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Theorem List for Metamath Proof Explorer - 30201-30300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
20.3.3.3  Set relations and operations - misc additions
 
Theoremelunsn 30201 Elementhood to a union with a singleton. (Contributed by Thierry Arnoux, 14-Dec-2023.)
(𝐴𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴𝐵𝐴 = 𝐶)))
 
Theoremnelun 30202 Negated membership for a union. (Contributed by Thierry Arnoux, 13-Dec-2023.)
(𝐴 = (𝐵𝐶) → (¬ 𝑋𝐴 ↔ (¬ 𝑋𝐵 ∧ ¬ 𝑋𝐶)))
 
Theoremdisjdifr 30203 A class and its relative complement are disjoint. (Contributed by Thierry Arnoux, 29-Nov-2023.)
((𝐵𝐴) ∩ 𝐴) = ∅
 
Theoremrabss3d 30204* Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)
((𝜑 ∧ (𝑥𝐴𝜓)) → 𝑥𝐵)       (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜓})
 
Theoreminin 30205 Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.)
(𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
 
Theoreminindif 30206 See inundif 4425. (Contributed by Thierry Arnoux, 13-Sep-2017.)
((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
 
Theoremdifininv 30207 Condition for the intersections of two sets with a given set to be equal. (Contributed by Thierry Arnoux, 28-Dec-2021.)
((((𝐴𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶𝐴) ∩ 𝐵) = ∅) → (𝐴𝐵) = (𝐶𝐵))
 
Theoremdifeq 30208 Rewriting an equation with class difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)
((𝐴𝐵) = 𝐶 ↔ ((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)))
 
Theoremeqdif 30209 If both set differences of two sets are empty, those sets are equal. (Contributed by Thierry Arnoux, 16-Nov-2023.)
(((𝐴𝐵) = ∅ ∧ (𝐵𝐴) = ∅) → 𝐴 = 𝐵)
 
Theoremdifxp1ss 30210 Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
((𝐴𝐶) × 𝐵) ⊆ (𝐴 × 𝐵)
 
Theoremdifxp2ss 30211 Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
(𝐴 × (𝐵𝐶)) ⊆ (𝐴 × 𝐵)
 
Theoremundifr 30212 Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023.)
(𝐴𝐵 ↔ ((𝐵𝐴) ∪ 𝐴) = 𝐵)
 
Theoremindifundif 30213 A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.)
(((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵)) = (𝐴 ∖ (𝐵𝐶))
 
Theoremelpwincl1 30214 Closure of intersection with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐴 ∈ 𝒫 𝐶)       (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)
 
Theoremelpwdifcl 30215 Closure of class difference with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐴 ∈ 𝒫 𝐶)       (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)
 
Theoremelpwiuncl 30216* Closure of indexed union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 27-May-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ 𝒫 𝐶)       (𝜑 𝑘𝐴 𝐵 ∈ 𝒫 𝐶)
 
20.3.3.4  Unordered pairs
 
Theoremeqsnd 30217* Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023.)
((𝜑𝑥𝐴) → 𝑥 = 𝐵)    &   (𝜑𝐵𝐴)       (𝜑𝐴 = {𝐵})
 
Theoremelpreq 30218 Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)
(𝜑𝑋 ∈ {𝐴, 𝐵})    &   (𝜑𝑌 ∈ {𝐴, 𝐵})    &   (𝜑 → (𝑋 = 𝐴𝑌 = 𝐴))       (𝜑𝑋 = 𝑌)
 
Theoremnelpr 30219 A set 𝐴 not in a pair is neither element of the pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝐴𝑉 → (¬ 𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴𝐵𝐴𝐶)))
 
Theoreminpr0 30220 Rewrite an empty intersection with a pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ (¬ 𝐵𝐴 ∧ ¬ 𝐶𝐴))
 
Theoremneldifpr1 30221 The first element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵})
 
Theoremneldifpr2 30222 The second element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})
 
Theoremunidifsnel 30223 The other element of a pair is an element of the pair. (Contributed by Thierry Arnoux, 26-Aug-2017.)
((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ 𝑃)
 
Theoremunidifsnne 30224 The other element of a pair is not the known element. (Contributed by Thierry Arnoux, 26-Aug-2017.)
((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
 
20.3.3.5  Conditional operator - misc additions
 
Theoremifeqeqx 30225* An equality theorem tailored for ballotlemsf1o 31671. (Contributed by Thierry Arnoux, 14-Apr-2017.)
(𝑥 = 𝑋𝐴 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝑎)    &   (𝑥 = 𝑋 → (𝜒𝜃))    &   (𝑥 = 𝑌 → (𝜒𝜓))    &   (𝜑𝑎 = 𝐶)    &   ((𝜑𝜓) → 𝜃)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑊)       ((𝜑𝑥 = if(𝜓, 𝑋, 𝑌)) → 𝑎 = if(𝜒, 𝐴, 𝐵))
 
