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Theorem List for Metamath Proof Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremottpos 7901 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 7902 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 7903 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 7904 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 7905 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹𝑉 → tpos 𝐹 ∈ V)

Theoremovtpos 7906 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 7907 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Fun 𝐹 → Fun tpos 𝐹)

Theoremdftpos2 7908* 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 7909* Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 5562. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom 𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥𝐹𝑧})

Theoremdftpos4 7910* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ((V × V) ∪ {∅}) ↦ {𝑥}))

Theoremtpostpos 7911 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 7912 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 7913 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn 𝐴))

Theoremtposfo2 7914 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹:𝐴onto𝐵 → tpos 𝐹:𝐴onto𝐵))

Theoremtposf2 7915 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹:𝐴𝐵 → tpos 𝐹:𝐴𝐵))

Theoremtposf12 7916 Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹:𝐴1-1𝐵 → tpos 𝐹:𝐴1-1𝐵))

Theoremtposf1o2 7917 Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel 𝐴 → (𝐹:𝐴1-1-onto𝐵 → tpos 𝐹:𝐴1-1-onto𝐵))

Theoremtposfo 7918 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(𝐹:(𝐴 × 𝐵)–onto𝐶 → tpos 𝐹:(𝐵 × 𝐴)–onto𝐶)

Theoremtposf 7919 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(𝐹:(𝐴 × 𝐵)⟶𝐶 → tpos 𝐹:(𝐵 × 𝐴)⟶𝐶)

Theoremtposfn 7920 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝐹 Fn (𝐴 × 𝐵) → tpos 𝐹 Fn (𝐵 × 𝐴))

Theoremtpos0 7921 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
tpos ∅ = ∅

Theoremtposco 7922 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos (𝐹𝐺) = (𝐹 ∘ tpos 𝐺)

Theoremtpossym 7923* 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 7924 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 = 𝐺       tpos 𝐹 = tpos 𝐺

Theoremtposex 7925 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 ∈ V       tpos 𝐹 ∈ V

Theoremnftpos 7926 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝑥𝐹       𝑥tpos 𝐹

Theoremtposoprab 7927* 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}

Theoremtposmpo 7928* Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       tpos 𝐹 = (𝑦𝐵, 𝑥𝐴𝐶)

Theoremtposconst 7929 The transposition of a constant operation using the relation representation. (Contributed by SO, 11-Jul-2018.)
tpos ((𝐴 × 𝐵) × {𝐶}) = ((𝐵 × 𝐴) × {𝐶})

2.4.12  Curry and uncurry

Syntaxccur 7930 Extend class notation to include the currying function.
class curry 𝐴

Syntaxcunc 7931 Extend class notation to include the uncurrying function.
class uncurry 𝐴

Definitiondf-cur 7932* 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 7933* 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}

Theoremmpocurryd 7934* 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)))

Theoremmpocurryvald 7935* 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))

Theoremfvmpocurryd 7936* 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))

2.4.13  Undefined values

Syntaxcund 7937 Extend class notation with undefined value function.
class Undef

Definitiondf-undef 7938 Define the undefined value function, whose value at set 𝑠 is guaranteed not to be a member of 𝑠 (see pwuninel 7940). (Contributed by NM, 15-Sep-2011.)
Undef = (𝑠 ∈ V ↦ 𝒫 𝑠)

Theorempwuninel2 7939 Direct proof of pwuninel 7940 avoiding functions and thus several ZF axioms. (Contributed by Stefan O'Rear, 22-Feb-2015.)
( 𝐴𝑉 → ¬ 𝒫 𝐴𝐴)

Theorempwuninel 7940 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 7939. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
¬ 𝒫 𝐴𝐴

Theoremundefval 7941 Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 7943 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝑆𝑉 → (Undef‘𝑆) = 𝒫 𝑆)

Theoremundefnel2 7942 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 7943 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 7944 The undefined value generated from a set is not empty. (Contributed by NM, 3-Sep-2018.)
(𝑆𝑉 → (Undef‘𝑆) ≠ ∅)

2.4.14  Well-founded recursion

Syntaxcwrecs 7945 Declare syntax for the well-founded recursive function generator.
class wrecs(𝑅, 𝐴, 𝐹)

Definitiondf-wrecs 7946* 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 7972, wfr2 7973, and wfr3 7974. (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 7947 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 7948 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 7949 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 7950 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 7951 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 7952* 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 7953* 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 7954* 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 7955* 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 7956* 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 7957* 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.) (Revised by AV, 18-Jul-2022.)
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 7958* 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 7959 The well-founded recursion generator generates a relationship. (Contributed by Scott Fenton, 8-Jun-2018.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)       Rel 𝐹

Theoremwfrdmss 7960 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 7961 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 7962 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 7963* 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 7964 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 7965* 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 7966* Lemma for well-founded recursion. From here through wfrlem16 7969, 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 7967* 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 7968* 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 7969* 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 7960), 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 7970 Without using ax-rep 5189, 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 7971 A weak version of wfr2 7973 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 7972 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 7973 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 7974* 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 7972 and wfr2 7973 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)

2.4.15  Functions on ordinals; strictly monotone ordinal functions

Theoremiunon 7975* 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)

