P. 391 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39001-39100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	refrelredund3 39001*	The naive version of the definition of reflexive relation (∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 ∧ Rel 𝑅) is redundant with respect to reflexive relation (see dfrefrel3 38876) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ redund ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)
21.26.16  Domain quotients
Definition	df-dmqss 39002*	Define the class of domain quotients. Domain quotients are pairs of sets, typically a relation and a set, where the quotient (see df-qs 8653) of the relation on its domain is equal to the set. See comments of df-ers 39028 for the motivation for this definition. (Contributed by Peter Mazsa, 16-Apr-2019.)
⊢ DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}
Definition	df-dmqs 39003	Define the domain quotient predicate. (Read: the domain quotient of 𝑅 is 𝐴.) If 𝐴 and 𝑅 are sets, the domain quotient binary relation and the domain quotient predicate are the same, see brdmqssqs 39011. (Contributed by Peter Mazsa, 9-Aug-2021.)
⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
Theorem	dmqseq 39004	Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
Theorem	dmqseqi 39005	Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆)
Theorem	dmqseqd 39006	Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.)
⊢ (𝜑 → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
Theorem	dmqseqeq1 39007	Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))
Theorem	dmqseqeq1i 39008	Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)
Theorem	dmqseqeq1d 39009	Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ (𝜑 → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))
Theorem	brdmqss 39010	The domain quotient binary relation. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))
Theorem	brdmqssqs 39011	If 𝐴 and 𝑅 are sets, the domain quotient binary relation and the domain quotient predicate are the same. (Contributed by Peter Mazsa, 14-Aug-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ 𝑅 DomainQs 𝐴))
Theorem	n0eldmqs 39012	The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 2-Mar-2018.)
⊢ ¬ ∅ ∈ (dom 𝑅 / 𝑅)
Theorem	qseq 39013*	The quotient set equal to a class. This theorem is used when a class 𝐴 is identified with a quotient (dom 𝑅 / 𝑅). In such a situation, every element 𝑢 ∈ 𝐴 is an 𝑅-coset [𝑣]𝑅 for some 𝑣 ∈ dom 𝑅, but there is no requirement that the "witness" 𝑣 be equal to its own block [𝑣]𝑅. 𝐴 is a set of blocks (equivalence classes), not a set of raw witnesses. In particular, when (dom 𝑅 / 𝑅) = 𝐴 is read together with a partition hypothesis 𝑅 Part 𝐴 (defined as dfpart2 39152), 𝐴 is being treated as the set of blocks [𝑣]𝑅; it does not assert any fixed-point condition 𝑣 = [𝑣]𝑅 such as would arise from the mistaken reading 𝑢 ∈ 𝐴 ↔ 𝑢 = [𝑢]𝑅. Cf. dmqsblocks 39247. (Contributed by Peter Mazsa, 19-Oct-2018.)
⊢ ((𝐵 / 𝑅) = 𝐴 ↔ ∀𝑢(𝑢 ∈ 𝐴 ↔ ∃𝑣 ∈ 𝐵 𝑢 = [𝑣]𝑅))
Theorem	n0eldmqseq 39014	The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 3-Nov-2018.)
⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴)
Theorem	n0elim 39015	Implication of that the empty set is not an element of a class. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ (¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)
Theorem	n0el3 39016	Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 27-May-2021.)
⊢ (¬ ∅ ∈ 𝐴 ↔ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)
Theorem	cnvepresdmqss 39017	The domain quotient binary relation of the restricted converse epsilon relation is equivalent to the negated elementhood of the empty set in the restriction. (Contributed by Peter Mazsa, 14-Aug-2021.)
⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) DomainQss 𝐴 ↔ ¬ ∅ ∈ 𝐴))
Theorem	cnvepresdmqs 39018	The domain quotient predicate for the restricted converse epsilon relation is equivalent to the negated elementhood of the empty set in the restriction. (Contributed by Peter Mazsa, 14-Aug-2021.)
⊢ ((◡ E ↾ 𝐴) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴)
Theorem	unidmqs 39019	The range of a relation is equal to the union of the domain quotient. (Contributed by Peter Mazsa, 13-Oct-2018.)
