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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dffunsALTV4 35801* | 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 35802* | 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 35803 | Alternate definition of the function relation predicate, cf. dfdisjALTV2 35829. (Contributed by Peter Mazsa, 8-Feb-2018.) |
⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹)) | ||
Theorem | dffunALTV3 35804* | Alternate definition of the function relation predicate, cf. dfdisjALTV3 35830. Reproduction of dffun2 6359. 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 35805* | Alternate definition of the function relation predicate, cf. dfdisjALTV4 35831. This is dffun6 6364. 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 35806* | Alternate definition of the function relation predicate, cf. dfdisjALTV5 35832. (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 35807 | Elementhood in the class of functions. (Contributed by Peter Mazsa, 24-Jul-2021.) |
⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels )) | ||
Theorem | elfunsALTV2 35808 | Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.) |
⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ⊆ I ∧ 𝐹 ∈ Rels )) | ||
Theorem | elfunsALTV3 35809* | 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 35810* | Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.) |
⊢ (𝐹 ∈ FunsALTV ↔ (∀𝑢∃*𝑥 𝑢𝐹𝑥 ∧ 𝐹 ∈ Rels )) | ||
Theorem | elfunsALTV5 35811* | 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 35812 | 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 35813 | Our definition of the function predicate df-funALTV 35797 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6351, are always the same and interchangeable. (Contributed by Peter Mazsa, 27-Jul-2021.) |
⊢ ( FunALTV 𝐹 ↔ Fun 𝐹) | ||
Theorem | funALTVss 35814 | 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 35815 | Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) |
⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵)) | ||
Theorem | funALTVeqi 35816 | Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ( FunALTV 𝐴 ↔ FunALTV 𝐵) | ||
Theorem | funALTVeqd 35817 | Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ( FunALTV 𝐴 ↔ FunALTV 𝐵)) | ||
Definition | df-disjss 35818 | Define the class of all disjoint sets (but not necessarily disjoint relations, cf. df-disjs 35819). It is used only by df-disjs 35819. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ Disjss = {𝑥 ∣ ≀ ◡𝑥 ∈ CnvRefRels } | ||
Definition | df-disjs 35819 |
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 35828).
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 35842. Alternate definitions are dfdisjs 35823, ... , dfdisjs5 35827. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ Disjs = ( Disjss ∩ Rels ) | ||
Definition | df-disjALTV 35820 |
Define the disjoint relation predicate, i.e., the disjoint predicate. A
disjoint relation is a converse function of the relation by dfdisjALTV 35828,
see the comment of df-disjs 35819 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 35842. Alternate definitions are dfdisjALTV 35828, ... , dfdisjALTV5 35832. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅)) | ||
Definition | df-eldisjs 35821 | 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 35844. (Contributed by Peter Mazsa, 28-Nov-2022.) |
⊢ ElDisjs = {𝑎 ∣ (◡ E ↾ 𝑎) ∈ Disjs } | ||
Definition | df-eldisj 35822 |
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 35844.
