P. 369 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36801-36900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dffunsALTV 36801	Alternate definition of the class of functions. (Contributed by Peter Mazsa, 18-Jul-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }
Theorem	dffunsALTV2 36802	Alternate definition of the class of functions. (Contributed by Peter Mazsa, 30-Aug-2021.)
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I }
Theorem	dffunsALTV3 36803*	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 36804*	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 36805*	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 36806	Alternate definition of the function relation predicate, cf. dfdisjALTV2 36832. (Contributed by Peter Mazsa, 8-Feb-2018.)
⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
Theorem	dffunALTV3 36807*	Alternate definition of the function relation predicate, cf. dfdisjALTV3 36833. Reproduction of dffun2 6447. 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 36808*	Alternate definition of the function relation predicate, cf. dfdisjALTV4 36834. This is dffun6 6449. 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 36809*	Alternate definition of the function relation predicate, cf. dfdisjALTV5 36835. (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 36810	Elementhood in the class of functions. (Contributed by Peter Mazsa, 24-Jul-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV2 36811	Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ⊆ I ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV3 36812*	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 36813*	Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ 𝐹 ∈ Rels ))
Theorem	elfunsALTV5 36814*	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 36815	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 36816	Our definition of the function predicate df-funALTV 36800 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6439, are always the same and interchangeable. (Contributed by Peter Mazsa, 27-Jul-2021.)
⊢ ( FunALTV 𝐹 ↔ Fun 𝐹)
Theorem	funALTVss 36817	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 36818	Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))
Theorem	funALTVeqi 36819	Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( FunALTV 𝐴 ↔ FunALTV 𝐵)
Theorem	funALTVeqd 36820	Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))
20.22.17  Disjoints vs. converse functions
Definition	df-disjss 36821	Define the class of all disjoint sets (but not necessarily disjoint relations, cf. df-disjs 36822). It is used only by df-disjs 36822. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ Disjss = {𝑥 ∣ ≀ ◡𝑥 ∈ CnvRefRels }
Definition	df-disjs 36822	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 36831). 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 36845. Alternate definitions are dfdisjs 36826, ... , dfdisjs5 36830. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ Disjs = ( Disjss ∩ Rels )
Definition	df-disjALTV 36823	Define the disjoint relation predicate, i.e., the disjoint predicate. A disjoint relation is a converse function of the relation by dfdisjALTV 36831, see the comment of df-disjs 36822 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 36845. Alternate definitions are dfdisjALTV 36831, ... , dfdisjALTV5 36835. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
Definition	df-eldisjs 36824	Define the disjoint elementhood 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 36847. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ ElDisjs = {𝑎 ∣ (◡ E ↾ 𝑎) ∈ Disjs }
Definition	df-eldisj 36825	Define the disjoint elementhood 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 36847. As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 36893 with dfeldisj5 36839. See also the comments of ~? dfmembpart2 and of ~? df-parts . (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
Theorem	dfdisjs 36826	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 18-Jul-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels }
Theorem	dfdisjs2 36827	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ⊆ I }
Theorem	dfdisjs3 36828*	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 36829*	Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑥∃*𝑢 𝑢𝑟𝑥}
Theorem	dfdisjs5 36830*	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 36831	Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 36822 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 36832	Alternate definition of the disjoint relation predicate, cf. dffunALTV2 36806. (Contributed by Peter Mazsa, 27-Jul-2021.)
⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅))
Theorem	dfdisjALTV3 36833*	Alternate definition of the disjoint relation predicate, cf. dffunALTV3 36807. (Contributed by Peter Mazsa, 28-Jul-2021.)
⊢ ( Disj 𝑅 ↔ (∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣) ∧ Rel 𝑅))
Theorem	dfdisjALTV4 36834*	Alternate definition of the disjoint relation predicate, cf. dffunALTV4 36808. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ ( Disj 𝑅 ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅))
Theorem	dfdisjALTV5 36835*	Alternate definition of the disjoint relation predicate, cf. dffunALTV5 36809. (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 36836	Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ≀ ◡(◡ E ↾ 𝐴) ⊆ I )
Theorem	dfeldisj3 36837*	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 36838*	Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢)
Theorem	dfeldisj5 36839*	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 36840	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021.)
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs2 36841	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ⊆ I ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs3 36842*	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 36843*	Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (𝑅 ∈ Disjs ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ 𝑅 ∈ Rels ))
Theorem	eldisjs5 36844*	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 36845	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 36846	Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))
Theorem	eleldisjseldisj 36847	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 36848	Disjoint relation is a relation. (Contributed by Peter Mazsa, 15-Sep-2021.)
