P. 59 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5801-5900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tposf2 5801	The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (Rel A → (𝐹:A⟶B → tpos 𝐹:◡A⟶B))
Theorem	tposf12 5802	Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (Rel A → (𝐹:A–1-1→B → tpos 𝐹:◡A–1-1→B))
Theorem	tposf1o2 5803	Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (Rel A → (𝐹:A–1-1-onto→B → tpos 𝐹:◡A–1-1-onto→B))
Theorem	tposfo 5804	The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (𝐹:(A × B)–onto→𝐶 → tpos 𝐹:(B × A)–onto→𝐶)
Theorem	tposf 5805	The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
⊢ (𝐹:(A × B)⟶𝐶 → tpos 𝐹:(B × A)⟶𝐶)
Theorem	tposfn 5806	Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝐹 Fn (A × B) → tpos 𝐹 Fn (B × A))
Theorem	tpos0 5807	Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
⊢ tpos ∅ = ∅
Theorem	tposco 5808	Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ tpos (𝐹 ∘ 𝐺) = (𝐹 ∘ tpos 𝐺)
Theorem	tpossym 5809*	Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝐹 Fn (A × A) → (tpos 𝐹 = 𝐹 ↔ ∀x ∈ A ∀y ∈ A (x𝐹y) = (y𝐹x)))
Theorem	tposeqi 5810	Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ 𝐹 = 𝐺    ⇒   ⊢ tpos 𝐹 = tpos 𝐺
Theorem	tposex 5811	A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ 𝐹 ∈ V    ⇒   ⊢ tpos 𝐹 ∈ V
Theorem	nftpos 5812	Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ Ⅎx𝐹    ⇒   ⊢ Ⅎxtpos 𝐹
Theorem	tposoprab 5813*	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}
Theorem	tposmpt2 5814*	Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ 𝐹 = (x ∈ A, y ∈ B ↦ 𝐶)    ⇒   ⊢ tpos 𝐹 = (y ∈ B, x ∈ A ↦ 𝐶)
2.6.17  Undefined values
Theorem	pwuninel2 5815	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)
Theorem	2pwuninelg 5816	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
Theorem	iunon 5817*	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)
Syntax	wsmo 5818	Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
wff Smo A
Definition	df-smo 5819*	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 A∀y ∈ dom A(x ∈ y → (A‘x) ∈ (A‘y))))
Theorem	dfsmo2 5820*	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 𝐹∀y ∈ x (𝐹‘y) ∈ (𝐹‘x)))
Theorem	issmo 5821*	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 → (A‘x) ∈ (A‘y)))    &   ⊢ dom A = B    ⇒   ⊢ Smo A
Theorem	issmo2 5822*	Alternative definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)
⊢ (𝐹:A⟶B → ((B ⊆ On ∧ Ord A ∧ ∀x ∈ A ∀y ∈ x (𝐹‘y) ∈ (𝐹‘x)) → Smo 𝐹))
Theorem	smoeq 5823	Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
⊢ (A = B → (Smo A ↔ Smo B))
Theorem	smodm 5824	The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
⊢ (Smo A → Ord dom A)
Theorem	smores 5825	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 (A ↾ B))
Theorem	smores3 5826	A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)
⊢ ((Smo (A ↾ B) ∧ 𝐶 ∈ (dom A ∩ B) ∧ Ord B) → Smo (A ↾ 𝐶))
Theorem	smores2 5827	A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
⊢ ((Smo 𝐹 ∧ Ord A) → Smo (𝐹 ↾ A))
Theorem	smodm2 5828	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)
Theorem	smofvon2dm 5829	The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
⊢ ((Smo 𝐹 ∧ B ∈ dom 𝐹) → (𝐹‘B) ∈ On)
Theorem	iordsmo 5830	The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)
⊢ Ord A    ⇒   ⊢ Smo ( I ↾ A)
Theorem	smo0 5831	The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
⊢ Smo ∅
Theorem	smofvon 5832	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) → (B‘A) ∈ On)
Theorem	smoel 5833	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) → (B‘𝐶) ∈ (B‘A))
Theorem	smoiun 5834*	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 (B‘x) ⊆ (B‘A))
Theorem	smoiso 5835	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)
Theorem	smoel2 5836	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)) → (𝐹‘𝐶) ∈ (𝐹‘B))
2.6.19  "Strong" transfinite recursion
Syntax	crecs 5837	Notation for a function defined by strong transfinite recursion.