Theoremelimifd 30226 Elimination of a conditional operator contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐴 → (𝜒𝜃)))    &   (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐵 → (𝜒𝜏)))       (𝜑 → (𝜒 ↔ ((𝜓𝜃) ∨ (¬ 𝜓𝜏))))
 
Theoremelim2if 30227 Elimination of two conditional operators contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒𝜃))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒𝜏))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒𝜂))       (𝜒 ↔ ((𝜑𝜃) ∨ (¬ 𝜑 ∧ ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))))
 
Theoremelim2ifim 30228 Elimination of two conditional operators for an implication. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒𝜃))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒𝜏))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒𝜂))    &   (𝜑𝜃)    &   ((¬ 𝜑𝜓) → 𝜏)    &   ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜂)       𝜒
 
Theoremifeq3da 30229 Given an expression 𝐶 containing if(𝜓, 𝐸, 𝐹), substitute (hypotheses .1 and .2) and evaluate (hypotheses .3 and .4) it for both cases at the same time. (Contributed by Thierry Arnoux, 13-Dec-2021.)
(if(𝜓, 𝐸, 𝐹) = 𝐸𝐶 = 𝐺)    &   (if(𝜓, 𝐸, 𝐹) = 𝐹𝐶 = 𝐻)    &   (𝜑𝐺 = 𝐴)    &   (𝜑𝐻 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)
 
20.3.3.6  Set union
 
Theoremuniinn0 30230* Sufficient and necessary condition for a union to intersect with a given set. (Contributed by Thierry Arnoux, 27-Jan-2020.)
(( 𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
 
Theoremuniin1 30231* Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)
𝑥𝐴 (𝑥𝐵) = ( 𝐴𝐵)
 
Theoremuniin2 30232* Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)
𝑥𝐵 (𝐴𝑥) = (𝐴 𝐵)
 
Theoremdifuncomp 30233 Express a class difference using unions and class complements. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝐴𝐶 → (𝐴𝐵) = (𝐶 ∖ ((𝐶𝐴) ∪ 𝐵)))
 
Theorempwuniss 30234 Condition for a class union to be a subset. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
 
Theoremelpwunicl 30235 Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝜑𝐵𝑉)    &   (𝜑𝐴 ∈ 𝒫 𝒫 𝐵)       (𝜑 𝐴 ∈ 𝒫 𝐵)
 
20.3.3.7  Indexed union - misc additions
 
Theoremcbviunf 30236* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 
Theoremiuneq12daf 30237 Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 13-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → 𝐶 = 𝐷)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
 
Theoremiunin1f 30238 Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4974 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) (Revised by Thierry Arnoux, 2-May-2020.)
𝑥𝐶        𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)
 
Theoremssiun3 30239* Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)
(∀𝑦𝐶𝑥𝐴 𝑦𝐵𝐶 𝑥𝐴 𝐵)
 
Theoremssiun2sf 30240 Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)
𝑥𝐴    &   𝑥𝐶    &   𝑥𝐷    &   (𝑥 = 𝐶𝐵 = 𝐷)       (𝐶𝐴𝐷 𝑥𝐴 𝐵)
 
Theoremiuninc 30241* The union of an increasing collection of sets is its last element. (Contributed by Thierry Arnoux, 22-Jan-2017.)
(𝜑𝐹 Fn ℕ)    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))       ((𝜑𝑖 ∈ ℕ) → 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖))
 
Theoremiundifdifd 30242* The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.)
(𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥))))
 
Theoremiundifdif 30243* The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 30242. (Contributed by Thierry Arnoux, 4-Sep-2016.)
𝑂 ∈ V    &   𝐴 ⊆ 𝒫 𝑂       (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))
 
Theoremiunrdx 30244* Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)
(𝜑𝐹:𝐴onto𝐶)    &   ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)       (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)
 
Theoremiunpreima 30245* Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.)
(Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))
 
Theoremiunrnmptss 30246* A subset relation for an indexed union over the range of function expressed as a mapping. (Contributed by Thierry Arnoux, 27-Mar-2018.)
(𝑦 = 𝐵𝐶 = 𝐷)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 𝑦 ∈ ran (𝑥𝐴𝐵)𝐶 𝑥𝐴 𝐷)
 
Theoremiunxunsn 30247* Appending a set to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑥 = 𝑋𝐵 = 𝐶)       (𝑋𝑉 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = ( 𝑥𝐴 𝐵𝐶))
 
Theoremiunxunpr 30248* Appending two sets to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑥 = 𝑋𝐵 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝐷)       ((𝑋𝑉𝑌𝑊) → 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = ( 𝑥𝐴 𝐵 ∪ (𝐶𝐷)))
 
