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Theorem List for Metamath Proof Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmpt2xopn0yelv 7101* If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)       ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → 𝐾𝑉))

Theoremmpt2xopynvov0g 7102* If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)       (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)

Theoremmpt2xopxnop0 7103* If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)       𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅)

Theoremmpt2xopx0ov0 7104* If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is the empty set, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)       (∅𝐹𝐾) = ∅

Theoremmpt2xopxprcov0 7105* If the components of the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, are not sets, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)       (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (⟨𝑉, 𝑊𝐹𝐾) = ∅)

Theoremmpt2xopynvov0 7106* If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ 𝐶)       (𝐾𝑉 → (⟨𝑉, 𝑊𝐹𝐾) = ∅)

Theoremmpt2xopoveq 7107* Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})       (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})

Theoremmpt2xopovel 7108* Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})       ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝐾𝑉𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))

Theoremmpt2xopoveqd 7109* Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})    &   (𝜓 → (𝑉𝑋𝑊𝑌))    &   ((𝜓 ∧ ¬ 𝐾𝑉) → {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} = ∅)       (𝜓 → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})

Theorembrovex 7110* A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶)    &   ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Rel (𝑉𝑂𝐸))       (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))

Theorembrovmpt2ex 7111* A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {⟨𝑧, 𝑤⟩ ∣ 𝜑})       (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))

Theoremsprmpt2d 7112* The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 20-Jun-2019.)
𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦𝜒)})    &   ((𝜑𝑣 = 𝑉𝑒 = 𝐸) → (𝜒𝜓))    &   (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V))    &   (𝜑 → ∀𝑥𝑦(𝑥(𝑉𝑅𝐸)𝑦𝜃))    &   (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V)       (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦𝜓)})

2.4.10  Function transposition

Syntaxctpos 7113 The transposition of a function.
class tpos 𝐹

Definitiondf-tpos 7114* Define the transposition of a function, which is a function 𝐺 = tpos 𝐹 satisfying 𝐺(𝑥, 𝑦) = 𝐹(𝑦, 𝑥). (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))

Theoremtposss 7115 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)

Theoremtposeq 7116 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)

Theoremtposeqd 7117 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐹 = 𝐺)       (𝜑 → tpos 𝐹 = tpos 𝐺)

Theoremtposssxp 7118 The transposition is a subset of a Cartesian product. (Contributed by Mario Carneiro, 12-Jan-2017.)
tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)

Theoremreltpos 7119 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
Rel tpos 𝐹

Theorembrtpos2 7120 Value of the transposition at a pair 𝐴, 𝐵. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐵𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))

Theorembrtpos0 7121 The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). This allows us to eliminate sethood hypotheses on 𝐴, 𝐵 in brtpos 7123. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))

Theoremreldmtpos 7122 Necessary and sufficient condition for dom tpos 𝐹 to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)

Theorembrtpos 7123 The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐶𝑉 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))

Theoremottpos 7124 The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
(𝐶𝑉 → (⟨𝐴, 𝐵, 𝐶⟩ ∈ tpos 𝐹 ↔ ⟨𝐵, 𝐴, 𝐶⟩ ∈ 𝐹))

Theoremrelbrtpos 7125 The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015.)
(Rel 𝐹 → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴𝐹𝐶))

Theoremdmtpos 7126 The domain of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)

Theoremrntpos 7127 The range of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)

Theoremtposexg 7128 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹𝑉 → tpos 𝐹 ∈ V)

Theoremovtpos 7129 The transposition swaps the arguments in a two-argument function. When 𝐹 is a matrix, which is to say a function from (1...𝑚) × (1...𝑛) to or some ring, tpos 𝐹 is the transposition of 𝐹, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴)

Theoremtposfun 7130 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Fun 𝐹 → Fun tpos 𝐹)

Theoremdftpos2 7131* Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥dom 𝐹 {𝑥})))

Theoremdftpos3 7132* Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 4940. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧})

Theoremdftpos4 7133* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))

Theoremtpostpos 7134 Value of the double transposition for a general class 𝐹. (Contributed by Mario Carneiro, 16-Sep-2015.)
tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))

Theoremtpostpos2 7135 Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.)
((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹)

Theoremtposfn2 7136 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn 𝐴))

Theoremtposfo2 7137 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹:𝐴onto𝐵 → tpos 𝐹:𝐴onto𝐵))

Theoremtposf2 7138 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹:𝐴𝐵 → tpos 𝐹:𝐴𝐵))

Theoremtposf12 7139 Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹:𝐴1-1𝐵 → tpos 𝐹:𝐴1-1𝐵))

Theoremtposf1o2 7140 Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹:𝐴1-1-onto𝐵 → tpos 𝐹:𝐴1-1-onto𝐵))

