P. 98 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9701-9800   *Has distinct variable group(s)

Type	Label	Description
Statement
2.6.3  Cantor normal form
Syntax	ccnf 9701	Extend class notation with the Cantor normal form function.
class CNF
Definition	df-cnf 9702*	Define the Cantor normal form function, which takes as input a finitely supported function from 𝑦 to 𝑥 and outputs the corresponding member of the ordinal exponential 𝑥 ↑o 𝑦. The content of the original Cantor Normal Form theorem is that for 𝑥 = ω this function is a bijection onto ω ↑o 𝑦 for any ordinal 𝑦 (or, since the function restricts naturally to different ordinals, the statement that the composite function is a bijection to On). More can be said about the function, however, and in particular it is an order isomorphism for a certain easily defined well-ordering of the finitely supported functions, which gives an alternate definition cantnffval2 9735 of this function in terms of df-oi 9550. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥 ↑m 𝑦) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)))
Theorem	cantnffval 9703*	The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅}    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵) = (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)))
Theorem	cantnfdm 9704*	The domain of the Cantor normal form function (in later lemmas we will use dom (𝐴 CNF 𝐵) to abbreviate "the set of finitely supported functions from 𝐵 to 𝐴"). (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅}    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆)
Theorem	cantnfvalf 9705*	Lemma for cantnf 9733. The function appearing in cantnfval 9708 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
⊢ 𝐹 = seqω((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)), ∅)    ⇒   ⊢ 𝐹:ω⟶On
Theorem	cantnfs 9706	Elementhood in the set of finitely supported functions from 𝐵 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
Theorem	cantnfcl 9707	Basic properties of the order isomorphism 𝐺 used later. The support of an 𝐹 ∈ 𝑆 is a finite subset of 𝐴, so it is well-ordered by E and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
Theorem	cantnfval 9708*	The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
Theorem	cantnfval2 9709*	Alternate expression for the value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)‘dom 𝐺))
Theorem	cantnfsuc 9710*	The value of the recursive function 𝐻 at a successor. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾)))
Theorem	cantnfle 9711*	A lower bound on the CNF function. Since ((𝐴 CNF 𝐵)‘𝐹) is defined as the sum of (𝐴 ↑o 𝑥) ·o (𝐹‘𝑥) over all 𝑥 in the support of 𝐹, it is larger than any of these terms (and all other terms are zero, so we can extend the statement to all 𝐶 ∈ 𝐵 instead of just those 𝐶 in the support). (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐴 ↑o 𝐶) ·o (𝐹‘𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹))
Theorem	cantnflt 9712*	An upper bound on the partial sums of the CNF function. Since each term dominates all previous terms, by induction we can bound the whole sum with any exponent 𝐴 ↑o 𝐶 where 𝐶 is larger than any exponent (𝐺‘𝑥), 𝑥 ∈ 𝐾 which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 ∈ suc dom 𝐺)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → (𝐺 “ 𝐾) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → (𝐻‘𝐾) ∈ (𝐴 ↑o 𝐶))
Theorem	cantnflt2 9713	An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶))
Theorem	cantnff 9714	The CNF function is a function from finitely supported functions from 𝐵 to 𝐴, to the ordinal exponential 𝐴 ↑o 𝐵. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑o 𝐵))
Theorem	cantnf0 9715	The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
Theorem	cantnfrescl 9716*	A function is finitely supported from 𝐵 to 𝐴 iff the extended function is finitely supported from 𝐷 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ 𝑇 = dom (𝐴 CNF 𝐷)    ⇒   ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇))
Theorem	cantnfres 9717*	The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ 𝑇 = dom (𝐴 CNF 𝐷)    &   ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋)))
Theorem	cantnfp1lem1 9718*	Lemma for cantnfp1 9721. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by AV, 30-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝑆)
