P. 84 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 8301-8400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tfrlem1 8301*	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	tfrlem3a 8302*	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 8303*	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	tfrlem4 8304*	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 8305*	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 8306*	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 8307*	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 8308*	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 8309*	Lemma for transfinite recursion. The domain of recs is an 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 8310*	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	tfrlem9a 8311*	Lemma for transfinite recursion. Without using ax-rep 5219, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}    ⇒   ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V)
Theorem	tfrlem10 8312*	Lemma for transfinite recursion. We define class 𝐶 by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to, On. Using this assumption we will prove facts about 𝐶 that will lead to a contradiction in tfrlem14 8316, thus showing the domain of recs does in fact equal On. Here we show (under the false assumption) that 𝐶 is a function extending the domain of recs(𝐹) by one. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}    &   ⊢ 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})    ⇒   ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
Theorem	tfrlem11 8313*	Lemma for transfinite recursion. Compute the value of 𝐶. (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}    &   ⊢ 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})    ⇒   ⊢ (dom recs(𝐹) ∈ On → (𝐵 ∈ suc dom recs(𝐹) → (𝐶‘𝐵) = (𝐹‘(𝐶 ↾ 𝐵))))
Theorem	tfrlem12 8314*	Lemma for transfinite recursion. Show 𝐶 is an acceptable function. (Contributed by NM, 15-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}    &   ⊢ 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})    ⇒   ⊢ (recs(𝐹) ∈ V → 𝐶 ∈ 𝐴)
Theorem	tfrlem13 8315*	Lemma for transfinite recursion. If recs is a set function, then 𝐶 is acceptable, and thus a subset of recs, but dom 𝐶 is bigger than dom recs. This is a contradiction, so recs must be a proper class function. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2014.)
⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}    ⇒   ⊢ ¬ recs(𝐹) ∈ V
Theorem	tfrlem14 8316*	Lemma for transfinite recursion. Assuming ax-rep 5219, dom recs ∈ V ↔ recs ∈ V, so since dom recs is an ordinal, it must be equal to On. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}    ⇒   ⊢ dom recs(𝐹) = On
Theorem	tfrlem15 8317*	Lemma for transfinite recursion. Without assuming ax-rep 5219, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 14-Nov-2014.)
⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}    ⇒   ⊢ (𝐵 ∈ On → (𝐵 ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ 𝐵) ∈ V))
Theorem	tfrlem16 8318*	Lemma for finite recursion. Without assuming ax-rep 5219, we can show that the domain of the constructed function is a limit ordinal, and hence contains all the finite ordinals. (Contributed by Mario Carneiro, 14-Nov-2014.)
⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}    ⇒   ⊢ Lim dom recs(𝐹)
Theorem	tfr1a 8319	A weak version of tfr1 8322 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    ⇒   ⊢ (Fun 𝐹 ∧ Lim dom 𝐹)
Theorem	tfr2a 8320	A weak version of tfr2 8323 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    ⇒   ⊢ (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))
Theorem	tfr2b 8321	Without assuming ax-rep 5219, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    ⇒   ⊢ (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V))
Theorem	tfr1 8322	Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47. We start with an arbitrary class 𝐺, normally a function, and 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 NM, 17-Aug-1994.) (Revised by Mario Carneiro, 18-Jan-2015.)
⊢ 𝐹 = recs(𝐺)    ⇒   ⊢ 𝐹 Fn On
Theorem	tfr2 8323	Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47. Here we show that the function 𝐹 has the property that for any function 𝐺 whatsoever, the "next" value of 𝐹 is 𝐺 recursively applied to all "previous" values of 𝐹. (Contributed by NM, 9-Apr-1995.) (Revised by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝐹 = recs(𝐺)    ⇒   ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))
Theorem	tfr3 8324*	Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47. 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 NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ 𝐹 = recs(𝐺)    ⇒   ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹)
Theorem	tfr1ALT 8325	Alternate proof of tfr1 8322 using well-ordered recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐹 = recs(𝐺)    ⇒   ⊢ 𝐹 Fn On
Theorem	tfr2ALT 8326	Alternate proof of tfr2 8323 using well-ordered recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐹 = recs(𝐺)    ⇒   ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))
Theorem	tfr3ALT 8327*	Alternate proof of tfr3 8324 using well-ordered recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐹 = recs(𝐺)    ⇒   ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹)
Theorem	recsfnon 8328	Strong transfinite recursion defines a function on ordinals. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ recs(𝐹) Fn On
Theorem	recsval 8329	Strong transfinite recursion in terms of all previous values. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ (𝐴 ∈ On → (recs(𝐹)‘𝐴) = (𝐹‘(recs(𝐹) ↾ 𝐴)))
Theorem	tz7.44lem1 8330*	The ordered pair abstraction 𝐺 defined in the hypothesis is a function. This was a lemma for tz7.44-1 8331, tz7.44-2 8332, and tz7.44-3 8333 when they used that definition of 𝐺. Now, they use the maps-to df-mpt 5175 idiom so this lemma is not needed anymore, but is kept in case other applications (for instance in intuitionistic set theory) need it. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))}    ⇒   ⊢ Fun 𝐺
Theorem	tz7.44-1 8331*	The value of 𝐹 at ∅. Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
⊢ 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥)))))    &   ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦)))    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴)
Theorem	tz7.44-2 8332*	The value of 𝐹 at a successor ordinal. Part 2 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 14-Nov-2014.)