Theoremtposfo 7141 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(𝐹:(𝐴 × 𝐵)–onto𝐶 → tpos 𝐹:(𝐵 × 𝐴)–onto𝐶)

Theoremtposf 7142 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(𝐹:(𝐴 × 𝐵)⟶𝐶 → tpos 𝐹:(𝐵 × 𝐴)⟶𝐶)

Theoremtposfn 7143 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝐹 Fn (𝐴 × 𝐵) → tpos 𝐹 Fn (𝐵 × 𝐴))

Theoremtpos0 7144 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
tpos ∅ = ∅

Theoremtposco 7145 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos (𝐹𝐺) = (𝐹 ∘ tpos 𝐺)

Theoremtpossym 7146* Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)))

Theoremtposeqi 7147 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 = 𝐺       tpos 𝐹 = tpos 𝐺

Theoremtposex 7148 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 ∈ V       tpos 𝐹 ∈ V

Theoremnftpos 7149 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝑥𝐹       𝑥tpos 𝐹

Theoremtposoprab 7150* Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}       tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}

Theoremtposmpt2 7151* Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       tpos 𝐹 = (𝑦𝐵, 𝑥𝐴𝐶)

Theoremtposconst 7152 The transposition of a constant operation using the relation representation. (Contributed by SO, 11-Jul-2018.)
tpos ((𝐴 × 𝐵) × {𝐶}) = ((𝐵 × 𝐴) × {𝐶})

2.4.11  Curry and uncurry

Syntaxccur 7153 Extend class notation to include the currying function.
class curry 𝐴

Syntaxcunc 7154 Extend class notation to include the uncurrying function.
class uncurry 𝐴

Definitiondf-cur 7155* Define the currying of 𝐹, which splits a function of two arguments into a function of the first argument, producing a function over the second argument. (Contributed by Mario Carneiro, 7-Jan-2017.)
curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧})

Definitiondf-unc 7156* Define the uncurrying of 𝐹, which takes a function producing functions, and transforms it into a two-argument function. (Contributed by Mario Carneiro, 7-Jan-2017.)
uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧}

Theoremmpt2curryd 7157* The currying of an operation given in maps-to notation, splitting the operation (function of two arguments) into a function of the first argument, producing a function over the second argument. (Contributed by AV, 27-Oct-2019.)
𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)    &   (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)    &   (𝜑𝑌 ≠ ∅)       (𝜑 → curry 𝐹 = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))

Theoremmpt2curryvald 7158* The value of a curried operation given in maps-to notation is a function over the second argument of the original operation. (Contributed by AV, 27-Oct-2019.)
𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)    &   (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)    &   (𝜑𝑌 ≠ ∅)    &   (𝜑𝑌𝑊)    &   (𝜑𝐴𝑋)       (𝜑 → (curry 𝐹𝐴) = (𝑦𝑌𝐴 / 𝑥𝐶))

Theoremfvmpt2curryd 7159* The value of the value of a curried operation given in maps-to notation is the operation value of the original operation. (Contributed by AV, 27-Oct-2019.)
𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)    &   (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)    &   (𝜑𝑌𝑊)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)       (𝜑 → ((curry 𝐹𝐴)‘𝐵) = (𝐴𝐹𝐵))

2.4.12  Undefined values

Syntaxcund 7160 Extend class notation with undefined value function.
class Undef

Definitiondf-undef 7161 Define the undefined value function, whose value at set 𝑠 is guaranteed not to be a member of 𝑠 (see pwuninel 7163). (Contributed by NM, 15-Sep-2011.)
Undef = (𝑠 ∈ V ↦ 𝒫 𝑠)

Theorempwuninel2 7162 Direct proof of pwuninel 7163 avoiding functions and thus several ZF axioms. (Contributed by Stefan O'Rear, 22-Feb-2015.)
( 𝐴𝑉 → ¬ 𝒫 𝐴𝐴)

Theorempwuninel 7163 The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. See also pwuninel2 7162. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
¬ 𝒫 𝐴𝐴

Theoremundefval 7164 Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 7166 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝑆𝑉 → (Undef‘𝑆) = 𝒫 𝑆)

Theoremundefnel2 7165 The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.)
(𝑆𝑉 → ¬ (Undef‘𝑆) ∈ 𝑆)

Theoremundefnel 7166 The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.)
(𝑆𝑉 → (Undef‘𝑆) ∉ 𝑆)

Theoremundefne0 7167 The undefined value generated from a set is not empty. (Contributed by NM, 3-Sep-2018.)
(𝑆𝑉 → (Undef‘𝑆) ≠ ∅)

2.4.13  Well-founded recursion

Syntaxcwrecs 7168 Declare syntax for the well-founded recursive function generator.
class wrecs(𝑅, 𝐴, 𝐹)