Theorem	cantnfp1lem2 9719*	Lemma for cantnfp1 9721. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 30-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    &   ⊢ (𝜑 → ∅ ∈ 𝑌)    &   ⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅))    ⇒   ⊢ (𝜑 → dom 𝑂 = suc ∪ dom 𝑂)
Theorem	cantnfp1lem3 9720*	Lemma for cantnfp1 9721. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    &   ⊢ (𝜑 → ∅ ∈ 𝑌)    &   ⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐹‘(𝑂‘𝑘))) +o 𝑧)), ∅)    &   ⊢ 𝐾 = OrdIso( E , (𝐺 supp ∅))    &   ⊢ 𝑀 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐾‘𝑘)) ·o (𝐺‘(𝐾‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))
Theorem	cantnfp1 9721*	If 𝐹 is created by adding a single term (𝐹‘𝑋) = 𝑌 to 𝐺, where 𝑋 is larger than any element of the support of 𝐺, then 𝐹 is also a finitely supported function and it is assigned the value ((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑧 where 𝑧 is the value of 𝐺. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))))
Theorem	oemapso 9722*	The relation 𝑇 is a strict order on 𝑆 (a corollary of wemapso2 9593). (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ (𝜑 → 𝑇 Or 𝑆)
Theorem	oemapval 9723*	Value of the relation 𝑇. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))))
Theorem	oemapvali 9724*	If 𝐹 < 𝐺, then there is some 𝑧 witnessing this, but we can say more and in fact there is a definable expression 𝑋 that also witnesses 𝐹 < 𝐺. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
Theorem	cantnflem1a 9725*	Lemma for cantnf 9733. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅))
Theorem	cantnflem1b 9726*	Lemma for cantnf 9733. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    &   ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))    ⇒   ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂‘𝑢))
Theorem	cantnflem1c 9727*	Lemma for cantnf 9733. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) (Proof shortened by AV, 4-Apr-2020.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    &   ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))    ⇒   ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑥 ∈ (𝐺 supp ∅))
Theorem	cantnflem1d 9728*	Lemma for cantnf 9733. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    &   ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐺‘(𝑂‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ (𝐻‘suc (◡𝑂‘𝑋)))
Theorem	cantnflem1 9729*	Lemma for cantnf 9733. This part of the proof is showing uniqueness of the Cantor normal form. We already know that the relation 𝑇 is a strict order, but we haven't shown it is a well-order yet. But being a strict order is enough to show that two distinct 𝐹, 𝐺 are 𝑇 -related as 𝐹 < 𝐺 or 𝐺 < 𝐹, and WLOG assuming that 𝐹 < 𝐺, we show that CNF respects this order and maps these two to different ordinals. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 2-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    &   ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐺‘(𝑂‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ ((𝐴 CNF 𝐵)‘𝐺))
Theorem	cantnflem2 9730*	Lemma for cantnf 9733. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   ⊢ (𝜑 → ∅ ∈ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
Theorem	cantnflem3 9731*	Lemma for cantnf 9733. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than 𝐶 has a normal form, we can use oeeu 8641 to factor 𝐶 into the form ((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍 where 0 < 𝑌 < 𝐴 and 𝑍 < (𝐴 ↑o 𝑋) (and a fortiori 𝑋 < 𝐵). Then since 𝑍 < (𝐴 ↑o 𝑋) ≤ (𝐴 ↑o 𝑋) ·o 𝑌 ≤ 𝐶, 𝑍 has a normal form, and by appending the term (𝐴 ↑o 𝑋) ·o 𝑌 using cantnfp1 9721 we get a normal form for 𝐶. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   ⊢ (𝜑 → ∅ ∈ 𝐶)    &   ⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)}    &   ⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))    &   ⊢ 𝑌 = (1st ‘𝑃)    &   ⊢ 𝑍 = (2nd ‘𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    ⇒   ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))
Theorem	cantnflem4 9732*	Lemma for cantnf 9733. Complete the induction step of cantnflem3 9731. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   ⊢ (𝜑 → ∅ ∈ 𝐶)    &   ⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)}    &   ⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))    &   ⊢ 𝑌 = (1st ‘𝑃)    &   ⊢ 𝑍 = (2nd ‘𝑃)    ⇒   ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))
