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Theorem List for Intuitionistic Logic Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-tpos 5801* 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 5802 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹𝐺 → tpos 𝐹 ⊆ tpos 𝐺)
 
Theoremtposeq 5803 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)
 
Theoremtposeqd 5804 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
(φ𝐹 = 𝐺)       (φ → tpos 𝐹 = tpos 𝐺)
 
Theoremtposssxp 5805 The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
 
Theoremreltpos 5806 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
Rel tpos 𝐹
 
Theorembrtpos2 5807 Value of the transposition at a pair A, B. (Contributed by Mario Carneiro, 10-Sep-2015.)
(B 𝑉 → (Atpos 𝐹B ↔ (A (dom 𝐹 ∪ {∅}) {A}𝐹B)))
 
Theorembrtpos0 5808 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). (Contributed by Mario Carneiro, 10-Sep-2015.)
(A 𝑉 → (∅tpos 𝐹A ↔ ∅𝐹A))
 
Theoremreldmtpos 5809 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)
 
Theorembrtposg 5810 The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.)
((A 𝑉 B 𝑊 𝐶 𝑋) → (⟨A, B⟩tpos 𝐹𝐶 ↔ ⟨B, A𝐹𝐶))
 
Theoremottposg 5811 The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
((A 𝑉 B 𝑊 𝐶 𝑋) → (⟨A, B, 𝐶 tpos 𝐹 ↔ ⟨B, A, 𝐶 𝐹))
 
Theoremdmtpos 5812 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 5813 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 5814 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
(𝐹 𝑉 → tpos 𝐹 V)
 
Theoremovtposg 5815 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 the reals or some ring, tpos 𝐹 is the transposition of 𝐹, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)
((A 𝑉 B 𝑊) → (Atpos 𝐹B) = (B𝐹A))
 
Theoremtposfun 5816 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Fun 𝐹 → Fun tpos 𝐹)
 
Theoremdftpos2 5817* Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (x dom 𝐹 {x})))
 
Theoremdftpos3 5818* Alternate definition of tpos when 𝐹 has relational domain. Compare df-cnv 4296. (Contributed by Mario Carneiro, 10-Sep-2015.)
(Rel dom 𝐹 → tpos 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ⟨y, x𝐹z})
 
Theoremdftpos4 5819* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos 𝐹 = (𝐹 ∘ (x ((V × V) ∪ {∅}) ↦ {x}))
 
Theoremtpostpos 5820 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 5821 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 5822 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
(Rel A → (𝐹 Fn A → tpos 𝐹 Fn A))
 
Theoremtposfo2 5823 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel A → (𝐹:AontoB → tpos 𝐹:AontoB))
 
Theoremtposf2 5824 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(Rel A → (𝐹:AB → tpos 𝐹:AB))
 
Theoremtposf12 5825 Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel A → (𝐹:A1-1B → tpos 𝐹:A1-1B))
 
Theoremtposf1o2 5826 Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
(Rel A → (𝐹:A1-1-ontoB → tpos 𝐹:A1-1-ontoB))
 
Theoremtposfo 5827 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(𝐹:(A × B)–onto𝐶 → tpos 𝐹:(B × A)–onto𝐶)
 
Theoremtposf 5828 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
(𝐹:(A × B)⟶𝐶 → tpos 𝐹:(B × A)⟶𝐶)
 
Theoremtposfn 5829 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝐹 Fn (A × B) → tpos 𝐹 Fn (B × A))
 
Theoremtpos0 5830 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
tpos ∅ = ∅
 
Theoremtposco 5831 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
tpos (𝐹𝐺) = (𝐹 ∘ tpos 𝐺)
 
Theoremtpossym 5832* Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝐹 Fn (A × A) → (tpos 𝐹 = 𝐹x A y A (x𝐹y) = (y𝐹x)))
 
Theoremtposeqi 5833 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 = 𝐺       tpos 𝐹 = tpos 𝐺
 
Theoremtposex 5834 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 V       tpos 𝐹 V
 
Theoremnftpos 5835 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
x𝐹       xtpos 𝐹
 
Theoremtposoprab 5836* 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 5837* Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
𝐹 = (x A, y B𝐶)       tpos 𝐹 = (y B, x A𝐶)
 
2.6.17  Undefined values
 
Theorempwuninel2 5838 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. (Contributed by Stefan O'Rear, 22-Feb-2015.)
( A 𝑉 → ¬ 𝒫 A A)
 