⊢ 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥)))))    &   ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦)))    &   ⊢ (𝑦 ∈ 𝑋 → (𝐹 ↾ 𝑦) ∈ V)    &   ⊢ 𝐹 Fn 𝑋    &   ⊢ Ord 𝑋    ⇒   ⊢ (suc 𝐵 ∈ 𝑋 → (𝐹‘suc 𝐵) = (𝐻‘(𝐹‘𝐵)))
Theorem	tz7.44-3 8333*	The value of 𝐹 at a limit ordinal. Part 3 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
⊢ 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥)))))    &   ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦)))    &   ⊢ (𝑦 ∈ 𝑋 → (𝐹 ↾ 𝑦) ∈ V)    &   ⊢ 𝐹 Fn 𝑋    &   ⊢ Ord 𝑋    ⇒   ⊢ ((𝐵 ∈ 𝑋 ∧ Lim 𝐵) → (𝐹‘𝐵) = ∪ (𝐹 “ 𝐵))
2.4.21  Recursive definition generator
Syntax	crdg 8334	Extend class notation with the recursive definition generator, with characteristic function 𝐹 and initial value 𝐼.
class rec(𝐹, 𝐼)
Definition	df-rdg 8335*	Define a recursive definition generator on On (the class of ordinal numbers) with characteristic function 𝐹 and initial value 𝐼. This combines functions 𝐹 in tfr1 8322 and 𝐺 in tz7.44-1 8331 into one definition. This rather amazing operation allows 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 8297 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple, as in for example oav 8432, from which we prove the recursive textbook definition as Theorems oa0 8437, oasuc 8445, and oalim 8453 (with the help of Theorems rdg0 8346, rdgsuc 8349, and rdglim2a 8358). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 8361 and frsuc 8362. Our rec operation apparently does not appear in published literature, although closely related is Definition 25.2 of [Quine] p. 177, which he uses to "turn...a recursion into a genuine or direct definition" (p. 174). Note that the if operations (see df-if 4475) select cases based on whether the domain of 𝑔 is zero, a successor, or a limit ordinal. An important use of this definition is in the recursive sequence generator df-seq 13911 on the natural numbers (as a subset of the complex numbers), allowing us to define, with direct definitions, recursive infinite sequences such as the factorial function df-fac 14183 and integer powers df-exp 13971. 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 NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
Theorem	rdgeq1 8336	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 8337	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	rdgeq12 8338	Equality theorem for the recursive definition generator. (Contributed by Scott Fenton, 28-Apr-2012.)
⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵))
Theorem	nfrdg 8339	Bound-variable hypothesis builder for the recursive definition generator. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥rec(𝐹, 𝐴)
Theorem	rdglem1 8340*	Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use. (Contributed by NM, 9-Apr-1995.)
⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))}
Theorem	rdgfun 8341	The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ Fun rec(𝐹, 𝐴)
Theorem	rdgdmlim 8342	The domain of the recursive definition generator is a limit ordinal. (Contributed by NM, 16-Nov-2014.)