Theorem	cantnf 9733*	The Cantor Normal Form theorem. The function (𝐴 CNF 𝐵), which maps a finitely supported function from 𝐵 to 𝐴 to the sum ((𝐴 ↑o 𝑓(𝑎1)) ∘ 𝑎1) +o ((𝐴 ↑o 𝑓(𝑎2)) ∘ 𝑎2) +o ... over all indices 𝑎 < 𝐵 such that 𝑓(𝑎) is nonzero, is an order isomorphism from the ordering 𝑇 of finitely supported functions to the set (𝐴 ↑o 𝐵) under the natural order. Setting 𝐴 = ω and letting 𝐵 be arbitrarily large, the surjectivity of this function implies that every ordinal has a Cantor normal form (and injectivity, together with coherence cantnfres 9717, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)))
Theorem	oemapwe 9734*	The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternate definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵)))
Theorem	cantnffval2 9735*	An alternate definition of df-cnf 9702 which relies on cantnf 9733. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 9704 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆))
Theorem	cantnff1o 9736	Simplify the isomorphism of cantnf 9733 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆–1-1-onto→(𝐴 ↑o 𝐵))
Theorem	wemapwe 9737*	Construct lexicographic order on a function space based on a reverse well-ordering of the indices and a well-ordering of the values. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ 𝑈 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑆 We 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ 𝐹 = OrdIso(𝑅, 𝐴)    &   ⊢ 𝐺 = OrdIso(𝑆, 𝐵)    &   ⊢ 𝑍 = (𝐺‘∅)    ⇒   ⊢ (𝜑 → 𝑇 We 𝑈)
Theorem	oef1o 9738*	A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 7322.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ (On ∖ 1o))    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → (𝐹‘∅) = ∅)    &   ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺)))    &   ⊢ 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵))    ⇒   ⊢ (𝜑 → 𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
Theorem	cnfcomlem 9739*	Lemma for cnfcom 9740. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ (𝜑 → 𝐼 ∈ dom 𝐺)    &   ⊢ (𝜑 → 𝑂 ∈ (ω ↑o (𝐺‘𝐼)))    &   ⊢ (𝜑 → (𝑇‘𝐼):(𝐻‘𝐼)–1-1-onto→𝑂)    ⇒   ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))
Theorem	cnfcom 9740*	Any ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. Here we show that bijection explicitly. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ (𝜑 → 𝐼 ∈ dom 𝐺)    ⇒   ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))
Theorem	cnfcom2lem 9741*	Lemma for cnfcom2 9742. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ (𝜑 → ∅ ∈ 𝐵)    ⇒   ⊢ (𝜑 → dom 𝐺 = suc ∪ dom 𝐺)
Theorem	cnfcom2 9742*	Any nonzero ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ (𝜑 → ∅ ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊)))
Theorem	cnfcom3lem 9743*	Lemma for cnfcom3 9744. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ (𝜑 → ω ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1o))
Theorem	cnfcom3 9744*	Any infinite ordinal 𝐵 is equinumerous to a power of ω. (We are being careful here to show explicit bijections rather than simple equinumerosity because we want a uniform construction for cnfcom3c 9746.) (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 4-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ (𝜑 → ω ⊆ 𝐵)    &   ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((𝐹‘𝑊) ·o 𝑣) +o 𝑢))    &   ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑢) +o 𝑣))    &   ⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺))    ⇒   ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→(ω ↑o 𝑊))
Theorem	cnfcom3clem 9745*	Lemma for cnfcom3c 9746. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝑏)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((𝐹‘𝑊) ·o 𝑣) +o 𝑢))    &   ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑢) +o 𝑣))    &   ⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺))    &   ⊢ 𝐿 = (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)    ⇒   ⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
Theorem	cnfcom3c 9746*	Wrap the construction of cnfcom3 9744 into an existential quantifier. For any ω ⊆ 𝑏, there is a bijection from 𝑏 to some power of ω. Furthermore, this bijection is canonical , which means that we can find a single function 𝑔 which will give such bijections for every 𝑏 less than some arbitrarily large bound 𝐴. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
2.6.4  Transitive closure of a relation
Syntax	cttrcl 9747	Declare the syntax for the transitive closure of a class.
