P. 63 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6201-6300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rnmpt0f 6201*	The range of a function in maps-to notation is empty if and only if its domain is empty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
Theorem	rnmptn0 6202*	The range of a function in maps-to notation is nonempty if the domain is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    ⇒   ⊢ (𝜑 → ran 𝐹 ≠ ∅)
Theorem	dfco2 6203*	Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
⊢ (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
Theorem	dfco2a 6204*	Generalization of dfco2 6203, where 𝐶 can have any value between dom 𝐴 ∩ ran 𝐵 and V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
Theorem	coundi 6205	Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶))
Theorem	coundir 6206	Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶))
Theorem	cores 6207	Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵))
Theorem	resco 6208	Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶))
Theorem	imaco 6209	Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) (Proof shortened by Wolf Lammen, 16-May-2025.)
⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶))
Theorem	rnco 6210	The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) Avoid ax-11 2162. (Revised by TM, 24-Jan-2026.)
⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
Theorem	rncoOLD 6211	Obsolete version of rnco 6210 as of 24-Jan-2026. (Contributed by NM, 12-Dec-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
Theorem	rnco2 6212	The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵)
Theorem	dmco 6213	The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴)
Theorem	coeq0 6214	A composition of two relations is empty iff there is no overlap between the range of the second and the domain of the first. Useful in combination with coundi 6205 and coundir 6206 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.)
⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
Theorem	coiun 6215*	Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
Theorem	cocnvcnv1 6216	A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)
Theorem	cocnvcnv2 6217	A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵)
Theorem	cores2 6218	Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵))
Theorem	co02 6219	Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
⊢ (𝐴 ∘ ∅) = ∅
Theorem	co01 6220	Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
⊢ (∅ ∘ 𝐴) = ∅
Theorem	coi1 6221	Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
⊢ (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
Theorem	coi2 6222	Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)
Theorem	coires1 6223	Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵)
Theorem	coass 6224	Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)
⊢ ((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶))
Theorem	relcnvtrg 6225	General form of relcnvtr 6226. (Contributed by Peter Mazsa, 17-Oct-2023.)
⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅 ∘ 𝑆) ⊆ 𝑇 ↔ (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇))
Theorem	relcnvtr 6226	A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Peter Mazsa, 17-Oct-2023.)
⊢ (Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))
Theorem	relssdmrn 6227	A relation is included in the Cartesian product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.) (Proof shortened by SN, 23-Dec-2024.)
⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
Theorem	resssxp 6228	If the 𝑅-image of a class 𝐴 is a subclass of 𝐵, then the restriction of 𝑅 to 𝐴 is a subset of the Cartesian product of 𝐴 and 𝐵. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵))
Theorem	cnvssrndm 6229	The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴)
Theorem	cossxp 6230	Composition as a subset of the Cartesian product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴)
Theorem	relrelss 6231	Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))
Theorem	unielrel 6232	The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∪ 𝐴 ∈ ∪ 𝑅)
Theorem	relfld 6233	The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
Theorem	relresfld 6234	Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)
Theorem	relcoi2 6235	Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
⊢ (Rel 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = 𝑅)
Theorem	relcoi1 6236	Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) (Proof shortened by OpenAI, 3-Jul-2020.)
⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅)
Theorem	unidmrn 6237	The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴)
Theorem	relcnvfld 6238	if 𝑅 is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)
Theorem	dfdm2 6239	Alternate definition of domain df-dm 5634 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴)
Theorem	unixp 6240	The double class union of a nonempty Cartesian product is the union of it members. (Contributed by NM, 17-Sep-2006.)
⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
Theorem	unixp0 6241	A Cartesian product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)
⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅)
Theorem	unixpid 6242	Field of a Cartesian square. (Contributed by FL, 10-Oct-2009.)
⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴
Theorem	ressn 6243	Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
Theorem	cnviin 6244*	The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)
⊢ (𝐴 ≠ ∅ → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)
Theorem	cnvpo 6245	The converse of a partial order is a partial order. (Contributed by NM, 15-Jun-2005.)
⊢ (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴)
Theorem	cnvso 6246	The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.)
⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
Theorem	xpco 6247	Composition of two Cartesian products. (Contributed by Thierry Arnoux, 17-Nov-2017.)
