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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cnvepresdmqs 37001 | 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 37002 | 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 37003 | 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 37004 | 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 37005 | Lemma for erimeq2 37026. (Contributed by Peter Mazsa, 29-Dec-2021.) |
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → (𝐵 ∈ ran 𝑅 ↔ 𝐵 ∈ ∪ 𝐴)))) | ||
Theorem | releldmqs 37006* | 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 37007* | 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 37008* | 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 37009 | 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 37010 | 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) | ||
Definition | df-ers 37011 |
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 8582 "is not standard", "somewhat cryptic", has no costant 0-ary class and does not follow the traditional transparent reflexive-symmetric-transitive relation way of definition of equivalence. Definitions df-eqvrels 36932, dfeqvrels2 36936, dfeqvrels3 36937 and df-eqvrel 36933, dfeqvrel2 36938, dfeqvrel3 36939 are fully transparent in this regard. However, they lack the domain component (dom 𝑅 = 𝐴) of the present df-er 8582. While we acknowledge the need of a domain component, the present df-er 8582 definition does not utilize the results revealed by the new theorems in the Partition-Equivalence Theorem part below (like pets 37200 and pet 37199). From those theorems follows that the natural domain of equivalence relations is not 𝑅Domain𝐴 (i.e. dom 𝑅 = 𝐴 see brdomaing 34406), but 𝑅 DomainQss 𝐴 (i.e. (dom 𝑅 / 𝑅) = 𝐴, see brdmqss 36994), see erimeq 37027 vs. prter3 37230. While I'm sure we need both equivalence relation df-eqvrels 36932 and equivalence relation on domain quotient df-ers 37011, 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 36932 and df-ers 37011 are enough and named the predicate version of the one on domain quotient as the alternate version df-erALTV 37012 of the present df-er 8582. (Contributed by Peter Mazsa, 26-Jun-2021.) |
⊢ Ers = ( DomainQss ↾ EqvRels ) | ||
Definition | df-erALTV 37012 | Equivalence relation with natural domain predicate, see also the comment of df-ers 37011. Alternate definition is dferALTV2 37016. 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 37025. (Contributed by Peter Mazsa, 12-Aug-2021.) |
⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴)) | ||
Definition | df-comembers 37013 | 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 37014 |
Define the comember equivalence relation on the class 𝐴 (or, the
restricted coelement equivalence relation on its domain quotient 𝐴.)
Alternate definitions are dfcomember2 37021 and dfcomember3 37022.
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 37015 | Binary equivalence relation with natural domain, see the comment of df-ers 37011. (Contributed by Peter Mazsa, 23-Jul-2021.) |
⊢ (𝐴 ∈ 𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴))) | ||
Theorem | dferALTV2 37016 | Equivalence relation with natural domain predicate, see the comment of df-ers 37011. (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 37017 | 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 37018 | 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 37019 | 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 37020 | Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.) |
⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴) | ||
Theorem | dfcomember2 37021 | Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.) |
⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴)) | ||
Theorem | dfcomember3 37022 | 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 37023 | 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 37024 | 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 37025 | 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 37026 | Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is prter3 37230 in a more convenient form , see also erimeq 37027). (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 37027 | 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 37230 and erimeq2 37026). (Contributed by Peter Mazsa, 7-Oct-2021.) (Revised by Peter Mazsa, 29-Dec-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ErALTV 𝐴 → ∼ 𝐴 = 𝑅)) | ||
Definition | df-funss 37028 | Define the class of all function sets (but not necessarily function relations, cf. df-funsALTV 37029). It is used only by df-funsALTV 37029. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ Funss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels } | ||
Definition | df-funsALTV 37029 | Define the function relations class, i.e., the class of functions. Alternate definitions are dffunsALTV 37031, ... , dffunsALTV5 37035. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ FunsALTV = ( Funss ∩ Rels ) | ||
Definition | df-funALTV 37030 |
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 6494, are always the same, that is
( FunALTV 𝐹 ↔ Fun 𝐹), see funALTVfun 37046.