As of now, disjoint elementhood is defined as "partition" in set.mm : compare df-prt 35890 with dfeldisj5 35836. See also the comments of ~? dfmembpart2 and of ~? df-parts . (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴)) | ||
Theorem | dfdisjs 35823 | Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 18-Jul-2021.) |
⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ∈ CnvRefRels } | ||
Theorem | dfdisjs2 35824 | Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ Disjs = {𝑟 ∈ Rels ∣ ≀ ◡𝑟 ⊆ I } | ||
Theorem | dfdisjs3 35825* | 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 35826* | Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ Disjs = {𝑟 ∈ Rels ∣ ∀𝑥∃*𝑢 𝑢𝑟𝑥} | ||
Theorem | dfdisjs5 35827* | 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 35828 | Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 35819 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 35829 | Alternate definition of the disjoint relation predicate, cf. dffunALTV2 35803. (Contributed by Peter Mazsa, 27-Jul-2021.) |
⊢ ( Disj 𝑅 ↔ ( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅)) | ||
Theorem | dfdisjALTV3 35830* | Alternate definition of the disjoint relation predicate, cf. dffunALTV3 35804. (Contributed by Peter Mazsa, 28-Jul-2021.) |
⊢ ( Disj 𝑅 ↔ (∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣) ∧ Rel 𝑅)) | ||
Theorem | dfdisjALTV4 35831* | Alternate definition of the disjoint relation predicate, cf. dffunALTV4 35805. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ ( Disj 𝑅 ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅)) | ||
Theorem | dfdisjALTV5 35832* | Alternate definition of the disjoint relation predicate, cf. dffunALTV5 35806. (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 35833 | Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) |
⊢ ( ElDisj 𝐴 ↔ ≀ ◡(◡ E ↾ 𝐴) ⊆ I ) | ||
Theorem | dfeldisj3 35834* | 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 35835* | Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) |
⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) | ||
Theorem | dfeldisj5 35836* | 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 35837 | Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021.) |
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ∈ CnvRefRels ∧ 𝑅 ∈ Rels )) | ||
Theorem | eldisjs2 35838 | Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ (𝑅 ∈ Disjs ↔ ( ≀ ◡𝑅 ⊆ I ∧ 𝑅 ∈ Rels )) | ||
Theorem | eldisjs3 35839* | 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 35840* | Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ (𝑅 ∈ Disjs ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ 𝑅 ∈ Rels )) | ||
Theorem | eldisjs5 35841* | 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 35842 | 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 35843 | Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs )) | ||
Theorem | eleldisjseldisj 35844 | 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 35845 | Disjoint relation is a relation. (Contributed by Peter Mazsa, 15-Sep-2021.) |
⊢ ( Disj 𝑅 → Rel 𝑅) | ||
Theorem | disjss 35846 | 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 35847 | Subclass theorem for disjoints, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ( Disj 𝐵 → Disj 𝐴) | ||
Theorem | disjssd 35848 | Subclass theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ( Disj 𝐵 → Disj 𝐴)) | ||
Theorem | disjeq 35849 | Equality theorem for disjoints. (Contributed by Peter Mazsa, 22-Sep-2021.) |
⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵)) | ||
Theorem | disjeqi 35850 | Equality theorem for disjoints, inference version. (Contributed by Peter Mazsa, 22-Sep-2021.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ( Disj 𝐴 ↔ Disj 𝐵) | ||
Theorem | disjeqd 35851 | Equality theorem for disjoints, deduction version. (Contributed by Peter Mazsa, 22-Sep-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ( Disj 𝐴 ↔ Disj 𝐵)) | ||
Theorem | disjdmqseqeq1 35852 | 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 35853 | Subclass theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝐴 ⊆ 𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴)) | ||
Theorem | eldisjssi 35854 | Subclass theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ( ElDisj 𝐵 → ElDisj 𝐴) | ||
Theorem | eldisjssd 35855 | Subclass theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 28-Sep-2021.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ( ElDisj 𝐵 → ElDisj 𝐴)) | ||
Theorem | eldisjeq 35856 | Equality theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵)) | ||
Theorem | eldisjeqi 35857 | Equality theorem for disjoint elementhood, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ( ElDisj 𝐴 ↔ ElDisj 𝐵) | ||
Theorem | eldisjeqd 35858 | Equality theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵)) | ||
Theorem | disjxrn 35859 | 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 35860 | 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 35861 | 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 35862 | Disjointness condition for restriction. (Contributed by Peter Mazsa, 27-Sep-2021.) |
⊢ ( Disj 𝑅 → Disj (𝑅 ↾ 𝐴)) | ||