⊢ (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
Theorem	xpcoid 6248	Composition of two Cartesian squares. (Contributed by Thierry Arnoux, 14-Jan-2018.)
⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)
Theorem	elsnxp 6249*	Membership in a Cartesian product with a singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.) (Proof shortened by JJ, 14-Jul-2021.)
⊢ (𝑋 ∈ 𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩))
Theorem	reu3op 6250*	There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 1-Jul-2023.)
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 𝜒 ∧ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
Theorem	reuop 6251*	There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 23-Jun-2023.)
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓 ↔ 𝜒))    &   ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜃))    ⇒   ⊢ (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝜃 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
Theorem	opreu2reurex 6252*	There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 24-Jun-2023.) (Revised by AV, 1-Jul-2023.)
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))    ⇒   ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒))
Theorem	opreu2reu 6253*	If there is a unique ordered pair fulfilling a wff, then there is a double restricted unique existential qualification fulfilling a corresponding wff. (Contributed by AV, 25-Jun-2023.) (Revised by AV, 2-Jul-2023.)
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))    ⇒   ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 → ∃!𝑎 ∈ 𝐴 ∃!𝑏 ∈ 𝐵 𝜒)
Theorem	dfpo2 6254	Quantifier-free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝑅 Po 𝐴 ↔ ((𝑅 ∩ ( I ↾ 𝐴)) = ∅ ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅))
Theorem	csbcog 6255	Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ∘ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∘ ⦋𝐴 / 𝑥⦌𝐶))
Theorem	snres0 6256	Condition for restriction of a singleton to be empty. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ ¬ 𝐴 ∈ 𝐶)
Theorem	imaindm 6257	The image is unaffected by intersection with the domain. (Contributed by Scott Fenton, 17-Dec-2021.)
⊢ (𝑅 “ 𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅))
2.3.11  The Predecessor Class
Syntax	cpred 6258	The predecessors symbol.
class Pred(𝑅, 𝐴, 𝑋)
Definition	df-pred 6259	Define the predecessor class of a binary relation. This is the class of all elements 𝑦 of 𝐴 such that 𝑦𝑅𝑋 (see elpred 6276). (Contributed by Scott Fenton, 29-Jan-2011.)
⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
Theorem	predeq123 6260	Equality theorem for the predecessor class. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))
Theorem	predeq1 6261	Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))
Theorem	predeq2 6262	Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
⊢ (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
Theorem	predeq3 6263	Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
Theorem	nfpred 6264	Bound-variable hypothesis builder for the predecessor class. (Contributed by Scott Fenton, 9-Jun-2018.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝑋    ⇒   ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑋)
Theorem	csbpredg 6265	Move class substitution in and out of the predecessor class of a relation. (Contributed by ML, 25-Oct-2020.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑋) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑋))
Theorem	predpredss 6266	If 𝐴 is a subset of 𝐵, then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.)
⊢ (𝐴 ⊆ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋))
Theorem	predss 6267	The predecessor class of 𝐴 is a subset of 𝐴. (Contributed by Scott Fenton, 2-Feb-2011.)
⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
Theorem	sspred 6268	Another subset/predecessor class relationship. (Contributed by Scott Fenton, 6-Feb-2011.)
⊢ ((𝐵 ⊆ 𝐴 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
Theorem	dfpred2 6269*	An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 8-Feb-2011.)
⊢ 𝑋 ∈ V    ⇒   ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑋})
Theorem	dfpred3 6270*	An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ 𝑋 ∈ V    ⇒   ⊢ Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
Theorem	dfpred3g 6271*	An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑋 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋})
Theorem	elpredgg 6272	Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) Generalize to closed form. (Revised by BJ, 16-Oct-2024.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
Theorem	elpredg 6273	Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) (Proof shortened by BJ, 16-Oct-2024.)
⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
Theorem	elpredimg 6274	Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.) (Proof shortened by BJ, 16-Oct-2024.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
Theorem	elpredim 6275	Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.) (Proof shortened by BJ, 16-Oct-2024.)
⊢ 𝑋 ∈ V    ⇒   ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)
Theorem	elpred 6276	Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.) (Proof shortened by BJ, 16-Oct-2024.)
⊢ 𝑌 ∈ V    ⇒   ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
Theorem	predexg 6277	The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.)
