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Theorem List for Metamath Proof Explorer - 16501-16600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremstructfun 16501 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

Theoremstructfn 16502 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐹 Struct ⟨𝑀, 𝑁       (Fun 𝐹 ∧ dom 𝐹 ⊆ (1...𝑁))

Theoremslotfn 16503 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝐸 = Slot 𝑁       𝐸 Fn V

Theoremstrfvnd 16504 Deduction version of strfvn 16507. (Contributed by Mario Carneiro, 15-Nov-2014.)
𝐸 = Slot 𝑁    &   (𝜑𝑆𝑉)       (𝜑 → (𝐸𝑆) = (𝑆𝑁))

Theorembasfn 16505 The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
Base Fn V

Theoremwunndx 16506 Closure of the index extractor in an infinite weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)       (𝜑 → ndx ∈ 𝑈)

Theoremstrfvn 16507 Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 16491) and 𝑁 is a fixed integer such as 1. 𝑆 is a structure, i.e. a specific member of a class of structures such as Poset (df-poset 17558) where 𝑆 ∈ Poset.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strfv 16533. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.)

𝑆 ∈ V    &   𝐸 = Slot 𝑁       (𝐸𝑆) = (𝑆𝑁)

Theoremstrfvss 16508 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       (𝐸𝑆) ⊆ ran 𝑆

Theoremwunstr 16509 Closure of a structure index in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐸 = Slot 𝑁    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝑆𝑈)       (𝜑 → (𝐸𝑆) ∈ 𝑈)

Theoremndxarg 16510 Get the numeric argument from a defined structure component extractor such as df-base 16491. (Contributed by Mario Carneiro, 6-Oct-2013.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸‘ndx) = 𝑁

Theoremndxid 16511 A structure component extractor is defined by its own index. This theorem, together with strfv 16533 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 16491 and the 10 in df-ple 16587, 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)

Theoremstrndxid 16512 The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.)
(𝜑𝑆𝑉)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸𝑆))

Theoremreldmsets 16513 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Rel dom sSet

Theoremsetsvalg 16514 Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝑆𝑉𝐴𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))

Theoremsetsval 16515 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝑆𝑉𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))

Theoremsetsidvald 16516 Value of the structure replacement function, deduction version. (Contributed by AV, 14-Mar-2020.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → (𝐸‘ndx) ∈ dom 𝑆)       (𝜑𝑆 = (𝑆 sSet ⟨(𝐸‘ndx), (𝐸𝑆)⟩))

Theoremfvsetsid 16517 The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.)
((𝐹𝑉𝑋𝑊𝑌𝑈) → ((𝐹 sSet ⟨𝑋, 𝑌⟩)‘𝑋) = 𝑌)

Theoremfsets 16518 The structure replacement function is a function. (Contributed by SO, 12-Jul-2018.)
(((𝐹𝑉𝐹:𝐴𝐵) ∧ 𝑋𝐴𝑌𝐵) → (𝐹 sSet ⟨𝑋, 𝑌⟩):𝐴𝐵)

Theoremsetsdm 16519 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.)
((𝐺𝑉𝐸𝑊) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))

Theoremsetsfun 16520 A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.)
(((𝐺𝑉 ∧ Fun 𝐺) ∧ (𝐼𝑈𝐸𝑊)) → Fun (𝐺 sSet ⟨𝐼, 𝐸⟩))

Theoremsetsfun0 16521 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 16520 is useful for proofs based on isstruct2 16495 which requires Fun (𝐹 ∖ {∅}) for 𝐹 to be an extensible structure. (Contributed by AV, 7-Jun-2021.)
(((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼𝑈𝐸𝑊)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))

Theoremsetsn0fun 16522 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.)
(𝜑𝑆 Struct 𝑋)    &   (𝜑𝐼𝑈)    &   (𝜑𝐸𝑊)       (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))

Theoremsetsstruct2 16523 An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
(((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌)

Theoremsetsexstruct2 16524* An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → ∃𝑦(𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑦)

Theoremsetsstruct 16525 An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 9-Jun-2021.) (Revised by AV, 14-Nov-2021.)
((𝐸𝑉𝐼 ∈ (ℤ𝑀) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩)

Theoremwunsets 16526 Closure of structure replacement in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝑆𝑈)    &   (𝜑𝐴𝑈)       (𝜑 → (𝑆 sSet 𝐴) ∈ 𝑈)

Theoremsetsres 16527 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.)
(𝑆𝑉 → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴})))

Theoremsetsabs 16528 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.)
((𝑆𝑉𝐶𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))

Theoremsetscom 16529 Component-setting is commutative when the x-values are different. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (((𝑆𝑉𝐴𝐵) ∧ (𝐶𝑊𝐷𝑋)) → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))