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ∪ (dom 𝑅 / 𝑅) = ran 𝑅))
Theorem	unidmqseq 39020	The union of the domain quotient of a relation is equal to the class 𝐴 if and only if the range is equal to it as well. (Contributed by Peter Mazsa, 21-Apr-2019.) (Revised by Peter Mazsa, 28-Dec-2021.)
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → (∪ (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴)))
Theorem	dmqseqim 39021	If the domain quotient of a relation is equal to the class 𝐴, then the range of the relation is the union of the class. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = ∪ 𝐴)))
Theorem	dmqseqim2 39022	Lemma for erimeq2 39043. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → (𝐵 ∈ ran 𝑅 ↔ 𝐵 ∈ ∪ 𝐴))))
Theorem	releldmqs 39023*	Elementhood in the domain quotient of a relation. (Contributed by Peter Mazsa, 24-Apr-2021.)
⊢ (𝐴 ∈ 𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅)))
Theorem	eldmqs1cossres 39024*	Elementhood in the domain quotient of the class of cosets by a restriction. (Contributed by Peter Mazsa, 4-May-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
Theorem	releldmqscoss 39025*	Elementhood in the domain quotient of the class of cosets by a relation. (Contributed by Peter Mazsa, 23-Apr-2021.)
⊢ (𝐴 ∈ 𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅)))
Theorem	dmqscoelseq 39026	Two ways to express the equality of the domain quotient of the coelements on the class 𝐴 with the class 𝐴. (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ ((dom ∼ 𝐴 / ∼ 𝐴) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)
Theorem	dmqs1cosscnvepreseq 39027	Two ways to express the equality of the domain quotient of the coelements on the class 𝐴 with the class 𝐴. (Contributed by Peter Mazsa, 26-Sep-2021.)
⊢ ((dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)
21.26.17  Equivalence relations on domain quotients
Definition	df-ers 39028	Define the class of equivalence relations on domain quotients (or: domain quotients restricted to equivalence relations). The present definition of equivalence relation in set.mm df-er 8647 "is not standard", "somewhat cryptic", has no constant 0-ary class and does not follow the traditional transparent reflexive-symmetric-transitive relation way of definition of equivalence. Definitions df-eqvrels 38948, dfeqvrels2 38952, dfeqvrels3 38953 and df-eqvrel 38949, dfeqvrel2 38954, dfeqvrel3 38955 are fully transparent in this regard. However, they lack the domain component (dom 𝑅 = 𝐴) of the present df-er 8647. While we acknowledge the need of a domain component, the present df-er 8647 definition does not utilize the results revealed by the new theorems in the Partition-Equivalence Theorem part below (like pets 39246 and pet 39245). From those theorems follows that the natural domain of equivalence relations is not 𝑅Domain𝐴 (i.e. dom 𝑅 = 𝐴 see brdomaing 36155), but 𝑅 DomainQss 𝐴 (i.e. (dom 𝑅 / 𝑅) = 𝐴, see brdmqss 39010), see erimeq 39044 vs. prter3 39287. While I'm sure we need both equivalence relation df-eqvrels 38948 and equivalence relation on domain quotient df-ers 39028, I'm not sure whether we need a third equivalence relation concept with the present dom 𝑅 = 𝐴 component as well: this needs further investigation. As a default I suppose that these two concepts df-eqvrels 38948 and df-ers 39028 are enough and named the predicate version of the one on domain quotient as the alternate version df-erALTV 39029 of the present df-er 8647. (Contributed by Peter Mazsa, 26-Jun-2021.)
⊢ Ers = ( DomainQss ↾ EqvRels )
Definition	df-erALTV 39029	Equivalence relation with natural domain predicate, see also the comment of df-ers 39028. Alternate definition is dferALTV2 39033. Binary equivalence relation with natural domain and the equivalence relation with natural domain predicate are the same when 𝐴 and 𝑅 are sets, see brerser 39042. (Contributed by Peter Mazsa, 12-Aug-2021.)
⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
Definition	df-comembers 39030	Define the class of comember equivalence relations on their domain quotients. (Contributed by Peter Mazsa, 28-Nov-2022.) (Revised by Peter Mazsa, 24-Jul-2023.)
⊢ CoMembErs = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
Definition	df-comember 39031	Define the comember equivalence relation on the class 𝐴 (or, the restricted coelement equivalence relation on its domain quotient 𝐴.) Alternate definitions are dfcomember2 39038 and dfcomember3 39039. Later on, in an application of set theory I make a distinction between the default elementhood concept and a special membership concept: membership equivalence relation will be an integral part of that membership concept. (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 28-Nov-2022.)
⊢ ( CoMembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
Theorem	brers 39032	Binary equivalence relation with natural domain, see the comment of df-ers 39028. (Contributed by Peter Mazsa, 23-Jul-2021.)
⊢ (𝐴 ∈ 𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴)))
Theorem	dferALTV2 39033	Equivalence relation with natural domain predicate, see the comment of df-ers 39028. (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 30-Aug-2021.)
⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
Theorem	erALTVeq1 39034	Equality theorem for equivalence relation on domain quotient. (Contributed by Peter Mazsa, 25-Sep-2021.)
⊢ (𝑅 = 𝑆 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴))
Theorem	erALTVeq1i 39035	Equality theorem for equivalence relation on domain quotient, inference version. (Contributed by Peter Mazsa, 25-Sep-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴)
Theorem	erALTVeq1d 39036	Equality theorem for equivalence relation on domain quotient, deduction version. (Contributed by Peter Mazsa, 25-Sep-2021.)
⊢ (𝜑 → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴))
Theorem	dfcomember 39037	Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)
Theorem	dfcomember2 39038	Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.)
⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	dfcomember3 39039	Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.)
⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	eqvreldmqs 39040	Two ways to express comember equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.)
⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	eqvreldmqs2 39041	Two ways to express comember equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	brerser 39042	Binary equivalence relation with natural domain and the equivalence relation with natural domain predicate are the same when 𝐴 and 𝑅 are sets. (Contributed by Peter Mazsa, 25-Aug-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Ers 𝐴 ↔ 𝑅 ErALTV 𝐴))
Theorem	erimeq2 39043	Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is prter3 39287 in a more convenient form , see also erimeq 39044). (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 29-Dec-2021.)
⊢ (𝑅 ∈ 𝑉 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ∼ 𝐴 = 𝑅))
Theorem	erimeq 39044	Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is the most convenient form of prter3 39287 and erimeq2 39043). (Contributed by Peter Mazsa, 7-Oct-2021.) (Revised by Peter Mazsa, 29-Dec-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ErALTV 𝐴 → ∼ 𝐴 = 𝑅))
21.26.18  Functions
Definition	df-funss 39045	Define the class of all function sets (but not necessarily function relations, cf. df-funsALTV 39046). It is used only by df-funsALTV 39046. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ Funss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels }
Definition	df-funsALTV 39046	Define the function relations class, i.e., the class of functions. Alternate definitions are dffunsALTV 39048, ... , dffunsALTV5 39052. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ FunsALTV = ( Funss ∩ Rels )
Definition	df-funALTV 39047	Define the function relation predicate, i.e., the function predicate. This definition of the function predicate (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6504, are always the same, that is ( FunALTV 𝐹 ↔ Fun 𝐹), see funALTVfun 39063. The element of the class of functions and the function predicate are the same, that is (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹) when 𝐹 is a set, see elfunsALTVfunALTV 39062. Alternate definitions are dffunALTV2 39053, ... , dffunALTV5 39056. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
Theorem	dffunsALTV 39048	Alternate definition of the class of functions. (Contributed by Peter Mazsa, 18-Jul-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
Theorem	dffunsALTV2 39049	Alternate definition of the class of functions. (Contributed by Peter Mazsa, 30-Aug-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I }
Theorem	dffunsALTV3 39050*	Alternate definition of the class of functions. For the 𝑋 axis and the 𝑌 axis you can convert the right side to {𝑓 ∈ Rels ∣ ∀ x1 ∀ y1 ∀ y2 (( x1 𝑓 y1 ∧ x1 𝑓 y2 ) → y1 = y2 )}. (Contributed by Peter Mazsa, 30-Aug-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦)}
Theorem	dffunsALTV4 39051*	Alternate definition of the class of functions. For the 𝑋 axis and the 𝑌 axis you can convert the right side to {𝑓 ∈ Rels ∣ ∀𝑥1∃*𝑦1𝑥1𝑓𝑦1}. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥}
Theorem	dffunsALTV5 39052*	Alternate definition of the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅)}
Theorem	dffunALTV2 39053	Alternate definition of the function relation predicate, cf. dfdisjALTV2 39079. (Contributed by Peter Mazsa, 8-Feb-2018.)
⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
Theorem	dffunALTV3 39054*	Alternate definition of the function relation predicate, cf. dfdisjALTV3 39080. Reproduction of dffun2 6512. For the 𝑋 axis and the 𝑌 axis you can convert the right side to (∀ x1 ∀ y1 ∀ y2 (( x1 𝑓 y1 ∧ x1 𝑓 y2 ) → y1 = y2 ) ∧ Rel 𝐹). (Contributed by NM, 29-Dec-1996.)
⊢ ( FunALTV 𝐹 ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ Rel 𝐹))
Theorem	dffunALTV4 39055*	Alternate definition of the function relation predicate, cf. dfdisjALTV4 39081. This is dffun6 6513. For the 𝑋 axis and the 𝑌 axis you can convert the right side to (∀𝑥1∃*𝑦1𝑥1𝐹𝑦1 ∧ Rel 𝐹). (Contributed by NM, 9-Mar-1995.)
⊢ ( FunALTV 𝐹 ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ Rel 𝐹))
Theorem	dffunALTV5 39056*	Alternate definition of the function relation predicate, cf. dfdisjALTV5 39082. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ ( FunALTV 𝐹 ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ Rel 𝐹))
Theorem	elfunsALTV 39057	Elementhood in the class of functions. (Contributed by Peter Mazsa, 24-Jul-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV2 39058	Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ⊆ I ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV3 39059*	Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝐹𝑥 ∧ 𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV4 39060*	Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV5 39061*	Elementhood in the class of functions. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTVfunALTV 39062	The element of the class of functions and the function predicate are the same when 𝐹 is a set. (Contributed by Peter Mazsa, 26-Jul-2021.)
⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹))
Theorem	funALTVfun 39063	Our definition of the function predicate df-funALTV 39047 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6504, are always the same and interchangeable. (Contributed by Peter Mazsa, 27-Jul-2021.)
⊢ ( FunALTV 𝐹 ↔ Fun 𝐹)
Theorem	funALTVss 39064	Subclass theorem for function. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) (Revised by Peter Mazsa, 22-Sep-2021.)
⊢ (𝐴 ⊆ 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
Theorem	funALTVeq 39065	Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))
Theorem	funALTVeqi 39066	Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( FunALTV 𝐴 ↔ FunALTV 𝐵)
Theorem	funALTVeqd 39067	Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))
21.26.19  Disjoints vs. converse functions
Definition	df-disjss 39068	Define the class of all disjoint sets (but not necessarily disjoint relations, cf. df-disjs 39069). It is used only by df-disjs 39069. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ Disjss = {𝑥 ∣ ≀ ◡𝑥 ∈ CnvRefRels }
Definition	df-disjs 39069	Define the disjoint relations class, i.e., the class of disjoints. We need Disjs for the definition of Parts and Part for the Partition-Equivalence Theorems: this need for Parts as disjoint relations on their domain quotients is the reason why we must define Disjs instead of simply using converse functions (cf. dfdisjALTV 39078). The element of the class of disjoints and the disjoint predicate are the same, that is (𝑅 ∈ Disjs ↔ Disj 𝑅) when 𝑅 is a set, see eldisjsdisj 39104. Alternate definitions are dfdisjs 39073, ... , dfdisjs5 39077. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ Disjs = ( Disjss ∩ Rels )
Definition	df-disjALTV 39070	Define the disjoint relation predicate, i.e., the disjoint predicate. A disjoint relation is a converse function of the relation by dfdisjALTV 39078, see the comment of df-disjs 39069 why we need disjoint relations instead of converse functions anyway. The element of the class of disjoints and the disjoint predicate are the same, that is (𝑅 ∈ Disjs ↔ Disj 𝑅) when 𝑅 is a set, see eldisjsdisj 39104. Alternate definitions are dfdisjALTV 39078, ... , dfdisjALTV5 39082. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