Definitiondf-wrecs 7169* Here we define the well-founded recursive function generator. This function takes the usual expressions from recursion theorems and forms a unified definition. Specifically, given a function 𝐹, a relationship 𝑅, and a base set 𝐴, this definition generates a function 𝐺 = wrecs(𝑅, 𝐴, 𝐹) that has property that, at any point 𝑥𝐴, (𝐺𝑥) = (𝐹‘(𝐺 ∣ ‘Pred(𝑅, 𝐴, 𝑥))). See wfr1 7195, wfr2 7196, and wfr3 7197. (Contributed by Scott Fenton, 7-Jun-2018.) (New usage is discouraged.)
wrecs(𝑅, 𝐴, 𝐹) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}

Theoremwrecseq123 7170 General equality theorem for the well-founded recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)
((𝑅 = 𝑆𝐴 = 𝐵𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

Theoremnfwrecs 7171 Bound-variable hypothesis builder for the well-founded recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018.)
𝑥𝑅    &   𝑥𝐴    &   𝑥𝐹       𝑥wrecs(𝑅, 𝐴, 𝐹)

Theoremwrecseq1 7172 Equality theorem for the well-founded recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)
(𝑅 = 𝑆 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐴, 𝐹))

Theoremwrecseq2 7173 Equality theorem for the well-founded recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)
(𝐴 = 𝐵 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐵, 𝐹))

Theoremwrecseq3 7174 Equality theorem for the well-founded recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)
(𝐹 = 𝐺 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐴, 𝐺))

Theoremwfr3g 7175* Functions defined by well-founded recursion are identical up to relation, domain, and characteristic function. (Contributed by Scott Fenton, 11-Feb-2011.)
(((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)

Theoremwfrlem1 7176* Lemma for well-founded recursion. The final item we are interested in is the union of acceptable functions 𝐵. This lemma just changes bound variables for later use. (Contributed by Scott Fenton, 21-Apr-2011.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}       𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}

Theoremwfrlem2 7177* Lemma for well-founded recursion. An acceptable function is a function. (Contributed by Scott Fenton, 21-Apr-2011.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}       (𝑔𝐵 → Fun 𝑔)

Theoremwfrlem3 7178* Lemma for well-founded recursion. An acceptable function's domain is a subset of 𝐴. (Contributed by Scott Fenton, 21-Apr-2011.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}       (𝑔𝐵 → dom 𝑔𝐴)

Theoremwfrlem3a 7179* Lemma for well-founded recursion. Show membership in the class of acceptable functions. (Contributed by Scott Fenton, 31-Jul-2020.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}    &   𝐺 ∈ V       (𝐺𝐵 ↔ ∃𝑧(𝐺 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))))

Theoremwfrlem4 7180* Lemma for well-founded recursion. Properties of the restriction of an acceptable function to the domain of another one. (Contributed by Scott Fenton, 21-Apr-2011.)
𝑅 We 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}       ((𝑔𝐵𝐵) → ((𝑔 ↾ (dom 𝑔 ∩ dom )) Fn (dom 𝑔 ∩ dom ) ∧ ∀𝑎 ∈ (dom 𝑔 ∩ dom )((𝑔 ↾ (dom 𝑔 ∩ dom ))‘𝑎) = (𝐹‘((𝑔 ↾ (dom 𝑔 ∩ dom )) ↾ Pred(𝑅, (dom 𝑔 ∩ dom ), 𝑎)))))

Theoremwfrlem5 7181* Lemma for well-founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}       ((𝑔𝐵𝐵) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))

Theoremwfrrel 7182 The well-founded recursion generator generates a relationship. (Contributed by Scott Fenton, 8-Jun-2018.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)       Rel 𝐹

Theoremwfrdmss 7183 The domain of the well-founded recursion generator is a subclass of 𝐴. (Contributed by Scott Fenton, 21-Apr-2011.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)       dom 𝐹𝐴

Theoremwfrlem8 7184 Lemma for well-founded recursion. Compute the prececessor class for an 𝑅 minimal element of (𝐴 ∖ dom 𝐹). (Contributed by Scott Fenton, 21-Apr-2011.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)       (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, dom 𝐹, 𝑋))

Theoremwfrdmcl 7185 Given 𝐹 = wrecs(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ dom 𝐹, then its predecessor class is a subset of dom 𝐹. (Contributed by Scott Fenton, 21-Apr-2011.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)       (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)

Theoremwfrlem10 7186* Lemma for well-founded recursion. When 𝑧 is an 𝑅 minimal element of (𝐴 ∖ dom 𝐹), then its predecessor class is equal to dom 𝐹. (Contributed by Scott Fenton, 21-Apr-2011.)
𝑅 We 𝐴    &   𝐹 = wrecs(𝑅, 𝐴, 𝐺)       ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) = dom 𝐹)