Theoremstrfvd 16530 Deduction version of strfv 16533. (Contributed by Mario Carneiro, 15-Nov-2014.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)       (𝜑𝐶 = (𝐸𝑆))

Theoremstrfv2d 16531 Deduction version of strfv2 16532. (Contributed by Mario Carneiro, 30-Apr-2015.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    &   (𝜑𝐶𝑊)       (𝜑𝐶 = (𝐸𝑆))

Theoremstrfv2 16532 A variation on strfv 16533 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       (𝐶𝑉𝐶 = (𝐸𝑆))

Theoremstrfv 16533 Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 (such as a member of Poset, df-poset 17558) with a component extractor 𝐸 (such as the base set extractor df-base 16491). By virtue of ndxid 16511, 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       (𝐶𝑉𝐶 = (𝐸𝑆))

Theoremstrfv3 16534 Variant on strfv 16533 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.)
(𝜑𝑈 = 𝑆)    &   𝑆 Struct 𝑋    &   𝐸 = Slot (𝐸‘ndx)    &   {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆    &   (𝜑𝐶𝑉)    &   𝐴 = (𝐸𝑈)       (𝜑𝐴 = 𝐶)

Theoremstrssd 16535 Deduction version of strss 16536. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑𝑇𝑉)    &   (𝜑 → Fun 𝑇)    &   (𝜑𝑆𝑇)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)       (𝜑 → (𝐸𝑇) = (𝐸𝑆))

Theoremstrss 16536 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    &   𝑆𝑇    &   𝐸 = Slot (𝐸‘ndx)    &   ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆       (𝐸𝑇) = (𝐸𝑆)

Theoremstr0 16537 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‘∅)

Theorembase0 16538 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
∅ = (Base‘∅)

Theoremstrfvi 16539 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 = (𝐸𝑆)       𝑋 = (𝐸‘( I ‘𝑆))

Theoremsetsid 16540 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)       ((𝑊𝐴𝐶𝑉) → 𝐶 = (𝐸‘(𝑊 sSet ⟨(𝐸‘ndx), 𝐶⟩)))

Theoremsetsnid 16541 Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ 𝐷       (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩))

Theoremsbcie2s 16542* A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝐴 = (𝐸𝑊)    &   𝐵 = (𝐹𝑊)    &   ((𝑎 = 𝐴𝑏 = 𝐵) → (𝜑𝜓))       (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏]𝜓𝜑))

Theoremsbcie3s 16543* A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 15-Mar-2019.)
𝐴 = (𝐸𝑊)    &   𝐵 = (𝐹𝑊)    &   𝐶 = (𝐺𝑊)    &   ((𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶) → (𝜑𝜓))       (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎][(𝐹𝑤) / 𝑏][(𝐺𝑤) / 𝑐]𝜓𝜑))

Theorembaseval 16544 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)

Theorembaseid 16545 Utility theorem: index-independent form of df-base 16491. (Contributed by NM, 20-Oct-2012.)
Base = Slot (Base‘ndx)

Theoremelbasfv 16546 Utility theorem: reverse closure for any structure defined as a function. (Contributed by Stefan O'Rear, 24-Aug-2015.)
𝑆 = (𝐹𝑍)    &   𝐵 = (Base‘𝑆)       (𝑋𝐵𝑍 ∈ V)

Theoremelbasov 16547 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)       (𝐴𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V))

Theoremstrov2rcl 16548 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       (𝑋𝐵𝐼 ∈ V)

Theorembasendx 16549 Index 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 Mario Carneiro, 2-Aug-2013.)
(Base‘ndx) = 1

Theorembasendxnn 16550 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.)
(Base‘ndx) ∈ ℕ

Theorembasprssdmsets 16551 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.)
(𝜑𝑆 Struct 𝑋)    &   (𝜑𝐼𝑈)    &   (𝜑𝐸𝑊)    &   (𝜑 → (Base‘ndx) ∈ dom 𝑆)       (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

Theoremreldmress 16552 The structure restriction is a proper operator, so it can be used with ovprc1 7190. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Rel dom ↾s

Theoremressval 16553 Value of structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩)))

Theoremressid2 16554 General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = 𝑊)

Theoremressval2 16555 Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))

Theoremressbas 16556 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       (𝐴𝑉 → (𝐴𝐵) = (Base‘𝑅))

Theoremressbas2 16557 Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       (𝐴𝐵𝐴 = (Base‘𝑅))

Theoremressbasss 16558 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.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       (Base‘𝑅) ⊆ 𝐵

Theoremresslem 16559 Other elements of a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐶 = (𝐸𝑊)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   1 < 𝑁       (𝐴𝑉𝐶 = (𝐸𝑅))

