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Theorem List for Metamath Proof Explorer - 35901-36000   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremdferALTV2 35901 Equivalence relation with natural domain predicate, see the comment of df-ers 35896. (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 30-Aug-2021.)
(𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))

TheoremerALTVeq1 35902 Equality theorem for equivalence relation on domain quotient. (Contributed by Peter Mazsa, 25-Sep-2021.)
(𝑅 = 𝑆 → (𝑅 ErALTV 𝐴𝑆 ErALTV 𝐴))

TheoremerALTVeq1i 35903 Equality theorem for equivalence relation on domain quotient, inference version. (Contributed by Peter Mazsa, 25-Sep-2021.)
𝑅 = 𝑆       (𝑅 ErALTV 𝐴𝑆 ErALTV 𝐴)

TheoremerALTVeq1d 35904 Equality theorem for equivalence relation on domain quotient, deduction version. (Contributed by Peter Mazsa, 25-Sep-2021.)
(𝜑𝑅 = 𝑆)       (𝜑 → (𝑅 ErALTV 𝐴𝑆 ErALTV 𝐴))

Theoremdfmember 35905 Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.)
( MembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)

Theoremdfmember2 35906 Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.)
( MembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 /𝐴) = 𝐴))

Theoremdfmember3 35907 Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.)
( MembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))

Theoremeqvreldmqs 35908 Two ways to express membership 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))

Theorembrerser 35909 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.)
((𝐴𝑉𝑅𝑊) → (𝑅 Ers 𝐴𝑅 ErALTV 𝐴))

Theoremerim2 35910 Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is prter3 36017 in a more convenient form , see also erim 35911). (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 29-Dec-2021.)
(𝑅𝑉 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ∼ 𝐴 = 𝑅))

Theoremerim 35911 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 36017 and erim2 35910). (Contributed by Peter Mazsa, 7-Oct-2021.) (Revised by Peter Mazsa, 29-Dec-2021.)
(𝑅𝑉 → (𝑅 ErALTV 𝐴 → ∼ 𝐴 = 𝑅))

20.22.16  Functions

Definitiondf-funss 35912 Define the class of all function sets (but not necessarily function relations, cf. df-funsALTV 35913). It is used only by df-funsALTV 35913. (Contributed by Peter Mazsa, 17-Jul-2021.)
Funss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels }

Definitiondf-funsALTV 35913 Define the function relations class, i.e., the class of functions. Alternate definitions are dffunsALTV 35915, ... , dffunsALTV5 35919. (Contributed by Peter Mazsa, 17-Jul-2021.)
FunsALTV = ( Funss ∩ Rels )

Definitiondf-funALTV 35914 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 6356, are always the same, that is ( FunALTV 𝐹 ↔ Fun 𝐹), see funALTVfun 35930.

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 35929. Alternate definitions are dffunALTV2 35920, ... , dffunALTV5 35923. (Contributed by Peter Mazsa, 17-Jul-2021.)

( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))

TheoremdffunsALTV 35915 Alternate definition of the class of functions. (Contributed by Peter Mazsa, 18-Jul-2021.)
FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels }

TheoremdffunsALTV2 35916 Alternate definition of the class of functions. (Contributed by Peter Mazsa, 30-Aug-2021.)
FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I }

TheoremdffunsALTV3 35917* 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 ∣ ∀𝑢𝑥𝑦((𝑢𝑓𝑥𝑢𝑓𝑦) → 𝑥 = 𝑦)}

TheoremdffunsALTV4 35918* 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}

TheoremdffunsALTV5 35919* Alternate definition of the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]𝑓 ∩ [𝑦]𝑓) = ∅)}

TheoremdffunALTV2 35920 Alternate definition of the function relation predicate, cf. dfdisjALTV2 35946. (Contributed by Peter Mazsa, 8-Feb-2018.)
( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))

TheoremdffunALTV3 35921* Alternate definition of the function relation predicate, cf. dfdisjALTV3 35947. Reproduction of dffun2 6364. 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 ↔ (∀𝑢𝑥𝑦((𝑢𝐹𝑥𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ Rel 𝐹))

TheoremdffunALTV4 35922* Alternate definition of the function relation predicate, cf. dfdisjALTV4 35948. This is dffun6 6369. 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))