class t++𝑅
Definition	df-ttrcl 9748*	Define the transitive closure of a class. This is the smallest relation containing 𝑅 (or more precisely, the relation (𝑅 ↾ V) induced by 𝑅) and having the transitive property. Definition from [Levy] p. 59, who denotes it as 𝑅∗ and calls it the "ancestral" of 𝑅. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚))}
Theorem	ttrcleq 9749	Equality theorem for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ (𝑅 = 𝑆 → t++𝑅 = t++𝑆)
Theorem	nfttrcld 9750	Bound variable hypothesis builder for transitive closure. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ (𝜑 → Ⅎ𝑥𝑅)    ⇒   ⊢ (𝜑 → Ⅎ𝑥t++𝑅)
Theorem	nfttrcl 9751	Bound variable hypothesis builder for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ Ⅎ𝑥𝑅    ⇒   ⊢ Ⅎ𝑥t++𝑅
Theorem	relttrcl 9752	The transitive closure of a class is a relation. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ Rel t++𝑅
Theorem	brttrcl 9753*	Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 18-Aug-2024.)
⊢ (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
Theorem	brttrcl2 9754*	Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 24-Aug-2024.)
⊢ (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ ω ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
Theorem	ssttrcl 9755	If 𝑅 is a relation, then it is a subclass of its transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ (Rel 𝑅 → 𝑅 ⊆ t++𝑅)
Theorem	ttrcltr 9756	The transitive closure of a class is transitive. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ (t++𝑅 ∘ t++𝑅) ⊆ t++𝑅
Theorem	ttrclresv 9757	The transitive closure of 𝑅 restricted to V is the same as the transitive closure of 𝑅 itself. (Contributed by Scott Fenton, 20-Oct-2024.)
⊢ t++(𝑅 ↾ V) = t++𝑅
Theorem	ttrclco 9758	Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024.)
⊢ (t++𝑅 ∘ 𝑅) ⊆ t++𝑅
Theorem	cottrcl 9759	Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024.)
⊢ (𝑅 ∘ t++𝑅) ⊆ t++𝑅
Theorem	ttrclss 9760	If 𝑅 is a subclass of 𝑆 and 𝑆 is transitive, then the transitive closure of 𝑅 is a subclass of 𝑆. (Contributed by Scott Fenton, 20-Oct-2024.)
⊢ ((𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → t++𝑅 ⊆ 𝑆)
Theorem	dmttrcl 9761	The domain of a transitive closure is the same as the domain of the original class. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ dom t++𝑅 = dom 𝑅
Theorem	rnttrcl 9762	The range of a transitive closure is the same as the range of the original class. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ ran t++𝑅 = ran 𝑅
Theorem	ttrclexg 9763	If 𝑅 is a set, then so is t++𝑅. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ (𝑅 ∈ 𝑉 → t++𝑅 ∈ V)
Theorem	dfttrcl2 9764*	When 𝑅 is a set and a relation, then its transitive closure can be defined by an intersection. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → t++𝑅 = ∩ {𝑧 ∣ (𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
Theorem	ttrclselem1 9765*	Lemma for ttrclse 9767. Show that all finite ordinal function values of 𝐹 are subsets of 𝐴. (Contributed by Scott Fenton, 31-Oct-2024.)