Definition	df-eldisjs 39071	Define the disjoint element relations class, i.e., the disjoint elements class. The element of the disjoint elements class and the disjoint elementhood predicate are the same, that is (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴) when 𝐴 is a set, see eleldisjseldisj 39109. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ ElDisjs = {𝑎 ∣ (◡ E ↾ 𝑎) ∈ Disjs }
Definition	df-eldisj 39072	Define the disjoint element relation predicate, i.e., the disjoint elementhood predicate. Read: the elements of 𝐴 are disjoint. The element of the disjoint elements class and the disjoint elementhood predicate are the same, that is (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴) when 𝐴 is a set, see eleldisjseldisj 39109. As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 39277 with dfeldisj5 39093. See also the comments of dfmembpart2 39153 and of df-parts 39148. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
Theorem	dfdisjs 39073	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 18-Jul-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }
Theorem	dfdisjs2 39074	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ⊆ I }
Theorem	dfdisjs3 39075*	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑢∀𝑣∀𝑥((𝑢𝑟𝑥 ∧ 𝑣𝑟𝑥) → 𝑢 = 𝑣)}
Theorem	dfdisjs4 39076*	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑥∃*𝑢 𝑢𝑟𝑥}
Theorem	dfdisjs5 39077*	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑢 ∈ dom 𝑟∀𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)}
Theorem	dfdisjALTV 39078	Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 39069 why we need disjoint relations instead of converse functions anyway. (Contributed by Peter Mazsa, 27-Jul-2021.)
⊢ ( Disj 𝑅 ↔ ( FunALTV ◡𝑅 ∧ Rel 𝑅))
Theorem	dfdisjALTV2 39079	Alternate definition of the disjoint relation predicate, cf. dffunALTV2 39053. (Contributed by Peter Mazsa, 27-Jul-2021.)
⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅))
Theorem	dfdisjALTV3 39080*	Alternate definition of the disjoint relation predicate, cf. dffunALTV3 39054. (Contributed by Peter Mazsa, 28-Jul-2021.)
⊢ ( Disj 𝑅 ↔ (∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣) ∧ Rel 𝑅))
Theorem	dfdisjALTV4 39081*	Alternate definition of the disjoint relation predicate, cf. dffunALTV4 39055. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ ( Disj 𝑅 ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅))
Theorem	dfdisjALTV5 39082*	Alternate definition of the disjoint relation predicate, cf. dffunALTV5 39056. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))
Theorem	dfdisjALTV5a 39083*	Alternate definition of the disjoint relation predicate. Disj 𝑅 means: different domain generators have disjoint cosets (unless the generators are equal), plus Rel 𝑅 for relation-typedness. This is the characterization that makes canonicity/uniqueness arguments modular. It is the starting point for the entire "Disj ↔ unique representative per block" pipeline that feeds into Disjs, see dfdisjs7 39223. (Contributed by Peter Mazsa, 3-Feb-2026.)
⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) ∧ Rel 𝑅))
Theorem	disjimeceqim 39084*	Disj implies coset-equality injectivity (domain-wise). Extracts the practical consequence of Disj: the map 𝑢 ↦ [𝑢]𝑅 is injective on dom 𝑅. This is exactly the "canonicity" property used repeatedly when turning ∃* into ∃! and when reasoning about uniqueness of representatives. (Contributed by Peter Mazsa, 3-Feb-2026.)
⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
Theorem	disjimeceqim2 39085	Disj implies injectivity (pairwise form). The same content as disjimeceqim 39084 but packaged for direct use with explicit hypotheses (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅). (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))
Theorem	disjimeceqbi 39086*	Disj gives biconditional injectivity (domain-wise). Strengthens injectivity to an iff. (Contributed by Peter Mazsa, 3-Feb-2026.)
⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 ↔ 𝑢 = 𝑣))
Theorem	disjimeceqbi2 39087	Injectivity of the block constructor under disjointness. suc11reg 9542 analogue: under disjointness, equal blocks force equal generators (on dom 𝑅). (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 ↔ 𝐴 = 𝐵)))
Theorem	disjimrmoeqec 39088*	Under Disj, every block has a unique generator (∃* form). If 𝑡 is a block in the quotient sense, then there is a uniquely determined 𝑢 in dom 𝑅 such that 𝑡 = [𝑢]𝑅. This is the existence+uniqueness engine behind Disjs and QMap characterizations: it is the "representative theorem" from which the ∃! forms are obtained. (Contributed by Peter Mazsa, 5-Feb-2026.)
⊢ ( Disj 𝑅 → ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)
Theorem	disjimdmqseq 39089*	Disjointness implies unique-generation of quotient blocks. Converts existence-quotient comprehension (see df-qs 8653) into a uniqueness-comprehension under disjointness; rewrites (dom 𝑅 / 𝑅) carriers as exactly the class of blocks with a unique representative. This is the "unique generator per block" content in a carrier-normal form. (Contributed by Peter Mazsa, 5-Feb-2026.)
⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = {𝑡 ∣ ∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅})
Theorem	dfeldisj2 39090	Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ≀ ◡(◡ E ↾ 𝐴) ⊆ I )
Theorem	dfeldisj3 39091*	Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ∀𝑥 ∈ (𝑢 ∩ 𝑣)𝑢 = 𝑣)
Theorem	dfeldisj4 39092*	Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
Theorem	dfeldisj5 39093*	Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅))
Theorem	dfeldisj5a 39094*	Alternate definition of the disjoint elementhood predicate. Members of 𝐴 are pairwise disjoint: if two members overlap, they are equal. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝑢 ∩ 𝑣) ≠ ∅ → 𝑢 = 𝑣))
Theorem	eldisjim3 39095	ElDisj elimination (two chosen elements). Standard specialization lemma: from ElDisj 𝐴 infer the disjointness condition for two specific elements. (Contributed by Peter Mazsa, 6-Feb-2026.)
⊢ ( ElDisj 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶)))
Theorem	eldisjdmqsim2 39096	ElDisj of quotient implies coset-disjointness (domain form). Converts element-disjointness of the quotient carrier into a usable "cosets don't overlap unless equal" rule. (Contributed by Peter Mazsa, 10-Feb-2026.)
⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)))
Theorem	eldisjdmqsim 39097*	Shared output implies equal cosets (under ElDisj of quotient): if 𝑢 and 𝑣 both relate to the same 𝑥, then their cosets intersect, hence must coincide under quotient ElDisj. (Contributed by Peter Mazsa, 10-Feb-2026.)
⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → [𝑢]𝑅 = [𝑣]𝑅))
Theorem	suceldisj 39098*	Disjointness of successor enforces element-carrier separation: If 𝐵 is the successor of 𝐴 and 𝐵 is element-disjoint as a family, then no element of 𝐴 can itself be a member of 𝐴 (equivalently, every 𝑥 ∈ 𝐴 has empty intersection with the carrier 𝐴). Provides a clean bridge between "disjoint family at the next grade" and "no block contains a block of the same family" at the previous grade: MembPart alone does not enforce this, see dfmembpart2 39153 (it gives disjoint blocks and excludes the empty block, but does not prevent 𝑢 ∈ 𝑚 from also being a member of the carrier 𝑚). This lemma is used to justify when grade-stability (via successor-shift) supplies the extra separation axioms needed in roof/root-style carrier reasoning. (Contributed by Peter Mazsa, 18-Feb-2026.)
⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → ∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
Theorem	eldisjs 39099	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021.)
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs2 39100	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ⊆ I ∧ 𝑅 ∈ Rels ))