Theoremress0 16560 All restrictions of the null set are trivial. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
(∅ ↾s 𝐴) = ∅

Theoremressid 16561 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝐵 = (Base‘𝑊)       (𝑊𝑋 → (𝑊s 𝐵) = 𝑊)

Theoremressinbas 16562 Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝐵 = (Base‘𝑊)       (𝐴𝑋 → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))

Theoremressval3d 16563 Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 3-Jul-2022.)
𝑅 = (𝑆s 𝐴)    &   𝐵 = (Base‘𝑆)    &   𝐸 = (Base‘ndx)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑𝐸 ∈ dom 𝑆)    &   (𝜑𝐴𝐵)       (𝜑𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Theoremressress 16564 Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)
((𝐴𝑋𝐵𝑌) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s (𝐴𝐵)))

Theoremressabs 16565 Restriction absorption law. (Contributed by Mario Carneiro, 12-Jun-2015.)
((𝐴𝑋𝐵𝐴) → ((𝑊s 𝐴) ↾s 𝐵) = (𝑊s 𝐵))

Theoremwunress 16566 Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝑊𝑈)       (𝜑 → (𝑊s 𝐴) ∈ 𝑈)

7.1.2  Slot definitions

Syntaxcplusg 16567 Extend class notation with group (addition) operation.
class +g

Syntaxcmulr 16568 Extend class notation with ring multiplication.
class .r

Syntaxcstv 16569 Extend class notation with involution.
class *𝑟

Syntaxcsca 16570 Extend class notation with scalar field.
class Scalar

Syntaxcvsca 16571 Extend class notation with scalar product.
class ·𝑠

Syntaxcip 16572 Extend class notation with Hermitian form (inner product).
class ·𝑖

Syntaxcts 16573 Extend class notation with the topology component of a topological space.
class TopSet

Syntaxcple 16574 Extend class notation with "less than or equal to" for posets.
class le

Syntaxcoc 16575 Extend class notation with the class of orthocomplementation extractors.
class oc

Syntaxcds 16576 Extend class notation with the metric space distance function.
class dist

Syntaxcunif 16577 Extend class notation with the uniform structure.
class UnifSet

Syntaxchom 16578 Extend class notation with the hom-set structure.
class Hom

Syntaxcco 16579 Extend class notation with the composition operation.
class comp

Definitiondf-plusg 16580 Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
+g = Slot 2

Definitiondf-mulr 16581 Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
.r = Slot 3

Definitiondf-starv 16582 Define the involution function of a *-ring. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
*𝑟 = Slot 4

Definitiondf-sca 16583 Define scalar field component of a vector space 𝑣. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Scalar = Slot 5

Definitiondf-vsca 16584 Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑠 = Slot 6

Definitiondf-ip 16585 Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
·𝑖 = Slot 8

Definitiondf-tset 16586 Define the topology component of a topological space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
TopSet = Slot 9

Definitiondf-ple 16587 Define "less than or equal to" ordering extractor for posets and related structures. We use 10 for the index to avoid conflict with 1 through 9 used for other purposes. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
le = Slot 10

Definitiondf-ocomp 16588 Define the orthocomplementation extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
oc = Slot 11

Definitiondf-ds 16589 Define the distance function component of a metric space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
dist = Slot 12

Definitiondf-unif 16590 Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
UnifSet = Slot 13

Definitiondf-hom 16591 Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hom = Slot 14

Definitiondf-cco 16592 Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp = Slot 15

Theoremstrleun 16593 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐹 Struct ⟨𝐴, 𝐵    &   𝐺 Struct ⟨𝐶, 𝐷    &   𝐵 < 𝐶       (𝐹𝐺) Struct ⟨𝐴, 𝐷

Theoremstrle1 16594 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐼 ∈ ℕ    &   𝐴 = 𝐼       {⟨𝐴, 𝑋⟩} Struct ⟨𝐼, 𝐼

Theoremstrle2 16595 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

Theoremstrle3 16596 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

Theoremplusgndx 16597 Index value of the df-plusg 16580 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(+g‘ndx) = 2

Theoremplusgid 16598 Utility theorem: index-independent form of df-plusg 16580. (Contributed by NM, 20-Oct-2012.)
+g = Slot (+g‘ndx)

Theoremopelstrbas 16599 The base set of a structure with a base set. (Contributed by AV, 10-Nov-2021.)
(𝜑𝑆 Struct 𝑋)    &   (𝜑𝑉𝑌)    &   (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝑆)       (𝜑𝑉 = (Base‘𝑆))

Theorem1strstr 16600 A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}       𝐺 Struct ⟨1, 1⟩

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