class recs(𝐹)
Definition	df-recs 5838*	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. If we were assuming the law of the excluded middle, we would then build on top of that a form of recursion which has separate cases for the empty set, successor ordinals, and limit ordinals. This version allows the update rule to use all previous values, which is why it is described as "strong". (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ recs(𝐹) = ∪ {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}
Theorem	recseq 5839	Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))
Theorem	nfrecs 5840	Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ Ⅎx𝐹    ⇒   ⊢ Ⅎxrecs(𝐹)
Theorem	tfrlem1 5841*	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.)
⊢ (φ → A ∈ On)    &   ⊢ (φ → (Fun 𝐹 ∧ A ⊆ dom 𝐹))    &   ⊢ (φ → (Fun 𝐺 ∧ A ⊆ dom 𝐺))    &   ⊢ (φ → ∀x ∈ A (𝐹‘x) = (B‘(𝐹 ↾ x)))    &   ⊢ (φ → ∀x ∈ A (𝐺‘x) = (B‘(𝐺 ↾ x)))    ⇒   ⊢ (φ → ∀x ∈ A (𝐹‘x) = (𝐺‘x))
Theorem	tfrlem3ag 5842*	Lemma for transfinite recursion. This lemma just changes some bound variables in A for later use. (Contributed by NM, 9-Apr-1995.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    ⇒   ⊢ (𝐺 ∈ V → (𝐺 ∈ A ↔ ∃z ∈ On (𝐺 Fn z ∧ ∀w ∈ z (𝐺‘w) = (𝐹‘(𝐺 ↾ w)))))
Theorem	tfrlem3a 5843*	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 = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    &   ⊢ 𝐺 ∈ V    ⇒   ⊢ (𝐺 ∈ A ↔ ∃z ∈ On (𝐺 Fn z ∧ ∀w ∈ z (𝐺‘w) = (𝐹‘(𝐺 ↾ w))))
Theorem	tfrlem3 5844*	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 = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    ⇒   ⊢ A = {g ∣ ∃z ∈ On (g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w)))}
Theorem	tfrlem3-2 5845*	Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 17-Apr-2019.)
⊢ (Fun 𝐹 ∧ (𝐹‘x) ∈ V)    ⇒   ⊢ (Fun 𝐹 ∧ (𝐹‘g) ∈ V)
Theorem	tfrlem3-2d 5846*	Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.)
⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))    ⇒   ⊢ (φ → (Fun 𝐹 ∧ (𝐹‘g) ∈ V))
Theorem	tfrlem4 5847*	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 = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    ⇒   ⊢ (g ∈ A → Fun g)
Theorem	tfrlem5 5848*	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 = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    ⇒   ⊢ ((g ∈ A ∧ ℎ ∈ A) → ((xgu ∧ xℎv) → u = v))
Theorem	recsfval 5849*	Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    ⇒   ⊢ recs(𝐹) = ∪ A
Theorem	tfrlem6 5850*	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 = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    ⇒   ⊢ Rel recs(𝐹)
Theorem	tfrlem7 5851*	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 = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    ⇒   ⊢ Fun recs(𝐹)
Theorem	tfrlem8 5852*	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 = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    ⇒   ⊢ Ord dom recs(𝐹)
Theorem	tfrlem9 5853*	Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    ⇒   ⊢ (B ∈ dom recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))
Theorem	tfr2a 5854	A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    ⇒   ⊢ (A ∈ dom 𝐹 → (𝐹‘A) = (𝐺‘(𝐹 ↾ A)))
Theorem	tfrlemisucfn 5855*	We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 5863. (Contributed by Jim Kingdon, 2-Jul-2019.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    &   ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))    &   ⊢ (φ → z ∈ On)    &   ⊢ (φ → g Fn z)    &   ⊢ (φ → g ∈ A)    ⇒   ⊢ (φ → (g ∪ {⟨z, (𝐹‘g)⟩}) Fn suc z)
Theorem	tfrlemisucaccv 5856*	We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 5863. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    &   ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))    &   ⊢ (φ → z ∈ On)    &   ⊢ (φ → g Fn z)    &   ⊢ (φ → g ∈ A)    ⇒   ⊢ (φ → (g ∪ {⟨z, (𝐹‘g)⟩}) ∈ A)
Theorem	tfrlemibacc 5857*	Each element of B is an acceptable function. Lemma for tfrlemi1 5863. (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    &   ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))    &   ⊢ B = {ℎ ∣ ∃z ∈ x ∃g(g Fn z ∧ g ∈ A ∧ ℎ = (g ∪ {⟨z, (𝐹‘g)⟩}))}    &   ⊢ (φ → x ∈ On)    &   ⊢ (φ → ∀z ∈ x ∃g(g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w))))    ⇒   ⊢ (φ → B ⊆ A)
Theorem	tfrlemibxssdm 5858*	The union of B is defined on all ordinals. Lemma for tfrlemi1 5863. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    &   ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))    &   ⊢ B = {ℎ ∣ ∃z ∈ x ∃g(g Fn z ∧ g ∈ A ∧ ℎ = (g ∪ {⟨z, (𝐹‘g)⟩}))}    &   ⊢ (φ → x ∈ On)    &   ⊢ (φ → ∀z ∈ x ∃g(g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w))))    ⇒   ⊢ (φ → x ⊆ dom ∪ B)
Theorem	tfrlemibfn 5859*	The union of B is a function defined on x. Lemma for tfrlemi1 5863. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    &   ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))    &   ⊢ B = {ℎ ∣ ∃z ∈ x ∃g(g Fn z ∧ g ∈ A ∧ ℎ = (g ∪ {⟨z, (𝐹‘g)⟩}))}    &   ⊢ (φ → x ∈ On)    &   ⊢ (φ → ∀z ∈ x ∃g(g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w))))    ⇒   ⊢ (φ → ∪ B Fn x)
Theorem	tfrlemibex 5860*	The set B exists. Lemma for tfrlemi1 5863. (Contributed by Jim Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    &   ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))    &   ⊢ B = {ℎ ∣ ∃z ∈ x ∃g(g Fn z ∧ g ∈ A ∧ ℎ = (g ∪ {⟨z, (𝐹‘g)⟩}))}    &   ⊢ (φ → x ∈ On)    &   ⊢ (φ → ∀z ∈ x ∃g(g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w))))    ⇒   ⊢ (φ → B ∈ V)
Theorem	tfrlemiubacc 5861*	The union of B satisfies the recursion rule (lemma for tfrlemi1 5863). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    &   ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))    &   ⊢ B = {ℎ ∣ ∃z ∈ x ∃g(g Fn z ∧ g ∈ A ∧ ℎ = (g ∪ {⟨z, (𝐹‘g)⟩}))}    &   ⊢ (φ → x ∈ On)    &   ⊢ (φ → ∀z ∈ x ∃g(g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w))))    ⇒   ⊢ (φ → ∀u ∈ x (∪ B‘u) = (𝐹‘(∪ B ↾ u)))
