P. 388 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38701-38800   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-erALTV 38701	Equivalence relation with natural domain predicate, see also the comment of df-ers 38700. Alternate definition is dferALTV2 38705. 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 38714. (Contributed by Peter Mazsa, 12-Aug-2021.)
⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
Definition	df-comembers 38702	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 38703	Define the comember equivalence relation on the class 𝐴 (or, the restricted coelement equivalence relation on its domain quotient 𝐴.) Alternate definitions are dfcomember2 38710 and dfcomember3 38711. 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 38704	Binary equivalence relation with natural domain, see the comment of df-ers 38700. (Contributed by Peter Mazsa, 23-Jul-2021.)
⊢ (𝐴 ∈ 𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴)))
Theorem	dferALTV2 38705	Equivalence relation with natural domain predicate, see the comment of df-ers 38700. (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 38706	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 38707	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 38708	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 38709	Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)
Theorem	dfcomember2 38710	Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.)
⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))
Theorem	dfcomember3 38711	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 38712	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 38713	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 38714	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 38715	Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is prter3 38920 in a more convenient form , see also erimeq 38716). (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 38716	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 38920 and erimeq2 38715). (Contributed by Peter Mazsa, 7-Oct-2021.) (Revised by Peter Mazsa, 29-Dec-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ErALTV 𝐴 → ∼ 𝐴 = 𝑅))
21.26.16  Functions
Definition	df-funss 38717	Define the class of all function sets (but not necessarily function relations, cf. df-funsALTV 38718). It is used only by df-funsALTV 38718. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ Funss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels }
Definition	df-funsALTV 38718	Define the function relations class, i.e., the class of functions. Alternate definitions are dffunsALTV 38720, ... , dffunsALTV5 38724. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ FunsALTV = ( Funss ∩ Rels )
Definition	df-funALTV 38719	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 6483, are always the same, that is ( FunALTV 𝐹 ↔ Fun 𝐹), see funALTVfun 38735. 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 38734. Alternate definitions are dffunALTV2 38725, ... , dffunALTV5 38728. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
Theorem	dffunsALTV 38720	Alternate definition of the class of functions. (Contributed by Peter Mazsa, 18-Jul-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
Theorem	dffunsALTV2 38721	Alternate definition of the class of functions. (Contributed by Peter Mazsa, 30-Aug-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I }
Theorem	dffunsALTV3 38722*	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 38723*	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 38724*	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 38725	Alternate definition of the function relation predicate, cf. dfdisjALTV2 38751. (Contributed by Peter Mazsa, 8-Feb-2018.)
⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
Theorem	dffunALTV3 38726*	Alternate definition of the function relation predicate, cf. dfdisjALTV3 38752. Reproduction of dffun2 6491. 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 38727*	Alternate definition of the function relation predicate, cf. dfdisjALTV4 38753. This is dffun6 6492. 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 38728*	Alternate definition of the function relation predicate, cf. dfdisjALTV5 38754. (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 38729	Elementhood in the class of functions. (Contributed by Peter Mazsa, 24-Jul-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV2 38730	Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ⊆ I ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV3 38731*	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 38732*	Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV5 38733*	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 38734	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 38735	Our definition of the function predicate df-funALTV 38719 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6483, are always the same and interchangeable. (Contributed by Peter Mazsa, 27-Jul-2021.)