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 37045. Alternate definitions are dffunALTV2 37036, ... , dffunALTV5 37039. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)) | ||
Theorem | dffunsALTV 37031 | Alternate definition of the class of functions. (Contributed by Peter Mazsa, 18-Jul-2021.) |
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels } | ||
Theorem | dffunsALTV2 37032 | Alternate definition of the class of functions. (Contributed by Peter Mazsa, 30-Aug-2021.) |
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I } | ||
Theorem | dffunsALTV3 37033* | 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 37034* | 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 37035* | 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 37036 | Alternate definition of the function relation predicate, cf. dfdisjALTV2 37062. (Contributed by Peter Mazsa, 8-Feb-2018.) |
⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹)) | ||
Theorem | dffunALTV3 37037* | Alternate definition of the function relation predicate, cf. dfdisjALTV3 37063. Reproduction of dffun2 6502. 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 37038* | Alternate definition of the function relation predicate, cf. dfdisjALTV4 37064. This is dffun6 6505. 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 37039* | Alternate definition of the function relation predicate, cf. dfdisjALTV5 37065. (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 37040 | Elementhood in the class of functions. (Contributed by Peter Mazsa, 24-Jul-2021.) |
⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels )) | ||
Theorem | elfunsALTV2 37041 | Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.) |
⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ⊆ I ∧ 𝐹 ∈ Rels )) | ||
Theorem | elfunsALTV3 37042* | 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 37043* | Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.) |
⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ 𝐹 ∈ Rels )) | ||
Theorem | elfunsALTV5 37044* | 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 37045 | 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 37046 | Our definition of the function predicate df-funALTV 37030 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6494, are always the same and interchangeable. (Contributed by Peter Mazsa, 27-Jul-2021.) |
⊢ ( FunALTV 𝐹 ↔ Fun 𝐹) | ||
Theorem | funALTVss 37047 | 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 37048 | Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) |
⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵)) | ||
Theorem | funALTVeqi 37049 | Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ( FunALTV 𝐴 ↔ FunALTV 𝐵) | ||
Theorem | funALTVeqd 37050 | Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ( FunALTV 𝐴 ↔ FunALTV 𝐵)) | ||
Definition | df-disjss 37051 | Define the class of all disjoint sets (but not necessarily disjoint relations, cf. df-disjs 37052). It is used only by df-disjs 37052. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ Disjss = {𝑥 ∣ ≀ ◡𝑥 ∈ CnvRefRels } | ||
Definition | df-disjs 37052 |
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 37061).
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 37075. Alternate definitions are dfdisjs 37056, ... , dfdisjs5 37060. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ Disjs = ( Disjss ∩ Rels ) | ||
Definition | df-disjALTV 37053 |
Define the disjoint relation predicate, i.e., the disjoint predicate. A
disjoint relation is a converse function of the relation by dfdisjALTV 37061,
see the comment of df-disjs 37052 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 37075. Alternate definitions are dfdisjALTV 37061, ... , dfdisjALTV5 37065. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅)) | ||
Definition | df-eldisjs 37054 | 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 37077. (Contributed by Peter Mazsa, 28-Nov-2022.) |
⊢ ElDisjs = {𝑎 ∣ (◡ E ↾ 𝑎) ∈ Disjs } | ||
Definition | df-eldisj 37055 |
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 37077.