⊢ 𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))    ⇒   ⊢ (𝑁 ∈ ω → (𝐹‘𝑁) ⊆ 𝐴)
Theorem	ttrclselem2 9766*	Lemma for ttrclse 9767. Show that a suc 𝑁 element long chain gives membership in the 𝑁-th predecessor class and vice-versa. (Contributed by Scott Fenton, 31-Oct-2024.)
⊢ 𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))    ⇒   ⊢ ((𝑁 ∈ ω ∧ 𝑅 Se 𝐴 ∧ 𝑋 ∈ 𝐴) → (∃𝑓(𝑓 Fn suc suc 𝑁 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘suc 𝑁) = 𝑋) ∧ ∀𝑎 ∈ suc 𝑁(𝑓‘𝑎)(𝑅 ↾ 𝐴)(𝑓‘suc 𝑎)) ↔ 𝑦 ∈ (𝐹‘𝑁)))
Theorem	ttrclse 9767	If 𝑅 is set-like over 𝐴, then the transitive closure of the restriction of 𝑅 to 𝐴 is set-like over 𝐴. This theorem requires the axioms of infinity and replacement for its proof. (Contributed by Scott Fenton, 31-Oct-2024.)
⊢ (𝑅 Se 𝐴 → t++(𝑅 ↾ 𝐴) Se 𝐴)
2.6.5  Transitive closure
Theorem	trcl 9768*	For any set 𝐴, show the properties of its transitive closure 𝐶. Similar to Theorem 9.1 of [TakeutiZaring] p. 73 except that we show an explicit expression for the transitive closure rather than just its existence. See tz9.1 9769 for an abbreviated version showing existence. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)    &   ⊢ 𝐶 = ∪ 𝑦 ∈ ω (𝐹‘𝑦)    ⇒   ⊢ (𝐴 ⊆ 𝐶 ∧ Tr 𝐶 ∧ ∀𝑥((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥))
Theorem	tz9.1 9769*	Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 9768 for an explicit expression for the transitive closure. Apparently open problems are whether this theorem can be proved without the Axiom of Infinity; if not, then whether it implies Infinity; and if not, what is the "property" that Infinity has that the other axioms don't have that is weaker than Infinity itself? (Added 22-Mar-2011) The following article seems to answer the first question, that it can't be proved without Infinity, in the affirmative: Mancini, Antonella and Zambella, Domenico (2001). "A note on recursive models of set theories." Notre Dame Journal of Formal Logic, 42(2):109-115. (Thanks to Scott Fenton.) (Contributed by NM, 15-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
Theorem	tz9.1c 9770*	Alternate expression for the existence of transitive closures tz9.1 9769: the intersection of all transitive sets containing 𝐴 is a set. (Contributed by Mario Carneiro, 22-Mar-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V
Theorem	epfrs 9771*	The strong form of the Axiom of Regularity (no sethood requirement on 𝐴), with the axiom itself present as an antecedent. See also zfregs 9772. (Contributed by Mario Carneiro, 22-Mar-2013.)
⊢ (( E Fr 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
Theorem	zfregs 9772*	The strong form of the Axiom of Regularity, which does not require that 𝐴 be a set. Axiom 6' of [TakeutiZaring] p. 21. See also epfrs 9771. (Contributed by NM, 17-Sep-2003.)
⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
Theorem	zfregs2 9773*	Alternate strong form of the Axiom of Regularity. Not every element of a nonempty class contains some element of that class. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)
⊢ (𝐴 ≠ ∅ → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
Theorem	setind 9774*	Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)
Theorem	setind2 9775	Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V)
Syntax	ctc 9776	Extend class notation to include the transitive closure function.