TheoremdffunALTV5 35923* Alternate definition of the function relation predicate, cf. dfdisjALTV5 35949. (Contributed by Peter Mazsa, 5-Sep-2021.)
( FunALTV 𝐹 ↔ (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ∧ Rel 𝐹))

TheoremelfunsALTV 35924 Elementhood in the class of functions. (Contributed by Peter Mazsa, 24-Jul-2021.)
(𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))

TheoremelfunsALTV2 35925 Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
(𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ⊆ I ∧ 𝐹 ∈ Rels ))

TheoremelfunsALTV3 35926* Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
(𝐹 ∈ FunsALTV ↔ (∀𝑢𝑥𝑦((𝑢𝐹𝑥𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ 𝐹 ∈ Rels ))

TheoremelfunsALTV4 35927* Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.)
(𝐹 ∈ FunsALTV ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥𝐹 ∈ Rels ))

TheoremelfunsALTV5 35928* Elementhood in the class of functions. (Contributed by Peter Mazsa, 5-Sep-2021.)
(𝐹 ∈ FunsALTV ↔ (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ∧ 𝐹 ∈ Rels ))

TheoremelfunsALTVfunALTV 35929 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.)
(𝐹𝑉 → (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹))

TheoremfunALTVfun 35930 Our definition of the function predicate df-funALTV 35914 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6356, are always the same and interchangeable. (Contributed by Peter Mazsa, 27-Jul-2021.)
( FunALTV 𝐹 ↔ Fun 𝐹)

TheoremfunALTVss 35931 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.)
(𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))

TheoremfunALTVeq 35932 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
(𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))

TheoremfunALTVeqi 35933 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐴 = 𝐵       ( FunALTV 𝐴 ↔ FunALTV 𝐵)

TheoremfunALTVeqd 35934 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))

20.22.17  Disjoints vs. converse functions

Definitiondf-disjss 35935 Define the class of all disjoint sets (but not necessarily disjoint relations, cf. df-disjs 35936). It is used only by df-disjs 35936. (Contributed by Peter Mazsa, 17-Jul-2021.)
Disjss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels }

Definitiondf-disjs 35936 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 35945).

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 35959. Alternate definitions are dfdisjs 35940, ... , dfdisjs5 35944. (Contributed by Peter Mazsa, 17-Jul-2021.)

Disjs = ( Disjss ∩ Rels )

Definitiondf-disjALTV 35937 Define the disjoint relation predicate, i.e., the disjoint predicate. A disjoint relation is a converse function of the relation by dfdisjALTV 35945, see the comment of df-disjs 35936 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 35959. Alternate definitions are dfdisjALTV 35945, ... , dfdisjALTV5 35949. (Contributed by Peter Mazsa, 17-Jul-2021.)

( Disj 𝑅 ↔ ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅))

Definitiondf-eldisjs 35938 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 35961. (Contributed by Peter Mazsa, 28-Nov-2022.)
ElDisjs = {𝑎 ∣ ( E ↾ 𝑎) ∈ Disjs }

Definitiondf-eldisj 35939 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 35961.

As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 36007 with dfeldisj5 35953. See also the comments of ~? dfmembpart2 and of ~? df-parts . (Contributed by Peter Mazsa, 17-Jul-2021.)

( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))

Theoremdfdisjs 35940 Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 18-Jul-2021.)
Disjs = {𝑟 ∈ Rels ∣ ≀ 𝑟 ∈ CnvRefRels }

Theoremdfdisjs2 35941 Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
Disjs = {𝑟 ∈ Rels ∣ ≀ 𝑟 ⊆ I }

Theoremdfdisjs3 35942* Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
Disjs = {𝑟 ∈ Rels ∣ ∀𝑢𝑣𝑥((𝑢𝑟𝑥𝑣𝑟𝑥) → 𝑢 = 𝑣)}

Theoremdfdisjs4 35943* Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
Disjs = {𝑟 ∈ Rels ∣ ∀𝑥∃*𝑢 𝑢𝑟𝑥}

Theoremdfdisjs5 35944* Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
Disjs = {𝑟 ∈ Rels ∣ ∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)}

TheoremdfdisjALTV 35945 Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 35936 why we need disjoint relations instead of converse functions anyway. (Contributed by Peter Mazsa, 27-Jul-2021.)
( Disj 𝑅 ↔ ( FunALTV 𝑅 ∧ Rel 𝑅))