Theorem2pwuninelg 5839 The power set of 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. (Contributed by Jim Kingdon, 14-Jan-2020.)
(A 𝑉 → ¬ 𝒫 𝒫 A A)
 
2.6.18  Functions on ordinals; strictly monotone ordinal functions
 
Theoremiunon 5840* 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.)
((A 𝑉 x A B On) → x A B On)
 
Syntaxwsmo 5841 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
wff Smo A
 
Definitiondf-smo 5842* 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(x y → (Ax) (Ay))))
 
Theoremdfsmo2 5843* 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 𝐹y x (𝐹y) (𝐹x)))
 
Theoremissmo 5844* Conditions for which A is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)
A:B⟶On    &   Ord B    &   ((x B y B) → (x y → (Ax) (Ay)))    &   dom A = B       Smo A
 
Theoremissmo2 5845* Alternative definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)
(𝐹:AB → ((B ⊆ On Ord A x A y x (𝐹y) (𝐹x)) → Smo 𝐹))
 
Theoremsmoeq 5846 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
(A = B → (Smo A ↔ Smo B))
 
Theoremsmodm 5847 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
(Smo A → Ord dom A)
 
Theoremsmores 5848 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 A B dom A) → Smo (AB))
 
Theoremsmores3 5849 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 5850 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 5851 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)
 
Theoremsmofvon2dm 5852 The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
((Smo 𝐹 B dom 𝐹) → (𝐹B) On)
 
Theoremiordsmo 5853 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 5854 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
Smo ∅
 
Theoremsmofvon 5855 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 B A dom B) → (BA) On)
 
Theoremsmoel 5856 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 B A dom B 𝐶 A) → (B𝐶) (BA))
 
Theoremsmoiun 5857* The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)
((Smo B A dom B) → x A (Bx) ⊆ (BA))
 
Theoremsmoiso 5858 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 A B ⊆ On) → Smo 𝐹)
 
Theoremsmoel2 5859 A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)
(((𝐹 Fn A Smo 𝐹) (B A 𝐶 B)) → (𝐹𝐶) (𝐹B))
 
2.6.19  "Strong" transfinite recursion
 
Syntaxcrecs 5860 Notation for a function defined by strong transfinite recursion.
class recs(𝐹)
 
Definitiondf-recs 5861* Define a function recs(𝐹) on On, the class of ordinal numbers, by transfinite recursion given a rule 𝐹 which sets the next value given all values so far. See df-irdg 5897 for more details on why this definition is desirable. Unlike df-irdg 5897 which restricts the update rule to use only the previous value, this version allows the update rule to use all previous values, which is why it is described as "strong", although it is actually more primitive. See tfri1d 5890 and tfri2d 5891 for the primary contract of this definition.

(Contributed by Stefan O'Rear, 18-Jan-2015.)

recs(𝐹) = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
 
Theoremrecseq 5862 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
(𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))
 
Theoremnfrecs 5863 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
x𝐹       xrecs(𝐹)
 
Theoremtfrlem1 5864* A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by Mario Carneiro, 24-May-2019.)
(φA On)    &   (φ → (Fun 𝐹 A ⊆ dom 𝐹))    &   (φ → (Fun 𝐺 A ⊆ dom 𝐺))    &   (φx A (𝐹x) = (B‘(𝐹x)))    &   (φx A (𝐺x) = (B‘(𝐺x)))       (φx A (𝐹x) = (𝐺x))
 
Theoremtfrlem3ag 5865* Lemma for transfinite recursion. This lemma just changes some bound variables in A for later use. (Contributed by NM, 9-Apr-1995.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}       (𝐺 V → (𝐺 Az On (𝐺 Fn z w z (𝐺w) = (𝐹‘(𝐺w)))))
 
Theoremtfrlem3a 5866* Lemma for transfinite recursion. Let A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in A for later use. (Contributed by NM, 9-Apr-1995.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}    &   𝐺 V       (𝐺 Az On (𝐺 Fn z w z (𝐺w) = (𝐹‘(𝐺w))))
 
Theoremtfrlem3 5867* Lemma for transfinite recursion. Let A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in A for later use. (Contributed by NM, 9-Apr-1995.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}       A = {gz On (g Fn z w z (gw) = (𝐹‘(gw)))}
 