⊢ ( FunALTV 𝐹 ↔ Fun 𝐹)
Theorem	funALTVss 38736	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 38737	Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))
Theorem	funALTVeqi 38738	Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( FunALTV 𝐴 ↔ FunALTV 𝐵)
Theorem	funALTVeqd 38739	Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))
21.26.17  Disjoints vs. converse functions
Definition	df-disjss 38740	Define the class of all disjoint sets (but not necessarily disjoint relations, cf. df-disjs 38741). It is used only by df-disjs 38741. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ Disjss = {𝑥 ∣ ≀ ◡𝑥 ∈ CnvRefRels }
Definition	df-disjs 38741	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 38750). 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 38764. Alternate definitions are dfdisjs 38745, ... , dfdisjs5 38749. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ Disjs = ( Disjss ∩ Rels )
Definition	df-disjALTV 38742	Define the disjoint relation predicate, i.e., the disjoint predicate. A disjoint relation is a converse function of the relation by dfdisjALTV 38750, see the comment of df-disjs 38741 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 38764. Alternate definitions are dfdisjALTV 38750, ... , dfdisjALTV5 38754. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
Definition	df-eldisjs 38743	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 38766. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ ElDisjs = {𝑎 ∣ (◡ E ↾ 𝑎) ∈ Disjs }
Definition	df-eldisj 38744	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 38766. As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 38910 with dfeldisj5 38758. See also the comments of dfmembpart2 38807 and of df-parts 38802. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
Theorem	dfdisjs 38745	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 18-Jul-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }
Theorem	dfdisjs2 38746	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ⊆ I }
Theorem	dfdisjs3 38747*	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 38748*	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑥∃*𝑢 𝑢𝑟𝑥}
Theorem	dfdisjs5 38749*	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 38750	Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 38741 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 38751	Alternate definition of the disjoint relation predicate, cf. dffunALTV2 38725. (Contributed by Peter Mazsa, 27-Jul-2021.)
⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅))
Theorem	dfdisjALTV3 38752*	Alternate definition of the disjoint relation predicate, cf. dffunALTV3 38726. (Contributed by Peter Mazsa, 28-Jul-2021.)
⊢ ( Disj 𝑅 ↔ (∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣) ∧ Rel 𝑅))
Theorem	dfdisjALTV4 38753*	Alternate definition of the disjoint relation predicate, cf. dffunALTV4 38727. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ ( Disj 𝑅 ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅))
Theorem	dfdisjALTV5 38754*	Alternate definition of the disjoint relation predicate, cf. dffunALTV5 38728. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))
Theorem	dfeldisj2 38755	Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ≀ ◡(◡ E ↾ 𝐴) ⊆ I )
Theorem	dfeldisj3 38756*	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 38757*	Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
Theorem	dfeldisj5 38758*	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	eldisjs 38759	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021.)
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs2 38760	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ⊆ I ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs3 38761*	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 ∈ Disjs ↔ (∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣) ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs4 38762*	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 ∈ Disjs ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs5 38763*	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels )))
Theorem	eldisjsdisj 38764	The element of the class of disjoint relations and the disjoint relation predicate are the same, that is (𝑅 ∈ Disjs ↔ Disj 𝑅) when 𝑅 is a set. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Disjs ↔ Disj 𝑅))
Theorem	eleldisjs 38765	Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))
Theorem	eleldisjseldisj 38766	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. (Contributed by Peter Mazsa, 23-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴))
Theorem	disjrel 38767	Disjoint relation is a relation. (Contributed by Peter Mazsa, 15-Sep-2021.)
⊢ ( Disj 𝑅 → Rel 𝑅)
Theorem	disjss 38768	Subclass theorem for disjoints. (Contributed by Peter Mazsa, 28-Oct-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)
⊢ (𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴))
Theorem	disjssi 38769	Subclass theorem for disjoints, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ ( Disj 𝐵 → Disj 𝐴)
Theorem	disjssd 38770	Subclass theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ( Disj 𝐵 → Disj 𝐴))
Theorem	disjeq 38771	Equality theorem for disjoints. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵))
Theorem	disjeqi 38772	Equality theorem for disjoints, inference version. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( Disj 𝐴 ↔ Disj 𝐵)
Theorem	disjeqd 38773	Equality theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( Disj 𝐴 ↔ Disj 𝐵))
Theorem	disjdmqseqeq1 38774	Lemma for the equality theorem for partition parteq1 38811. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ (𝑅 = 𝑆 → (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( Disj 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴)))
Theorem	eldisjss 38775	Subclass theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 ⊆ 𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴))
Theorem	eldisjssi 38776	Subclass theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ ( ElDisj 𝐵 → ElDisj 𝐴)
Theorem	eldisjssd 38777	Subclass theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ( ElDisj 𝐵 → ElDisj 𝐴))
Theorem	eldisjeq 38778	Equality theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
Theorem	eldisjeqi 38779	Equality theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( ElDisj 𝐴 ↔ ElDisj 𝐵)
Theorem	eldisjeqd 38780	Equality theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
Theorem	disjres 38781*	Disjoint restriction. (Contributed by Peter Mazsa, 25-Aug-2023.)