Theoremwfrfun 7187 The well-founded function generator generates a function. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   𝐹 = wrecs(𝑅, 𝐴, 𝐺)       Fun 𝐹

Theoremwfrlem12 7188* Lemma for well-founded recursion. Here, we compute the value of the recursive definition generator. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   𝐹 = wrecs(𝑅, 𝐴, 𝐺)       (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))

Theoremwfrlem13 7189* Lemma for well-founded recursion. From here through wfrlem16 7192, we aim to prove that dom 𝐹 = 𝐴. We do this by supposing that there is an element 𝑧 of 𝐴 that is not in dom 𝐹. We then define 𝐶 by extending dom 𝐹 with the appropriate value at 𝑧. We then show that 𝑧 cannot be an 𝑅 minimal element of (𝐴 ∖ dom 𝐹), meaning that (𝐴 ∖ dom 𝐹) must be empty, so dom 𝐹 = 𝐴. Here, we show that 𝐶 is a function extending the domain of 𝐹 by one. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   𝐹 = wrecs(𝑅, 𝐴, 𝐺)    &   𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})       (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))

Theoremwfrlem14 7190* Lemma for well-founded recursion. Compute the value of 𝐶. (Contributed by Scott Fenton, 21-Apr-2011.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   𝐹 = wrecs(𝑅, 𝐴, 𝐺)    &   𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})       (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))

Theoremwfrlem15 7191* Lemma for well-founded recursion. When 𝑧 is 𝑅 minimal, 𝐶 is an acceptable function. This step is where the Axiom of Replacement becomes required. (Contributed by Scott Fenton, 21-Apr-2011.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   𝐹 = wrecs(𝑅, 𝐴, 𝐺)    &   𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})       ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})

Theoremwfrlem16 7192* Lemma for well-founded recursion. If 𝑧 is 𝑅 minimal in (𝐴 ∖ dom 𝐹), then 𝐶 is acceptable and thus a subset of 𝐹, but dom 𝐶 is bigger than dom 𝐹. Thus, 𝑧 cannot be minimal, so (𝐴 ∖ dom 𝐹) must be empty, and (due to wfrdmss 7183), dom 𝐹 = 𝐴. (Contributed by Scott Fenton, 21-Apr-2011.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   𝐹 = wrecs(𝑅, 𝐴, 𝐺)    &   𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})       dom 𝐹 = 𝐴

Theoremwfrlem17 7193 Without using ax-rep 4597, show that all restrictions of wrecs are sets. (Contributed by Scott Fenton, 31-Jul-2020.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   𝐹 = wrecs(𝑅, 𝐴, 𝐺)       (𝑋 ∈ dom 𝐹 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)

Theoremwfr2a 7194 A weak version of wfr2 7196 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Scott Fenton, 30-Jul-2020.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   𝐹 = wrecs(𝑅, 𝐴, 𝐺)       (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Theoremwfr1 7195 The Principle of Well-Founded Recursion, part 1 of 3. We start with an arbitrary function 𝐺. Then, using a base class 𝐴 and a well-ordering 𝑅 of 𝐴, we define a function 𝐹. This function is said to be defined by "well-founded recursion." The purpose of these three theorems is to demonstrate the properties of 𝐹. We begin by showing that 𝐹 is a function over 𝐴. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   𝐹 = wrecs(𝑅, 𝐴, 𝐺)       𝐹 Fn 𝐴

Theoremwfr2 7196 The Principle of Well-Founded Recursion, part 2 of 3. Next, we show that the value of 𝐹 at any 𝑋𝐴 is 𝐺 recursively applied to all "previous" values of 𝐹. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   𝐹 = wrecs(𝑅, 𝐴, 𝐺)       (𝑋𝐴 → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Theoremwfr3 7197* The principle of Well-Founded Recursion, part 3 of 3. Finally, we show that 𝐹 is unique. We do this by showing that any function 𝐻 with the same properties we proved of 𝐹 in wfr1 7195 and wfr2 7196 is identical to 𝐹. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   𝐹 = wrecs(𝑅, 𝐴, 𝐺)       ((𝐻 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐻𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧)))) → 𝐹 = 𝐻)

2.4.14  Functions on ordinals; strictly monotone ordinal functions

Theoremiunon 7198* The indexed union of a set of ordinal numbers 𝐵(𝑥) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)
((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → 𝑥𝐴 𝐵 ∈ On)

TheoremiunonOLD 7199* The indexed union of a set of ordinal numbers 𝐵(𝑥) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (∀𝑥𝐴 𝐵 ∈ On → 𝑥𝐴 𝐵 ∈ On)

Theoremiinon 7200* The nonempty indexed intersection of a class of ordinal numbers 𝐵(𝑥) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
((∀𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 ∈ On)

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
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