Theorem | disjimin 35863 | 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 35864 | Disjointness condition for intersection with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.) |
⊢ ( Disj 𝑆 → Disj (𝑅 ∩ (𝑆 ↾ 𝐴))) | ||
Theorem | disjimxrnres 35865 | Disjointness condition for range Cartesian product with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.) |
⊢ ( Disj 𝑆 → Disj (𝑅 ⋉ (𝑆 ↾ 𝐴))) | ||
Theorem | disjALTV0 35866 | The null class is disjoint. (Contributed by Peter Mazsa, 27-Sep-2021.) |
⊢ Disj ∅ | ||
Theorem | disjALTVid 35867 | The class of identity relations is disjoint. (Contributed by Peter Mazsa, 20-Jun-2021.) |
⊢ Disj I | ||
Theorem | disjALTVidres 35868 | 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 35869 | The intersection with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.) |
⊢ Disj (𝑅 ∩ ( I ↾ 𝐴)) | ||
Theorem | disjALTVxrnidres 35870 | 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 | prtlem60 35871 | Lemma for prter3 35900. (Contributed by Rodolfo Medina, 9-Oct-2010.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜓 → (𝜃 → 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) | ||
Theorem | bicomdd 35872 | 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 35873 | 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 35874 | 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 35875 | Lemma for prter3 35900: 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 35876 | Reverse of ibd 270. (Contributed by Rodolfo Medina, 30-Sep-2010.) |
⊢ (𝜑 → (𝜒 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (𝜒 → 𝜓)) | ||
Theorem | prtlem100 35877 | Lemma for prter3 35900. (Contributed by Rodolfo Medina, 19-Oct-2010.) |
⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵 ∈ 𝑥 ∧ 𝜑)) | ||
Theorem | prtlem5 35878* | Lemma for prter1 35897, prter2 35899, prter3 35900 and prtex 35898. (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 35879 | Lemma for prter2 35899. (Contributed by Rodolfo Medina, 17-Oct-2010.) |
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴})) | ||
Theorem | brabsb2 35880* | A closed form of brabsb 5410. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)) | ||
Theorem | eqbrrdv2 35881* | Other version of eqbrrdiv 5661. (Contributed by Rodolfo Medina, 30-Sep-2010.) |
⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) ⇒ ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) | ||
Theorem | prtlem9 35882* | Lemma for prter3 35900. (Contributed by Rodolfo Medina, 25-Sep-2010.) |
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ ) | ||
Theorem | prtlem10 35883* | Lemma for prter3 35900. (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 35884 | Lemma for prter2 35899. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ )))) | ||
Theorem | prtlem12 35885* | Lemma for prtex 35898 and prter3 35900. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ ( ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} → Rel ∼ ) | ||
Theorem | prtlem13 35886* | Lemma for prter1 35897, prter2 35899, prter3 35900 and prtex 35898. (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 35887* | Lemma for prtex 35898, prter2 35899 and prter3 35900. (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 35888* | Lemma for prter2 35899 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 35889 | Extend the definition of a wff to include the partition predicate. |
wff Prt 𝐴 | ||
Definition | df-prt 35890* | Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) | ||
Theorem | erprt 35891 | The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ ( ∼ Er 𝑋 → Prt (𝐴 / ∼ )) | ||
Theorem | prtlem14 35892* | Lemma for prter1 35897, prter2 35899 and prtex 35898. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑤 ∈ 𝑥 ∧ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦))) | ||
Theorem | prtlem15 35893* | Lemma for prter1 35897 and prtex 35898. (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 35894* | Lemma for prter2 35899. (Contributed by Rodolfo Medina, 15-Oct-2010.) |
⊢ (Prt 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥))) | ||
Theorem | prtlem18 35895* | Lemma for prter2 35899. (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 35896* | Lemma for prter2 35899. (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 35897* | 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) | ||
Theorem | prtex 35898* | The equivalence relation generated by a partition is a set if and only if the partition itself is a set. (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 ∈ V)) | ||
Theorem | prter2 35899* | The quotient set of the equivalence relation generated by a partition equals the partition itself. (Contributed by Rodolfo Medina, 17-Oct-2010.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ (Prt 𝐴 → (∪ 𝐴 / ∼ ) = (𝐴 ∖ {∅})) | ||
Theorem | prter3 35900* | For every partition there exists a unique equivalence relation whose quotient set equals the partition. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) |
⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} ⇒ ⊢ ((𝑆 Er ∪ 𝐴 ∧ (∪ 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → ∼ = 𝑆) |
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