class TC
Definition	df-tc 9777*	The transitive closure function. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)})
Theorem	tcvalg 9778*	Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9678; see tz9.1 9769.) (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
Theorem	tcid 9779	Defining property of the transitive closure function: it contains its argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (TC‘𝐴))
Theorem	tctr 9780	Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ Tr (TC‘𝐴)
Theorem	tcmin 9781	Defining property of the transitive closure function: it is a subset of any transitive class containing 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵))
Theorem	tc2 9782*	A variant of the definition of the transitive closure function, using instead the smallest transitive set containing 𝐴 as a member, gives almost the same set, except that 𝐴 itself must be added because it is not usually a member of (TC‘𝐴) (and it is never a member if 𝐴 is well-founded). (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ((TC‘𝐴) ∪ {𝐴}) = ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
Theorem	tcsni 9783	The transitive closure of a singleton. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (TC‘{𝐴}) = ((TC‘𝐴) ∪ {𝐴})
Theorem	tcss 9784	The transitive closure function inherits the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ⊆ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))
Theorem	tcel 9785	The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))
Theorem	tcidm 9786	The transitive closure function is idempotent. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (TC‘(TC‘𝐴)) = (TC‘𝐴)
Theorem	tc0 9787	The transitive closure of the empty set. (Contributed by Mario Carneiro, 4-Jun-2015.)
⊢ (TC‘∅) = ∅
Theorem	tc00 9788	The transitive closure is empty iff its argument is. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)
⊢ (𝐴 ∈ 𝑉 → ((TC‘𝐴) = ∅ ↔ 𝐴 = ∅))
2.6.6  Well-Founded Induction
Theorem	frmin 9789*	Every (possibly proper) subclass of a class 𝐴 with a well-founded set-like relation 𝑅 has a minimal element. This is a very strong generalization of tz6.26 6368 and tz7.5 6405. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 27-Nov-2024.)
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
Theorem	frind 9790*	A subclass of a well-founded class 𝐴 with the property that whenever it contains all predecessors of an element of 𝐴 it also contains that element, is equal to 𝐴. Compare wfi 6371 and tfi 7874, which are special cases of this theorem that do not require the axiom of infinity. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)
Theorem	frinsg 9791*	Well-Founded Induction Schema. If a property passes from all elements less than 𝑦 of a well-founded class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. Theorem 5.6(ii) of [Levy] p. 64. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))    ⇒   ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)
Theorem	frins 9792*	Well-Founded Induction Schema. If a property passes from all elements less than 𝑦 of a well-founded class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ 𝑅 Fr 𝐴    &   ⊢ 𝑅 Se 𝐴    &   ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))    ⇒   ⊢ (𝑦 ∈ 𝐴 → 𝜑)
Theorem	frins2f 9793*	Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑))    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)
Theorem	frins2 9794*	Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 8-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑))    &   ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)
Theorem	frins3 9795*	Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑))    ⇒   ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝐵 ∈ 𝐴) → 𝜒)
2.6.7  Well-Founded Recursion
Theorem	frr3g 9796*	Functions defined by well-founded recursion are identical up to relation, domain, and characteristic function. General version of frr3 9801. (Contributed by Scott Fenton, 10-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)
Theorem	frrlem15 9797*	Lemma for general well-founded recursion. Two acceptable functions are compatible. (Contributed by Scott Fenton, 11-Sep-2023.)
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}    &   ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
Theorem	frrlem16 9798*	Lemma for general well-founded recursion. Establish a subset relation. (Contributed by Scott Fenton, 11-Sep-2023.) Revised notion of transitive closure. (Revised by Scott Fenton, 1-Dec-2024.)
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ Pred (t++(𝑅 ↾ 𝐴), 𝐴, 𝑧)Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧))
Theorem	frr1 9799	Law of general well-founded recursion, part one. This theorem and the following two drop the partial order requirement from fpr1 8328, fpr2 8329, and fpr3 8330, which requires using the axiom of infinity (Contributed by Scott Fenton, 11-Sep-2023.)
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
Theorem	frr2 9800	Law of general well-founded recursion, part two. Now we establish the value of 𝐹 within 𝐴. (Contributed by Scott Fenton, 11-Sep-2023.)
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))