⊢ ( Disj 𝑅 → Rel 𝑅)
Theorem	disjss 36849	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 36850	Subclass theorem for disjoints, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ ( Disj 𝐵 → Disj 𝐴)
Theorem	disjssd 36851	Subclass theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ( Disj 𝐵 → Disj 𝐴))
Theorem	disjeq 36852	Equality theorem for disjoints. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵))
Theorem	disjeqi 36853	Equality theorem for disjoints, inference version. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( Disj 𝐴 ↔ Disj 𝐵)
Theorem	disjeqd 36854	Equality theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 22-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( Disj 𝐴 ↔ Disj 𝐵))
Theorem	disjdmqseqeq1 36855	Lemma for the equality theorem for partition ~? parteq1 . (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ (𝑅 = 𝑆 → (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( Disj 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴)))
Theorem	eldisjss 36856	Subclass theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 ⊆ 𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴))
Theorem	eldisjssi 36857	Subclass theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ ( ElDisj 𝐵 → ElDisj 𝐴)
Theorem	eldisjssd 36858	Subclass theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ( ElDisj 𝐵 → ElDisj 𝐴))
Theorem	eldisjeq 36859	Equality theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
Theorem	eldisjeqi 36860	Equality theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ( ElDisj 𝐴 ↔ ElDisj 𝐵)
Theorem	eldisjeqd 36861	Equality theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))
Theorem	disjxrn 36862	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	disjorimxrn 36863	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 36864	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 36865	Disjointness condition for restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑅 → Disj (𝑅 ↾ 𝐴))
Theorem	disjimin 36866	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 36867	Disjointness condition for intersection with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ∩ (𝑆 ↾ 𝐴)))
Theorem	disjimxrnres 36868	Disjointness condition for range Cartesian product with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ( Disj 𝑆 → Disj (𝑅 ⋉ (𝑆 ↾ 𝐴)))
Theorem	disjALTV0 36869	The null class is disjoint. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ Disj ∅
Theorem	disjALTVid 36870	The class of identity relations is disjoint. (Contributed by Peter Mazsa, 20-Jun-2021.)
⊢ Disj I
Theorem	disjALTVidres 36871	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 36872	The intersection with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ Disj (𝑅 ∩ ( I ↾ 𝐴))
Theorem	disjALTVxrnidres 36873	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))
20.23  Mathbox for Rodolfo Medina
20.23.1  Partitions
Theorem	prtlem60 36874	Lemma for prter3 36903. (Contributed by Rodolfo Medina, 9-Oct-2010.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜓 → (𝜃 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
Theorem	bicomdd 36875	Commute two sides of a biconditional in a deduction. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜒)))
Theorem	jca2r 36876	Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜓 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 ∧ 𝜒)))
Theorem	jca3 36877	Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → 𝜏)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜒 ∧ 𝜏))))
Theorem	prtlem70 36878	Lemma for prter3 36903: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.)
⊢ ((((𝜓 ∧ 𝜂) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜏))) ∧ 𝜑) ↔ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ (𝜃 ∧ 𝜏)))) ∧ 𝜂))
Theorem	ibdr 36879	Reverse of ibd 268. (Contributed by Rodolfo Medina, 30-Sep-2010.)
⊢ (𝜑 → (𝜒 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))
Theorem	prtlem100 36880	Lemma for prter3 36903. (Contributed by Rodolfo Medina, 19-Oct-2010.)
⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵 ∈ 𝑥 ∧ 𝜑))
Theorem	prtlem5 36881*	Lemma for prter1 36900, prter2 36902, prter3 36903 and prtex 36901. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
⊢ ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥))
Theorem	prtlem80 36882	Lemma for prter2 36902. (Contributed by Rodolfo Medina, 17-Oct-2010.)
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))
Theorem	brabsb2 36883*	A closed form of brabsb 5445. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Theorem	eqbrrdv2 36884*	Other version of eqbrrdiv 5706. (Contributed by Rodolfo Medina, 30-Sep-2010.)
⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))    ⇒   ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
Theorem	prtlem9 36885*	Lemma for prter3 36903. (Contributed by Rodolfo Medina, 25-Sep-2010.)
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )
Theorem	prtlem10 36886*	Lemma for prter3 36903. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ( ∼ Er 𝐴 → (𝑧 ∈ 𝐴 → (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ [𝑣] ∼ ∧ 𝑤 ∈ [𝑣] ∼ ))))
Theorem	prtlem11 36887	Lemma for prter2 36902. (Contributed by Rodolfo Medina, 12-Oct-2010.)
⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ ))))
Theorem	prtlem12 36888*	Lemma for prtex 36901 and prter3 36903. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ ( ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} → Rel ∼ )
Theorem	prtlem13 36889*	Lemma for prter1 36900, prter2 36902, prter3 36903 and prtex 36901. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ (𝑧 ∼ 𝑤 ↔ ∃𝑣 ∈ 𝐴 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))
Theorem	prtlem16 36890*	Lemma for prtex 36901, prter2 36902 and prter3 36903. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ dom ∼ = ∪ 𝐴
Theorem	prtlem400 36891*	Lemma for prter2 36902 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ ¬ ∅ ∈ (∪ 𝐴 / ∼ )
Syntax	wprt 36892	Extend the definition of a wff to include the partition predicate.
wff Prt 𝐴
Definition	df-prt 36893*	Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
Theorem	erprt 36894	The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ ))
Theorem	prtlem14 36895*	Lemma for prter1 36900, prter2 36902 and prtex 36901. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦)))
Theorem	prtlem15 36896*	Lemma for prter1 36900 and prtex 36901. (Contributed by Rodolfo Medina, 13-Oct-2010.)
⊢ (Prt 𝐴 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝑢 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) ∧ (𝑤 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦)) → ∃𝑧 ∈ 𝐴 (𝑢 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧)))
Theorem	prtlem17 36897*	Lemma for prter2 36902. (Contributed by Rodolfo Medina, 15-Oct-2010.)
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥)))
Theorem	prtlem18 36898*	Lemma for prter2 36902. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → (𝑤 ∈ 𝑣 ↔ 𝑧 ∼ 𝑤)))
Theorem	prtlem19 36899*	Lemma for prter2 36902. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ (Prt 𝐴 → ((𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣) → 𝑣 = [𝑧] ∼ ))
Theorem	prter1 36900*	Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}    ⇒   ⊢ (Prt 𝐴 → ∼ Er ∪ 𝐴)