⊢ (Rel 𝑅 → ( Disj (𝑅 ↾ 𝐴) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅)))
Theorem	eldisjn0elb 38782	Two forms of disjoint elements when the empty set is not an element of the class. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( Disj (◡ E ↾ 𝐴) ∧ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴))
Theorem	disjxrn 38783	Two ways of saying that a range Cartesian product is disjoint. (Contributed by Peter Mazsa, 17-Jun-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
⊢ ( Disj (𝑅 ⋉ 𝑆) ↔ ( ≀ ◡𝑅 ∩ ≀ ◡𝑆) ⊆ I )
Theorem	disjxrnres5 38784*	Disjoint range Cartesian product. (Contributed by Peter Mazsa, 25-Aug-2023.)
⊢ ( Disj (𝑅 ⋉ (𝑆 ↾ 𝐴)) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ 𝑆) ∩ [𝑣](𝑅 ⋉ 𝑆)) = ∅))
Theorem	disjorimxrn 38785	Disjointness condition for range Cartesian product. (Contributed by Peter Mazsa, 12-Jul-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)
⊢ (( Disj 𝑅 ∨ Disj 𝑆) → Disj (𝑅 ⋉ 𝑆))
Theorem	disjimxrn 38786	Disjointness condition for range Cartesian product. (Contributed by Peter Mazsa, 15-Dec-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ⋉ 𝑆))
Theorem	disjimres 38787	Disjointness condition for restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑅 → Disj (𝑅 ↾ 𝐴))
Theorem	disjimin 38788	Disjointness condition for intersection. (Contributed by Peter Mazsa, 11-Jun-2021.) (Revised by Peter Mazsa, 28-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ∩ 𝑆))
Theorem	disjiminres 38789	Disjointness condition for intersection with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ∩ (𝑆 ↾ 𝐴)))
Theorem	disjimxrnres 38790	Disjointness condition for range Cartesian product with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ⋉ (𝑆 ↾ 𝐴)))
Theorem	disjALTV0 38791	The null class is disjoint. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ Disj ∅
Theorem	disjALTVid 38792	The class of identity relations is disjoint. (Contributed by Peter Mazsa, 20-Jun-2021.)
⊢ Disj I
Theorem	disjALTVidres 38793	The class of identity relations restricted is disjoint. (Contributed by Peter Mazsa, 28-Jun-2020.) (Revised by Peter Mazsa, 27-Sep-2021.)
⊢ Disj ( I ↾ 𝐴)
Theorem	disjALTVinidres 38794	The intersection with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ Disj (𝑅 ∩ ( I ↾ 𝐴))
Theorem	disjALTVxrnidres 38795	The class of range Cartesian product with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 25-Jun-2020.) (Revised by Peter Mazsa, 27-Sep-2021.)
⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴))
Theorem	disjsuc 38796*	Disjoint range Cartesian product, special case. (Contributed by Peter Mazsa, 25-Aug-2023.)
⊢ (𝐴 ∈ 𝑉 → ( Disj (𝑅 ⋉ (◡ E ↾ suc 𝐴)) ↔ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
21.26.18  Antisymmetry
Definition	df-antisymrel 38797	Define the antisymmetric relation predicate. (Read: 𝑅 is an antisymmetric relation.) (Contributed by Peter Mazsa, 24-Jun-2024.)
⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ◡𝑅) ∧ Rel 𝑅))
Theorem	dfantisymrel4 38798	Alternate definition of the antisymmetric relation predicate. (Contributed by Peter Mazsa, 24-Jun-2024.)
⊢ ( AntisymRel 𝑅 ↔ ((𝑅 ∩ ◡𝑅) ⊆ I ∧ Rel 𝑅))
Theorem	dfantisymrel5 38799*	Alternate definition of the antisymmetric relation predicate. (Contributed by Peter Mazsa, 24-Jun-2024.)
⊢ ( AntisymRel 𝑅 ↔ (∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ Rel 𝑅))
Theorem	antisymrelres 38800*	(Contributed by Peter Mazsa, 25-Jun-2024.)
⊢ ( AntisymRel (𝑅 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))