Theorem	tfrlemiex 5862*	Lemma for tfrlemi1 5863. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    &   ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))    &   ⊢ B = {ℎ ∣ ∃z ∈ x ∃g(g Fn z ∧ g ∈ A ∧ ℎ = (g ∪ {⟨z, (𝐹‘g)⟩}))}    &   ⊢ (φ → x ∈ On)    &   ⊢ (φ → ∀z ∈ x ∃g(g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w))))    ⇒   ⊢ (φ → ∃f(f Fn x ∧ ∀u ∈ x (f‘u) = (𝐹‘(f ↾ u))))
Theorem	tfrlemi1 5863*	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 = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    &   ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))    ⇒   ⊢ ((φ ∧ 𝐶 ∈ On) → ∃g(g Fn 𝐶 ∧ ∀u ∈ 𝐶 (g‘u) = (𝐹‘(g ↾ u))))
Theorem	tfrlemi14d 5864*	The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    &   ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))    ⇒   ⊢ (φ → dom recs(𝐹) = On)
Theorem	tfrlemi14 5865*	The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 4-May-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    &   ⊢ (Fun 𝐹 ∧ (𝐹‘x) ∈ V)    ⇒   ⊢ dom recs(𝐹) = On
Theorem	tfrexlem 5866*	The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}    &   ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))    ⇒   ⊢ ((φ ∧ 𝐶 ∈ 𝑉) → (recs(𝐹)‘𝐶) ∈ V)
Theorem	tfri1d 5867*	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 (𝐺‘x) ∈ V. Alternatively ∀x ∈ On∀f(f Fn x → f ∈ 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)    &   ⊢ (φ → ∀x(Fun 𝐺 ∧ (𝐺‘x) ∈ V))    ⇒   ⊢ (φ → 𝐹 Fn On)
Theorem	tfri2d 5868*	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 5869). 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)    &   ⊢ (φ → ∀x(Fun 𝐺 ∧ (𝐺‘x) ∈ V))    ⇒   ⊢ ((φ ∧ A ∈ On) → (𝐹‘A) = (𝐺‘(𝐹 ↾ A)))
Theorem	tfri1 5869*	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 (𝐺‘x) ∈ V. Alternatively ∀x ∈ On∀f(f Fn x → f ∈ 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 ∧ (𝐺‘x) ∈ V)    ⇒   ⊢ 𝐹 Fn On
Theorem	tfri2 5870*	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 5869). 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 ∧ (𝐺‘x) ∈ V)    ⇒   ⊢ (A ∈ On → (𝐹‘A) = (𝐺‘(𝐹 ↾ A)))
Theorem	tfri3 5871*	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 5869). 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 ∧ (𝐺‘x) ∈ V)    ⇒   ⊢ ((B Fn On ∧ ∀x ∈ On (B‘x) = (𝐺‘(B ↾ x))) → B = 𝐹)
Theorem	tfrex 5872*	The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (φ → ∀x(Fun 𝐺 ∧ (𝐺‘x) ∈ V))    ⇒   ⊢ ((φ ∧ A ∈ 𝑉) → (𝐹‘A) ∈ V)
2.6.20  Recursive definition generator
Syntax	crdg 5873	Extend class notation with the recursive definition generator, with characteristic function 𝐹 and initial value 𝐼.
class rec(𝐹, 𝐼)
Definition	df-irdg 5874*	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 5838 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 5894 and for suitable characteristic functions df-frec 5894 yields the same result as rec restricted to 𝜔, as seen at frecrdg 5904. 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‘(g‘x)))))
Theorem	rdgeq1 5875	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))
Theorem	rdgeq2 5876	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))
Theorem	rdgfun 5877	The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ Fun rec(𝐹, A)
Theorem	rdgruledefgg 5878*	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → (Fun (g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x)))) ∧ ((g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))))‘f) ∈ V))
Theorem	rdgruledefg 5879*	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
⊢ 𝐹 Fn V    ⇒   ⊢ (A ∈ 𝑉 → (Fun (g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x)))) ∧ ((g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))))‘f) ∈ V))
Theorem	rdgexggg 5880	The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
⊢ ((𝐹 Fn V ∧ A ∈ 𝑉 ∧ B ∈ 𝑊) → (rec(𝐹, A)‘B) ∈ V)
Theorem	rdgexgg 5881	The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
⊢ 𝐹 Fn V    ⇒   ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (rec(𝐹, A)‘B) ∈ V)
Theorem	rdgi0g 5882	The initial value of the recursive definition generator. (Contributed by Jim Kingdon, 10-Jul-2019.)
⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → (rec(𝐹, A)‘∅) = A)
Theorem	rdgifnon 5883	The recursive definition generator is a function on ordinal numbers. The 𝐹 Fn V condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough, but being defined for all sets will generally hold for the characteristic functions we need to use this with). (Contributed by Jim Kingdon, 13-Jul-2019.)
⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → rec(𝐹, A) Fn On)
Theorem	rdgivallem 5884*	Value of the recursive definition generator. Lemma for rdgival 5885 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)
⊢ ((𝐹 Fn V ∧ A ∈ 𝑉 ∧ B ∈ On) → (rec(𝐹, A)‘B) = (A ∪ ∪ x ∈ B (𝐹‘((rec(𝐹, A) ↾ B)‘x))))
Theorem	rdgival 5885*	Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.)
⊢ ((𝐹 Fn V ∧ A ∈ 𝑉 ∧ B ∈ On) → (rec(𝐹, A)‘B) = (A ∪ ∪ x ∈ B (𝐹‘(rec(𝐹, A)‘x))))
Theorem	rdgss 5886	Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)
⊢ (φ → 𝐹 Fn V)    &   ⊢ (φ → 𝐼 ∈ 𝑉)    &   ⊢ (φ → A ∈ On)    &   ⊢ (φ → B ∈ On)    &   ⊢ (φ → A ⊆ B)    ⇒   ⊢ (φ → (rec(𝐹, 𝐼)‘A) ⊆ (rec(𝐹, 𝐼)‘B))
Theorem	rdgisuc1 5887*	One way of describing the value of the recursive definition generator at a successor. There is no condition on the characteristic function 𝐹 other than 𝐹 Fn V. Given that, the resulting expression encompasses both the expected successor term (𝐹‘(rec(𝐹, A)‘B)) but also terms that correspond to the initial value A and to limit ordinals ∪ x ∈ B(𝐹‘(rec(𝐹, A)‘x)). If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 5888. (Contributed by Jim Kingdon, 9-Jun-2019.)
⊢ (φ → 𝐹 Fn V)    &   ⊢ (φ → A ∈ 𝑉)    &   ⊢ (φ → B ∈ On)    ⇒   ⊢ (φ → (rec(𝐹, A)‘suc B) = (A ∪ (∪ x ∈ B (𝐹‘(rec(𝐹, A)‘x)) ∪ (𝐹‘(rec(𝐹, A)‘B)))))
Theorem	rdgisucinc 5888*	Value of the recursive definition generator at a successor. This can be thought of as a generalization of oasuc 5955 and omsuc 5962. (Contributed by Jim Kingdon, 29-Aug-2019.)
⊢ (φ → 𝐹 Fn V)    &   ⊢ (φ → A ∈ 𝑉)    &   ⊢ (φ → B ∈ On)    &   ⊢ (φ → ∀x x ⊆ (𝐹‘x))    ⇒   ⊢ (φ → (rec(𝐹, A)‘suc B) = (𝐹‘(rec(𝐹, A)‘B)))
Theorem	rdgon 5889*	Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.)
⊢ (φ → 𝐹 Fn V)    &   ⊢ (φ → A ∈ On)    &   ⊢ (φ → ∀x ∈ On (𝐹‘x) ∈ On)    ⇒   ⊢ ((φ ∧ B ∈ On) → (rec(𝐹, A)‘B) ∈ On)
Theorem	rdgruledef 5890*	The recursion rule for the recursive definition generator is defined everywhere. Lemma for rdg0 5891. (Contributed by Jim Kingdon, 29-May-2019.)