Theoremtfrlem3-2 5868* Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 17-Apr-2019.)
(Fun 𝐹 (𝐹x) V)       (Fun 𝐹 (𝐹g) V)
 
Theoremtfrlem3-2d 5869* Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.)
(φx(Fun 𝐹 (𝐹x) V))       (φ → (Fun 𝐹 (𝐹g) V))
 
Theoremtfrlem4 5870* Lemma for transfinite recursion. A is the class of all "acceptable" functions, and 𝐹 is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}       (g A → Fun g)
 
Theoremtfrlem5 5871* Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}       ((g A A) → ((xgu xv) → u = v))
 
Theoremrecsfval 5872* Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}       recs(𝐹) = A
 
Theoremtfrlem6 5873* Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}       Rel recs(𝐹)
 
Theoremtfrlem7 5874* Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}       Fun recs(𝐹)
 
Theoremtfrlem8 5875* Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}       Ord dom recs(𝐹)
 
Theoremtfrlem9 5876* Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}       (B dom recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))
 
Theoremtfr2a 5877 A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)       (A dom 𝐹 → (𝐹A) = (𝐺‘(𝐹A)))
 
Theoremtfr0 5878 Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.)
𝐹 = recs(𝐺)       ((𝐺‘∅) 𝑉 → (𝐹‘∅) = (𝐺‘∅))
 
Theoremtfrlemisucfn 5879* We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 2-Jul-2019.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}    &   (φx(Fun 𝐹 (𝐹x) V))    &   (φz On)    &   (φg Fn z)    &   (φg A)       (φ → (g ∪ {⟨z, (𝐹g)⟩}) Fn suc z)
 
Theoremtfrlemisucaccv 5880* We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}    &   (φx(Fun 𝐹 (𝐹x) V))    &   (φz On)    &   (φg Fn z)    &   (φg A)       (φ → (g ∪ {⟨z, (𝐹g)⟩}) A)
 
Theoremtfrlemibacc 5881* Each element of B is an acceptable function. Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}    &   (φx(Fun 𝐹 (𝐹x) V))    &   B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}    &   (φx On)    &   (φz x g(g Fn z w z (gw) = (𝐹‘(gw))))       (φBA)
 
Theoremtfrlemibxssdm 5882* The union of B is defined on all ordinals. Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}    &   (φx(Fun 𝐹 (𝐹x) V))    &   B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}    &   (φx On)    &   (φz x g(g Fn z w z (gw) = (𝐹‘(gw))))       (φx ⊆ dom B)
 
Theoremtfrlemibfn 5883* The union of B is a function defined on x. Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}    &   (φx(Fun 𝐹 (𝐹x) V))    &   B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}    &   (φx On)    &   (φz x g(g Fn z w z (gw) = (𝐹‘(gw))))       (φ B Fn x)
 
Theoremtfrlemibex 5884* The set B exists. Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}    &   (φx(Fun 𝐹 (𝐹x) V))    &   B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}    &   (φx On)    &   (φz x g(g Fn z w z (gw) = (𝐹‘(gw))))       (φB V)
 
Theoremtfrlemiubacc 5885* The union of B satisfies the recursion rule (lemma for tfrlemi1 5887). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}    &   (φx(Fun 𝐹 (𝐹x) V))    &   B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}    &   (φx On)    &   (φz x g(g Fn z w z (gw) = (𝐹‘(gw))))       (φu x ( Bu) = (𝐹‘( Bu)))
 
Theoremtfrlemiex 5886* Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}    &   (φx(Fun 𝐹 (𝐹x) V))    &   B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}    &   (φx On)    &   (φz x g(g Fn z w z (gw) = (𝐹‘(gw))))       (φf(f Fn x u x (fu) = (𝐹‘(fu))))
 
Theoremtfrlemi1 5887* We can define an acceptable function on any ordinal.

As with many of the transfinite recursion theorems, we have a hypothesis that states that 𝐹 is a function and that it is defined for all ordinals. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}    &   (φx(Fun 𝐹 (𝐹x) V))       ((φ 𝐶 On) → g(g Fn 𝐶 u 𝐶 (gu) = (𝐹‘(gu))))
 
Theoremtfrlemi14d 5888* The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}    &   (φx(Fun 𝐹 (𝐹x) V))       (φ → dom recs(𝐹) = On)
 
Theoremtfrexlem 5889* The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}    &   (φx(Fun 𝐹 (𝐹x) V))       ((φ 𝐶 𝑉) → (recs(𝐹)‘𝐶) V)
 
Theoremtfri1d 5890* Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition.