TheoremdfdisjALTV2 35946 Alternate definition of the disjoint relation predicate, cf. dffunALTV2 35920. (Contributed by Peter Mazsa, 27-Jul-2021.)
( Disj 𝑅 ↔ ( ≀ 𝑅 ⊆ I ∧ Rel 𝑅))

TheoremdfdisjALTV3 35947* Alternate definition of the disjoint relation predicate, cf. dffunALTV3 35921. (Contributed by Peter Mazsa, 28-Jul-2021.)
( Disj 𝑅 ↔ (∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣) ∧ Rel 𝑅))

TheoremdfdisjALTV4 35948* Alternate definition of the disjoint relation predicate, cf. dffunALTV4 35922. (Contributed by Peter Mazsa, 5-Sep-2021.)
( Disj 𝑅 ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅))

TheoremdfdisjALTV5 35949* Alternate definition of the disjoint relation predicate, cf. dffunALTV5 35923. (Contributed by Peter Mazsa, 5-Sep-2021.)
( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))

Theoremdfeldisj2 35950 Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
( ElDisj 𝐴 ↔ ≀ ( E ↾ 𝐴) ⊆ I )

Theoremdfeldisj3 35951* Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
( ElDisj 𝐴 ↔ ∀𝑢𝐴𝑣𝐴𝑥 ∈ (𝑢𝑣)𝑢 = 𝑣)

Theoremdfeldisj4 35952* Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢𝐴 𝑥𝑢)

Theoremdfeldisj5 35953* Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
( ElDisj 𝐴 ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ (𝑢𝑣) = ∅))

Theoremeldisjs 35954 Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021.)
(𝑅 ∈ Disjs ↔ ( ≀ 𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels ))

Theoremeldisjs2 35955 Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
(𝑅 ∈ Disjs ↔ ( ≀ 𝑅 ⊆ I ∧ 𝑅 ∈ Rels ))

Theoremeldisjs3 35956* Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
(𝑅 ∈ Disjs ↔ (∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣) ∧ 𝑅 ∈ Rels ))

Theoremeldisjs4 35957* Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
(𝑅 ∈ Disjs ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥𝑅 ∈ Rels ))

Theoremeldisjs5 35958* Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
(𝑅𝑉 → (𝑅 ∈ Disjs ↔ (∀𝑢 ∈ dom 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ 𝑅 ∈ Rels )))

Theoremeldisjsdisj 35959 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.)
(𝑅𝑉 → (𝑅 ∈ Disjs ↔ Disj 𝑅))

Theoremeleldisjs 35960 Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.)
(𝐴𝑉 → (𝐴 ∈ ElDisjs ↔ ( E ↾ 𝐴) ∈ Disjs ))

Theoremeleldisjseldisj 35961 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.)
(𝐴𝑉 → (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴))

Theoremdisjrel 35962 Disjoint relation is a relation. (Contributed by Peter Mazsa, 15-Sep-2021.)
( Disj 𝑅 → Rel 𝑅)

Theoremdisjss 35963 Subclass theorem for disjoints. (Contributed by Peter Mazsa, 28-Oct-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)
(𝐴𝐵 → ( Disj 𝐵 → Disj 𝐴))

Theoremdisjssi 35964 Subclass theorem for disjoints, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.)
𝐴𝐵       ( Disj 𝐵 → Disj 𝐴)

Theoremdisjssd 35965 Subclass theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.)
(𝜑𝐴𝐵)       (𝜑 → ( Disj 𝐵 → Disj 𝐴))

Theoremdisjeq 35966 Equality theorem for disjoints. (Contributed by Peter Mazsa, 22-Sep-2021.)
(𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵))

Theoremdisjeqi 35967 Equality theorem for disjoints, inference version. (Contributed by Peter Mazsa, 22-Sep-2021.)
𝐴 = 𝐵       ( Disj 𝐴 ↔ Disj 𝐵)

Theoremdisjeqd 35968 Equality theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 22-Sep-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → ( Disj 𝐴 ↔ Disj 𝐵))

Theoremdisjdmqseqeq1 35969 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)))

Theoremeldisjss 35970 Subclass theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
(𝐴𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴))

Theoremeldisjssi 35971 Subclass theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.)
𝐴𝐵       ( ElDisj 𝐵 → ElDisj 𝐴)