⊢ (𝐴 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V)
Theorem	dffr4 6278*	Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.)
⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅))
Theorem	predel 6279	Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.)
⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌 ∈ 𝐴)
Theorem	predtrss 6280	If 𝑅 is transitive over 𝐴 and 𝑌𝑅𝑋, then Pred(𝑅, 𝐴, 𝑌) is a subclass of Pred(𝑅, 𝐴, 𝑋). (Contributed by Scott Fenton, 28-Oct-2024.)
⊢ ((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋))
Theorem	predpo 6281	Property of the predecessor class for partial orders. (Contributed by Scott Fenton, 28-Apr-2012.) (Proof shortened by Scott Fenton, 28-Oct-2024.)
⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋)))
Theorem	predso 6282	Property of the predecessor class for strict total orders. (Contributed by Scott Fenton, 11-Feb-2011.)
⊢ ((𝑅 Or 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋)))
Theorem	setlikespec 6283	If 𝑅 is set-like in 𝐴, then all predecessor classes of elements of 𝐴 exist. (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
Theorem	predidm 6284	Idempotent law for the predecessor class. (Contributed by Scott Fenton, 29-Mar-2011.)
⊢ Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = Pred(𝑅, 𝐴, 𝑋)
Theorem	predin 6285	Intersection law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)
⊢ Pred(𝑅, (𝐴 ∩ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋))
Theorem	predun 6286	Union law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)
⊢ Pred(𝑅, (𝐴 ∪ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋))
Theorem	preddif 6287	Difference law for predecessor classes. (Contributed by Scott Fenton, 14-Apr-2011.)
⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋))
Theorem	predep 6288	The predecessor under the membership relation is equivalent to an intersection. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝑋 ∈ 𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋))
Theorem	trpred 6289	The class of predecessors of an element of a transitive class for the membership relation is that element. (Contributed by BJ, 12-Oct-2024.)
⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)
Theorem	preddowncl 6290*	A property of classes that are downward closed under predecessor. (Contributed by Scott Fenton, 13-Apr-2011.)
⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑋 ∈ 𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋)))
Theorem	predpoirr 6291	Given a partial ordering, a class is not a member of its predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)
⊢ (𝑅 Po 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
Theorem	predfrirr 6292	Given a well-founded relation, a class is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011.)
⊢ (𝑅 Fr 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
Theorem	pred0 6293	The predecessor class over ∅ is always ∅. (Contributed by Scott Fenton, 16-Apr-2011.) (Proof shortened by AV, 11-Jun-2021.)
⊢ Pred(𝑅, ∅, 𝑋) = ∅
Theorem	dfse3 6294*	Alternate definition of set-like relationships. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 Pred(𝑅, 𝐴, 𝑥) ∈ V)
Theorem	predrelss 6295	Subset carries from relation to predecessor class. (Contributed by Scott Fenton, 25-Nov-2024.)
⊢ (𝑅 ⊆ 𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋))
Theorem	predprc 6296	The predecessor of a proper class is empty. (Contributed by Scott Fenton, 25-Nov-2024.)
⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅)
Theorem	predres 6297	Predecessor class is unaffected by restriction to the base class. (Contributed by Scott Fenton, 25-Nov-2024.)
⊢ Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋)
2.3.12  Well-founded induction (variant)
Theorem	frpomin 6298*	Every nonempty (possibly proper) subclass of a class 𝐴 with a well-founded set-like partial order 𝑅 has a minimal element. The additional condition of partial order over frmin 9661 enables avoiding the axiom of infinity. (Contributed by Scott Fenton, 11-Feb-2022.)
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
Theorem	frpomin2 6299*	Every nonempty (possibly proper) subclass of a class 𝐴 with a well-founded set-like partial order 𝑅 has a minimal element. The additional condition of partial order over frmin 9661 enables avoiding the axiom of infinity. (Contributed by Scott Fenton, 11-Feb-2022.)
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅)
Theorem	frpoind 6300*	The principle of well-founded induction over a partial order. This theorem is a version of frind 9662 that does not require the axiom of infinity and can be used to prove wfi 6307 and tfi 7795. (Contributed by Scott Fenton, 11-Feb-2022.)
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)