As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 37220 with dfeldisj5 37069. See also the comments of dfmembpart2 37118 and of df-parts 37113. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴)) | ||
Theorem | dfdisjs 37056 | Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 18-Jul-2021.) |
⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels } | ||
Theorem | dfdisjs2 37057 | Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ⊆ I } | ||
Theorem | dfdisjs3 37058* | 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 37059* | Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑥∃*𝑢 𝑢𝑟𝑥} | ||
Theorem | dfdisjs5 37060* | 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 37061 | Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 37052 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 37062 | Alternate definition of the disjoint relation predicate, cf. dffunALTV2 37036. (Contributed by Peter Mazsa, 27-Jul-2021.) |
⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅)) | ||
Theorem | dfdisjALTV3 37063* | Alternate definition of the disjoint relation predicate, cf. dffunALTV3 37037. (Contributed by Peter Mazsa, 28-Jul-2021.) |
⊢ ( Disj 𝑅 ↔ (∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣) ∧ Rel 𝑅)) | ||
Theorem | dfdisjALTV4 37064* | Alternate definition of the disjoint relation predicate, cf. dffunALTV4 37038. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ ( Disj 𝑅 ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅)) | ||
Theorem | dfdisjALTV5 37065* | Alternate definition of the disjoint relation predicate, cf. dffunALTV5 37039. (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 37066 | Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) |
⊢ ( ElDisj 𝐴 ↔ ≀ ◡(◡ E ↾ 𝐴) ⊆ I ) | ||
Theorem | dfeldisj3 37067* | 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 37068* | Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) |
⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) | ||
Theorem | dfeldisj5 37069* | 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 37070 | Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021.) |
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) | ||
Theorem | eldisjs2 37071 | Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ⊆ I ∧ 𝑅 ∈ Rels )) | ||
Theorem | eldisjs3 37072* | 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 37073* | Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ (𝑅 ∈ Disjs ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ 𝑅 ∈ Rels )) | ||
Theorem | eldisjs5 37074* | 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 37075 | 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 37076 | Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs )) | ||
Theorem | eleldisjseldisj 37077 | 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 37078 | Disjoint relation is a relation. (Contributed by Peter Mazsa, 15-Sep-2021.) |
⊢ ( Disj 𝑅 → Rel 𝑅) | ||
Theorem | disjss 37079 | 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 37080 | Subclass theorem for disjoints, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ( Disj 𝐵 → Disj 𝐴) | ||
Theorem | disjssd 37081 | Subclass theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ( Disj 𝐵 → Disj 𝐴)) | ||
Theorem | disjeq 37082 | Equality theorem for disjoints. (Contributed by Peter Mazsa, 22-Sep-2021.) |
⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵)) | ||
Theorem | disjeqi 37083 | Equality theorem for disjoints, inference version. (Contributed by Peter Mazsa, 22-Sep-2021.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ( Disj 𝐴 ↔ Disj 𝐵) | ||
Theorem | disjeqd 37084 | Equality theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 22-Sep-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ( Disj 𝐴 ↔ Disj 𝐵)) | ||
Theorem | disjdmqseqeq1 37085 | Lemma for the equality theorem for partition parteq1 37122. (Contributed by Peter Mazsa, 5-Oct-2021.) |
⊢ (𝑅 = 𝑆 → (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( Disj 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴))) | ||
Theorem | eldisjss 37086 | Subclass theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝐴 ⊆ 𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴)) | ||
Theorem | eldisjssi 37087 | Subclass theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ( ElDisj 𝐵 → ElDisj 𝐴) | ||
Theorem | eldisjssd 37088 | Subclass theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ( ElDisj 𝐵 → ElDisj 𝐴)) | ||
Theorem | eldisjeq 37089 | Equality theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵)) | ||
Theorem | eldisjeqi 37090 | Equality theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ( ElDisj 𝐴 ↔ ElDisj 𝐵) | ||
Theorem | eldisjeqd 37091 | Equality theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵)) | ||
Theorem | disjres 37092* | 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 37093 | 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 37094 | 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 37095* | 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 37096 | 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 37097 | 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 37098 | Disjointness condition for restriction. (Contributed by Peter Mazsa, 27-Sep-2021.) |
⊢ ( Disj 𝑅 → Disj (𝑅 ↾ 𝐴)) | ||
Theorem | disjimin 37099 | 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 37100 | Disjointness condition for intersection with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.) |
⊢ ( Disj 𝑆 → Disj (𝑅 ∩ (𝑆 ↾ 𝐴))) |
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