⊢ Lim dom rec(𝐹, 𝐴)
Theorem	rdgfnon 8343	The recursive definition generator is a function on ordinal numbers. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ rec(𝐹, 𝐴) Fn On
Theorem	rdgvalg 8344*	Value of the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))
Theorem	rdgval 8345*	Value of the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
⊢ (𝐵 ∈ On → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))
Theorem	rdg0 8346	The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (rec(𝐹, 𝐴)‘∅) = 𝐴
Theorem	rdgseg 8347	The initial segments of the recursive definition generator are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ (𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
Theorem	rdgsucg 8348	The value of the recursive definition generator at a successor. (Contributed by NM, 16-Nov-2014.)
⊢ (𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
Theorem	rdgsuc 8349	The value of the recursive definition generator at a successor. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
⊢ (𝐵 ∈ On → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
Theorem	rdglimg 8350	The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 16-Nov-2014.)
⊢ ((𝐵 ∈ dom rec(𝐹, 𝐴) ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ (rec(𝐹, 𝐴) “ 𝐵))
Theorem	rdglim 8351	The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ (rec(𝐹, 𝐴) “ 𝐵))
Theorem	rdg0g 8352	The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
⊢ (𝐴 ∈ 𝐶 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
Theorem	rdgsucmptf 8353	The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)    &   ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Theorem	rdgsucmptnf 8354	The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered-pair class abstraction and where the mapping class 𝐷 is a proper class). This is a technical lemma that can be used together with rdgsucmptf 8353 to help eliminate redundant sethood antecedents. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)    &   ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)
Theorem	rdgsucmpt2 8355*	This version of rdgsucmpt 8356 avoids the not-free hypothesis of rdgsucmptf 8353 by using two substitutions instead of one. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)    &   ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶)    &   ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷)    ⇒   ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Theorem	rdgsucmpt 8356*	The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (Contributed by Mario Carneiro, 9-Sep-2013.)
⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)    &   ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Theorem	rdglim2 8357*	The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values. (Contributed by NM, 23-Apr-1995.)
⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = (rec(𝐹, 𝐴)‘𝑥)})
Theorem	rdglim2a 8358*	The value of the recursive definition generator at a limit ordinal, in terms of indexed union of all smaller values. (Contributed by NM, 28-Jun-1998.)
⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ 𝑥 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑥))
Theorem	rdg0n 8359	If 𝐴 is a proper class, then the recursive function generator at ∅ is the empty set. (Contributed by Scott Fenton, 31-Oct-2024.)
⊢ (¬ 𝐴 ∈ V → (rec(𝐹, 𝐴)‘∅) = ∅)
2.4.22  Finite recursion
Theorem	frfnom 8360	The function generated by finite recursive definition generation is a function on omega. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 14-Nov-2014.)
⊢ (rec(𝐹, 𝐴) ↾ ω) Fn ω
Theorem	fr0g 8361	The initial value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.)
⊢ (𝐴 ∈ 𝐵 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴)
Theorem	frsuc 8362	The successor value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ (𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝐵) = (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝐵)))
Theorem	frsucmpt 8363	The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation). (Contributed by NM, 14-Sep-2003.) (Revised by Scott Fenton, 2-Nov-2011.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)    &   ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Theorem	frsucmptn 8364	The value of the finite recursive definition generator at a successor (special case where the characteristic function is a mapping abstraction and where the mapping class 𝐷 is a proper class). This is a technical lemma that can be used together with frsucmpt 8363 to help eliminate redundant sethood antecedents. (Contributed by Scott Fenton, 19-Feb-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)    &   ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)
Theorem	frsucmpt2 8365*	The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation), using double-substitution instead of a bound variable condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)    &   ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶)    &   ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷)    ⇒   ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Theorem	tz7.48lem 8366*	A way of showing an ordinal function is one-to-one. (Contributed by NM, 9-Feb-1997.)
⊢ 𝐹 Fn On    ⇒   ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ 𝐴))
Theorem	tz7.48-2 8367*	Proposition 7.48(2) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) (Revised by David Abernethy, 5-May-2013.)
⊢ 𝐹 Fn On    ⇒   ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹)
Theorem	tz7.48-1 8368*	Proposition 7.48(1) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
⊢ 𝐹 Fn On    ⇒   ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴)
Theorem	tz7.48-3 8369*	Proposition 7.48(3) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
⊢ 𝐹 Fn On    ⇒   ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V)
Theorem	tz7.49 8370*	Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 10-Jan-2013.)
⊢ 𝐹 Fn On    &   ⊢ (𝜑 ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))
Theorem	tz7.49c 8371*	Corollary of Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 19-Jan-2013.)