Theoremiinon 7976* 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)

Theoremonfununi 7977* A property of functions on ordinal numbers. Generalization of Theorem Schema 8E of [Enderton] p. 218. (Contributed by Eric Schmidt, 26-May-2009.)
(Lim 𝑦 → (𝐹𝑦) = 𝑥𝑦 (𝐹𝑥))    &   ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦))       ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐹 𝑆) = 𝑥𝑆 (𝐹𝑥))

Theoremonovuni 7978* A variant of onfununi 7977 for operations. (Contributed by Eric Schmidt, 26-May-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
(Lim 𝑦 → (𝐴𝐹𝑦) = 𝑥𝑦 (𝐴𝐹𝑥))    &   ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → (𝐴𝐹𝑥) ⊆ (𝐴𝐹𝑦))       ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹 𝑆) = 𝑥𝑆 (𝐴𝐹𝑥))

Theoremonoviun 7979* A variant of onovuni 7978 with indexed unions. (Contributed by Eric Schmidt, 26-May-2009.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
(Lim 𝑦 → (𝐴𝐹𝑦) = 𝑥𝑦 (𝐴𝐹𝑥))    &   ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → (𝐴𝐹𝑥) ⊆ (𝐴𝐹𝑦))       ((𝐾𝑇 ∧ ∀𝑧𝐾 𝐿 ∈ On ∧ 𝐾 ≠ ∅) → (𝐴𝐹 𝑧𝐾 𝐿) = 𝑧𝐾 (𝐴𝐹𝐿))

Theoremonnseq 7980* There are no length ω decreasing sequences in the ordinals. See also noinfep 9122 for a stronger version assuming Regularity. (Contributed by Mario Carneiro, 19-May-2015.)
((𝐹‘∅) ∈ On → ∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹𝑥))

Syntaxwsmo 7981 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
wff Smo 𝐴

Definitiondf-smo 7982* Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)
(Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))

Theoremdfsmo2 7983* Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)
(Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))

Theoremissmo 7984* Conditions for which 𝐴 is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) Avoid ax-13 2386. (Revised by Gino Giotto, 19-May-2023.)
𝐴:𝐵⟶On    &   Ord 𝐵    &   ((𝑥𝐵𝑦𝐵) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))    &   dom 𝐴 = 𝐵       Smo 𝐴

Theoremissmo2 7985* Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)
(𝐹:𝐴𝐵 → ((𝐵 ⊆ On ∧ Ord 𝐴 ∧ ∀𝑥𝐴𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)) → Smo 𝐹))

Theoremsmoeq 7986 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
(𝐴 = 𝐵 → (Smo 𝐴 ↔ Smo 𝐵))

Theoremsmodm 7987 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
(Smo 𝐴 → Ord dom 𝐴)

Theoremsmores 7988 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
((Smo 𝐴𝐵 ∈ dom 𝐴) → Smo (𝐴𝐵))

Theoremsmores3 7989 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)
((Smo (𝐴𝐵) ∧ 𝐶 ∈ (dom 𝐴𝐵) ∧ Ord 𝐵) → Smo (𝐴𝐶))

Theoremsmores2 7990 A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
((Smo 𝐹 ∧ Ord 𝐴) → Smo (𝐹𝐴))

Theoremsmodm2 7991 The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)
((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)

Theoremsmofvon2 7992 The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
(Smo 𝐹 → (𝐹𝐵) ∈ On)

Theoremiordsmo 7993 The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)
Ord 𝐴       Smo ( I ↾ 𝐴)

Theoremsmo0 7994 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
Smo ∅

Theoremsmofvon 7995 If 𝐵 is a strictly monotone ordinal function, and 𝐴 is in the domain of 𝐵, then the value of the function at 𝐴 is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.)
((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐵𝐴) ∈ On)

Theoremsmoel 7996 If 𝑥 is less than 𝑦 then a strictly monotone function's value will be strictly less at 𝑥 than at 𝑦. (Contributed by Andrew Salmon, 22-Nov-2011.)
((Smo 𝐵𝐴 ∈ dom 𝐵𝐶𝐴) → (𝐵𝐶) ∈ (𝐵𝐴))

Theoremsmoiun 7997* The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)
((Smo 𝐵𝐴 ∈ dom 𝐵) → 𝑥𝐴 (𝐵𝑥) ⊆ (𝐵𝐴))

Theoremsmoiso 7998 If 𝐹 is an isomorphism from an ordinal 𝐴 onto 𝐵, which is a subset of the ordinals, then 𝐹 is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)
((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴𝐵 ⊆ On) → Smo 𝐹)

Theoremsmoel2 7999 A strictly monotone ordinal function preserves the membership relation. (Contributed by Mario Carneiro, 12-Mar-2013.)
(((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐵𝐴𝐶𝐵)) → (𝐹𝐶) ∈ (𝐹𝐵))

Theoremsmo11 8000 A strictly monotone ordinal function is one-to-one. (Contributed by Mario Carneiro, 28-Feb-2013.)
((𝐹:𝐴𝐵 ∧ Smo 𝐹) → 𝐹:𝐴1-1𝐵)

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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 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