The condition is that 𝐺 is defined "everywhere" and here is stated as (𝐺x) V. Alternatively x Onf(f Fn xf dom 𝐺) would suffice.

Given a function 𝐺 satisfying that condition, we define a class A of all "acceptable" functions. The final function we're interested in is the union 𝐹 = recs(𝐺) of them. 𝐹 is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of 𝐹. In this first part we show that 𝐹 is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)

𝐹 = recs(𝐺)    &   (φx(Fun 𝐺 (𝐺x) V))       (φ𝐹 Fn On)
 
Theoremtfri2d 5891* Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule 𝐺 ( as described at tfri1 5892). Here we show that the function 𝐹 has the property that for any function 𝐺 satisfying that condition, the "next" value of 𝐹 is 𝐺 recursively applied to all "previous" values of 𝐹. (Contributed by Jim Kingdon, 4-May-2019.)
𝐹 = recs(𝐺)    &   (φx(Fun 𝐺 (𝐺x) V))       ((φ A On) → (𝐹A) = (𝐺‘(𝐹A)))
 
Theoremtfri1 5892* Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition.

The condition is that 𝐺 is defined "everywhere" and here is stated as (𝐺x) V. Alternatively x Onf(f Fn xf dom 𝐺) would suffice.

Given a function 𝐺 satisfying that condition, we define a class A of all "acceptable" functions. The final function we're interested in is the union 𝐹 = recs(𝐺) of them. 𝐹 is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of 𝐹. In this first part we show that 𝐹 is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)

𝐹 = recs(𝐺)    &   (Fun 𝐺 (𝐺x) V)       𝐹 Fn On
 
Theoremtfri2 5893* Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule 𝐺 ( as described at tfri1 5892). Here we show that the function 𝐹 has the property that for any function 𝐺 satisfying that condition, the "next" value of 𝐹 is 𝐺 recursively applied to all "previous" values of 𝐹. (Contributed by Jim Kingdon, 4-May-2019.)
𝐹 = recs(𝐺)    &   (Fun 𝐺 (𝐺x) V)       (A On → (𝐹A) = (𝐺‘(𝐹A)))
 
Theoremtfri3 5894* Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule 𝐺 ( as described at tfri1 5892). Finally, we show that 𝐹 is unique. We do this by showing that any class B with the same properties of 𝐹 that we showed in parts 1 and 2 is identical to 𝐹. (Contributed by Jim Kingdon, 4-May-2019.)
𝐹 = recs(𝐺)    &   (Fun 𝐺 (𝐺x) V)       ((B Fn On x On (Bx) = (𝐺‘(Bx))) → B = 𝐹)
 
Theoremtfrex 5895* The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
𝐹 = recs(𝐺)    &   (φx(Fun 𝐺 (𝐺x) V))       ((φ A 𝑉) → (𝐹A) V)
 
2.6.20  Recursive definition generator
 
Syntaxcrdg 5896 Extend class notation with the recursive definition generator, with characteristic function 𝐹 and initial value 𝐼.
class rec(𝐹, 𝐼)
 
Definitiondf-irdg 5897* Define a recursive definition generator on On (the class of ordinal numbers) with characteristic function 𝐹 and initial value 𝐼. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our rec operation (especially when df-recs 5861 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple. In classical logic it would be easier to divide this definition into cases based on whether the domain of g is zero, a successor, or a limit ordinal. Cases do not (in general) work that way in intuitionistic logic, so instead we choose a definition which takes the union of all the results of the characteristic function for ordinals in the domain of g. This means that this definition has the expected properties for increasing and continuous ordinal functions, which include ordinal addition and multiplication.

For finite recursion we also define df-frec 5918 and for suitable characteristic functions df-frec 5918 yields the same result as rec restricted to 𝜔, as seen at frecrdg 5931.

Note: We introduce rec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Jim Kingdon, 19-May-2019.)

rec(𝐹, 𝐼) = recs((g V ↦ (𝐼 x dom g(𝐹‘(gx)))))
 
Theoremrdgeq1 5898 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
(𝐹 = 𝐺 → rec(𝐹, A) = rec(𝐺, A))
 
Theoremrdgeq2 5899 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
(A = B → rec(𝐹, A) = rec(𝐹, B))
 
Theoremrdgfun 5900 The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
Fun rec(𝐹, A)
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