⊢ 𝐹 Fn On    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
Syntax	cseqom 8372	Extend class notation to include index-aware recursive definitions.
class seqω(𝐹, 𝐼)
Definition	df-seqom 8373*	Index-aware recursive definitions over ω. A mashup of df-rdg 8335 and df-seq 13911, this allows for recursive definitions that use an index in the recursion in cases where Infinity is not admitted. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ seqω(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
Theorem	seqomlem0 8374*	Lemma for seqω. Change bound variables. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩), ⟨∅, ( I ‘𝐼)⟩)
Theorem	seqomlem1 8375*	Lemma for seqω. The underlying recursion generates a sequence of pairs with the expected first values. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)    ⇒   ⊢ (𝐴 ∈ ω → (𝑄‘𝐴) = ⟨𝐴, (2nd ‘(𝑄‘𝐴))⟩)
Theorem	seqomlem2 8376*	Lemma for seqω. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)    ⇒   ⊢ (𝑄 “ ω) Fn ω
Theorem	seqomlem3 8377*	Lemma for seqω. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)    ⇒   ⊢ ((𝑄 “ ω)‘∅) = ( I ‘𝐼)
Theorem	seqomlem4 8378*	Lemma for seqω. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)    ⇒   ⊢ (𝐴 ∈ ω → ((𝑄 “ ω)‘suc 𝐴) = (𝐴𝐹((𝑄 “ ω)‘𝐴)))
Theorem	seqomeq12 8379	Equality theorem for seqω. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → seqω(𝐴, 𝐶) = seqω(𝐵, 𝐷))
Theorem	fnseqom 8380	An index-aware recursive definition defines a function on the natural numbers. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ 𝐺 = seqω(𝐹, 𝐼)    ⇒   ⊢ 𝐺 Fn ω
Theorem	seqom0g 8381	Value of an index-aware recursive definition at 0. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by AV, 17-Sep-2021.)
⊢ 𝐺 = seqω(𝐹, 𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → (𝐺‘∅) = 𝐼)
Theorem	seqomsuc 8382	Value of an index-aware recursive definition at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ 𝐺 = seqω(𝐹, 𝐼)    ⇒   ⊢ (𝐴 ∈ ω → (𝐺‘suc 𝐴) = (𝐴𝐹(𝐺‘𝐴)))
Theorem	omsucelsucb 8383	Membership is inherited by successors for natural numbers. (Contributed by AV, 15-Sep-2023.)
⊢ (𝑁 ∈ ω ↔ suc 𝑁 ∈ suc ω)
2.4.23  Ordinal arithmetic
Syntax	c1o 8384	Extend the definition of a class to include the ordinal number 1.
class 1o
Syntax	c2o 8385	Extend the definition of a class to include the ordinal number 2.
class 2o
Syntax	c3o 8386	Extend the definition of a class to include the ordinal number 3.
class 3o
Syntax	c4o 8387	Extend the definition of a class to include the ordinal number 4.
class 4o
Syntax	coa 8388	Extend the definition of a class to include the ordinal addition operation.
class +o
Syntax	comu 8389	Extend the definition of a class to include the ordinal multiplication operation.
class ·o
Syntax	coe 8390	Extend the definition of a class to include the ordinal exponentiation operation.
class ↑o
Definition	df-1o 8391	Define the ordinal number 1. Definition 2.1 of [Schloeder] p. 4. (Contributed by NM, 29-Oct-1995.)
⊢ 1o = suc ∅
Definition	df-2o 8392	Define the ordinal number 2. Lemma 3.17 of [Schloeder] p. 10. (Contributed by NM, 18-Feb-2004.)
⊢ 2o = suc 1o
Definition	df-3o 8393	Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ 3o = suc 2o
Definition	df-4o 8394	Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ 4o = suc 3o
Definition	df-oadd 8395*	Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)
⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
Definition	df-omul 8396*	Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)
⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
Definition	df-oexp 8397*	Define the ordinal exponentiation operation. (Contributed by NM, 30-Dec-2004.)
⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
Theorem	df1o2 8398	Expanded value of the ordinal number 1. Definition 2.1 of [Schloeder] p. 4. (Contributed by NM, 4-Nov-2002.)
⊢ 1o = {∅}
Theorem	df2o3 8399	Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 2o = {∅, 1o}
Theorem	df2o2 8400	Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
⊢ 2o = {∅, {∅}}