalities, it is recommended to provide them explicitly as theorems like basendxnplusgndx 17257, 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 17256 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 17268), it is not recommended to provide explicit theorems for all of them, but to use the (discouraged) *ndx theorems as in lmodstr 17295. Therefore, *str theorems generally depend on the hard-coded values of the indices.
Syntax	cnx 17170	Extend class notation with the structure component index extractor.
class ndx
Definition	df-ndx 17171	Define the structure component index extractor. See Theorem ndxarg 17173 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 17187) another set such as the set of finite ordinals ω (df-om 7846). (Contributed by NM, 4-Sep-2011.)
⊢ ndx = ( I ↾ ℕ)
Theorem	wunndx 17172	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 17173	Get the numeric argument from a defined structure component extractor such as df-base 17187. (Contributed by Mario Carneiro, 6-Oct-2013.)
⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    ⇒   ⊢ (𝐸‘ndx) = 𝑁
Theorem	ndxid 17174	A structure component extractor is defined by its own index. This theorem, together with strfv 17180 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 17187 and the ;10 in df-ple 17247, 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 17175	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 17162 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 17176	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	strfvd 17177	Deduction version of strfv 17180. (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 17178	Deduction version of strfv2 17179. (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 17179	A variation on strfv 17180 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 17180	Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 (such as a member of Poset, df-poset 18281) with a component extractor 𝐸 (such as the base set extractor df-base 17187). By virtue of ndxid 17174, 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 17181	Variant on strfv 17180 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 17182	Deduction version of strss 17183. (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 17183	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 17184	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 17185	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⟩))
7.1.1.5  Base sets
Syntax	cbs 17186	Extend class notation with the class of all base set extractors.
class Base
Definition	df-base 17187	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 17189 instead. (New usage is discouraged.)
⊢ Base = Slot 1
Theorem	baseval 17188	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 17189	Utility theorem: index-independent form of df-base 17187. (Contributed by NM, 20-Oct-2012.)
⊢ Base = Slot (Base‘ndx)
Theorem	basfn 17190	The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
⊢ Base Fn V
Theorem	base0 17191	The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ ∅ = (Base‘∅)
Theorem	elbasfv 17192	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 17193	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 17194	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 17195	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 17196	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	basndxelwund 17197	The index of the base set is an element in a weak universe containing the natural numbers. Formerly part of proof for 1strwun 17203. (Contributed by AV, 27-Mar-2020.) (Revised by AV, 17-Oct-2024.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
Theorem	basprssdmsets 17198	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 17199	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 17200	A constructed one-slot structure. (Contributed by AV, 15-Nov-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   ⊢ 𝐺 Struct ⟨(Base‘ndx), (Base‘ndx)⟩