⊢ A ∈ V    &   ⊢ 𝐹 Fn V    ⇒   ⊢ (Fun (g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x)))) ∧ ((g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))))‘f) ∈ V)
Theorem	rdg0 5891	The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
⊢ A ∈ V    &   ⊢ 𝐹 Fn V    ⇒   ⊢ (rec(𝐹, A)‘∅) = A
Theorem	rdgexg 5892	The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ A ∈ V    &   ⊢ 𝐹 Fn V    ⇒   ⊢ (B ∈ 𝑉 → (rec(𝐹, A)‘B) ∈ V)
2.6.21  Finite recursion
Syntax	cfrec 5893	Extend class notation with the fnite recursive definition generator, with characteristic function 𝐹 and initial value 𝐼.
class frec(𝐹, 𝐼)
Definition	df-frec 5894*	Define a recursive definition generator on 𝜔 (the class of finite ordinals) 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 frec operation (especially when df-recs 5838 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple; see frec0g 5898 and frecsuc 5903. Unlike with transfinite recursion, finite recurson can readily divide definitions and proofs into zero and successor cases, because even without excluded middle we have theorems such as nn0suc 4250. The analogous situation with transfinite recursion - being able to say that an ordinal is zero, successor, or limit - is enabled by excluded middle and thus is not available to us. For the characteristic functions which satisfy the conditions given at frecrdg 5904, this definition and df-irdg 5874 restricted to 𝜔 produce the same result. Note: We introduce frec 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 Mario Carneiro and Jim Kingdon, 10-Aug-2019.)
⊢ frec(𝐹, 𝐼) = (recs((g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ 𝐼))})) ↾ 𝜔)
Theorem	frecabex 5895*	The class abstraction from df-frec 5894 exists. This is a lemma for several other finite recursion proofs. (Contributed by Jim Kingdon, 16-Aug-2019.)
⊢ (φ → 𝑆 ∈ 𝑉)    &   ⊢ (φ → 𝐹 Fn V)    &   ⊢ (φ → A ∈ 𝑊)    ⇒   ⊢ (φ → {x ∣ (∃𝑚 ∈ 𝜔 (dom 𝑆 = suc 𝑚 ∧ x ∈ (𝐹‘(𝑆‘𝑚))) ∨ (dom 𝑆 = ∅ ∧ x ∈ A))} ∈ V)
Theorem	frectfr 5896*	Lemma to connect transfinite recursion theorems with finite recursion. That is, given the conditions 𝐹 Fn V and A ∈ 𝑉 on frec(𝐹, A), we want to be able to apply tfri1d 5867 or tfri2d 5868, and this lemma lets us satisfy hypotheses of those theorems. (Contributed by Jim Kingdon, 15-Aug-2019.)
⊢ 𝐺 = (g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})    ⇒   ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → ∀y(Fun 𝐺 ∧ (𝐺‘y) ∈ V))
Theorem	frecfnom 5897	The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 11-Aug-2019.)
⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → frec(𝐹, A) Fn 𝜔)
Theorem	frec0g 5898	The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 11-Aug-2019.)
⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → (frec(𝐹, A)‘∅) = A)
Theorem	frecsuclem1 5899*	Lemma for frecsuc 5903. (Contributed by Jim Kingdon, 13-Aug-2019.)
⊢ 𝐺 = (g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})    ⇒   ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉 ∧ B ∈ 𝜔) → (frec(𝐹, A)‘suc B) = (𝐺‘(recs(𝐺) ↾ suc B)))
Theorem	frecsuclemdm 5900*	Lemma for frecsuc 5903. (Contributed by Jim Kingdon, 15-Aug-2019.)
⊢ 𝐺 = (g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})    ⇒   ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉 ∧ B ∈ 𝜔) → dom (recs(𝐺) ↾ suc B) = suc B)