Theoremeldisjssd 35972 Subclass theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.)
(𝜑𝐴𝐵)       (𝜑 → ( ElDisj 𝐵 → ElDisj 𝐴))

Theoremeldisjeq 35973 Equality theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.)
(𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))

Theoremeldisjeqi 35974 Equality theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
𝐴 = 𝐵       ( ElDisj 𝐴 ↔ ElDisj 𝐵)

Theoremeldisjeqd 35975 Equality theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵))

Theoremdisjxrn 35976 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 (𝑅𝑆) ↔ ( ≀ 𝑅 ∩ ≀ 𝑆) ⊆ I )

Theoremdisjorimxrn 35977 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 (𝑅𝑆))

Theoremdisjimxrn 35978 Disjointness condition for range Cartesian product. (Contributed by Peter Mazsa, 15-Dec-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)
( Disj 𝑆 → Disj (𝑅𝑆))

Theoremdisjimres 35979 Disjointness condition for restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
( Disj 𝑅 → Disj (𝑅𝐴))

Theoremdisjimin 35980 Disjointness condition for intersection. (Contributed by Peter Mazsa, 11-Jun-2021.) (Revised by Peter Mazsa, 28-Sep-2021.)
( Disj 𝑆 → Disj (𝑅𝑆))

Theoremdisjiminres 35981 Disjointness condition for intersection with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
( Disj 𝑆 → Disj (𝑅 ∩ (𝑆𝐴)))

Theoremdisjimxrnres 35982 Disjointness condition for range Cartesian product with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)
( Disj 𝑆 → Disj (𝑅 ⋉ (𝑆𝐴)))

TheoremdisjALTV0 35983 The null class is disjoint. (Contributed by Peter Mazsa, 27-Sep-2021.)
Disj ∅

TheoremdisjALTVid 35984 The class of identity relations is disjoint. (Contributed by Peter Mazsa, 20-Jun-2021.)
Disj I

TheoremdisjALTVidres 35985 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)

TheoremdisjALTVinidres 35986 The intersection with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)
Disj (𝑅 ∩ ( I ↾ 𝐴))

TheoremdisjALTVxrnidres 35987 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

Theoremprtlem60 35988 Lemma for prter3 36017. (Contributed by Rodolfo Medina, 9-Oct-2010.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜓 → (𝜃𝜏))       (𝜑 → (𝜓 → (𝜒𝜏)))

Theorembicomdd 35989 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 → (𝜒𝜃)))       (𝜑 → (𝜓 → (𝜃𝜒)))

Theoremjca2r 35990 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)       (𝜑 → (𝜓 → (𝜃𝜒)))

Theoremjca3 35991 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜏)       (𝜑 → (𝜓 → (𝜃 → (𝜒𝜏))))

Theoremprtlem70 35992 Lemma for prter3 36017: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.)
((((𝜓𝜂) ∧ ((𝜑𝜃) ∧ (𝜒𝜏))) ∧ 𝜑) ↔ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ (𝜃𝜏)))) ∧ 𝜂))

Theoremibdr 35993 Reverse of ibd 271. (Contributed by Rodolfo Medina, 30-Sep-2010.)
(𝜑 → (𝜒 → (𝜓𝜒)))       (𝜑 → (𝜒𝜓))

Theoremprtlem100 35994 Lemma for prter3 36017. (Contributed by Rodolfo Medina, 19-Oct-2010.)
(∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))

Theoremprtlem5 35995* Lemma for prter1 36014, prter2 36016, prter3 36017 and prtex 36015. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥))

Theoremprtlem80 35996 Lemma for prter2 36016. (Contributed by Rodolfo Medina, 17-Oct-2010.)
(𝐴𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))

Theorembrabsb2 35997* A closed form of brabsb 5417. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))

Theoremeqbrrdv2 35998* Other version of eqbrrdiv 5666. (Contributed by Rodolfo Medina, 30-Sep-2010.)
(((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦𝑥𝐵𝑦))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

Theoremprtlem9 35999* Lemma for prter3 36017. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(𝐴𝐵 → ∃𝑥𝐵 [𝑥] = [𝐴] )

Theoremprtlem10 36000* Lemma for prter3 36017. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
( Er 𝐴 → (𝑧𝐴 → (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] ))))

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