20.3.3.8  Disjointness - misc additions
 
Theoremdisjnf 30249* In case 𝑥 is not free in 𝐵, disjointness is not so interesting since it reduces to cases where 𝐴 is a singleton. (Google Groups discussion with Peter Mazsa.) (Contributed by Thierry Arnoux, 26-Jul-2018.)
(Disj 𝑥𝐴 𝐵 ↔ (𝐵 = ∅ ∨ ∃*𝑥 𝑥𝐴))
 
Theoremcbvdisjf 30250* Change bound variables in a disjoint collection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
𝑥𝐴    &   𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
 
Theoremdisjss1f 30251 A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
 
Theoremdisjeq1f 30252 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
 
Theoremdisjxun0 30253* Simplify a disjoint union. (Contributed by Thierry Arnoux, 27-Nov-2023.)
((𝜑𝑥𝐵) → 𝐶 = ∅)       (𝜑 → (Disj 𝑥 ∈ (𝐴𝐵)𝐶Disj 𝑥𝐴 𝐶))
 
Theoremdisjdifprg 30254* A trivial partition into a subset and its complement. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝐴𝑉𝐵𝑊) → Disj 𝑥 ∈ {(𝐵𝐴), 𝐴}𝑥)
 
Theoremdisjdifprg2 30255* A trivial partition of a set into its difference and intersection with another set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝐴𝑉Disj 𝑥 ∈ {(𝐴𝐵), (𝐴𝐵)}𝑥)
 
Theoremdisji2f 30256* Property of a disjoint collection: if 𝐵(𝑥) = 𝐶 and 𝐵(𝑌) = 𝐷, and 𝑥𝑌, then 𝐵 and 𝐶 are disjoint. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑥𝐶    &   (𝑥 = 𝑌𝐵 = 𝐶)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ 𝑥𝑌) → (𝐵𝐶) = ∅)
 
Theoremdisjif 30257* Property of a disjoint collection: if 𝐵(𝑥) and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑥 = 𝑌. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑥𝐶    &   (𝑥 = 𝑌𝐵 = 𝐶)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)
 
Theoremdisjorf 30258* Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑖𝐴    &   𝑗𝐴    &   (𝑖 = 𝑗𝐵 = 𝐶)       (Disj 𝑖𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅))
 
Theoremdisjorsf 30259* Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝐴       (Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
 
Theoremdisjif2 30260* Property of a disjoint collection: if 𝐵(𝑥) and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑥 = 𝑌. (Contributed by Thierry Arnoux, 6-Apr-2017.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝑌𝐵 = 𝐶)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑌𝐴) ∧ (𝑍𝐵𝑍𝐶)) → 𝑥 = 𝑌)
 
Theoremdisjabrex 30261* Rewriting a disjoint collection into a partition of its image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
(Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
 
Theoremdisjabrexf 30262* Rewriting a disjoint collection into a partition of its image set. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Revised by Thierry Arnoux, 9-Mar-2017.)
𝑥𝐴       (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
 
Theoremdisjpreima 30263* A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)
((Fun 𝐹Disj 𝑥𝐴 𝐵) → Disj 𝑥𝐴 (𝐹𝐵))
 
Theoremdisjrnmpt 30264* Rewriting a disjoint collection using the range of a mapping. (Contributed by Thierry Arnoux, 27-May-2020.)
(Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ ran (𝑥𝐴𝐵)𝑦)
 
Theoremdisjin 30265 If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐶𝐴))
 
Theoremdisjin2 30266 If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(Disj 𝑥𝐵 𝐶Disj 𝑥𝐵 (𝐴𝐶))
 
Theoremdisjxpin 30267* Derive a disjunction over a Cartesian product from the disjunctions over its first and second elements. (Contributed by Thierry Arnoux, 9-Mar-2018.)
(𝑥 = (1st𝑝) → 𝐶 = 𝐸)    &   (𝑦 = (2nd𝑝) → 𝐷 = 𝐹)    &   (𝜑Disj 𝑥𝐴 𝐶)    &   (𝜑Disj 𝑦𝐵 𝐷)       (𝜑Disj 𝑝 ∈ (𝐴 × 𝐵)(𝐸𝐹))
 
Theoremiundisjf 30268* Rewrite a countable union as a disjoint union. Cf. iundisj 24078. (Contributed by Thierry Arnoux, 31-Dec-2016.)
𝑘𝐴    &   𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)        𝑛 ∈ ℕ 𝐴 = 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
 
Theoremiundisj2f 30269* A disjoint union is disjoint. Cf. iundisj2 24079. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑘𝐴    &   𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)       Disj 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
 
Theoremdisjrdx 30270* Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017.)
(𝜑𝐹:𝐴1-1-onto𝐶)    &   ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)       (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑦𝐶 𝐷))
 
Theoremdisjex 30271* Two ways to say that two classes are disjoint (or equal). (Contributed by Thierry Arnoux, 4-Oct-2016.)
((∃𝑧(𝑧𝐴𝑧𝐵) → 𝐴 = 𝐵) ↔ (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
 
Theoremdisjexc 30272* A variant of disjex 30271, applicable for more generic families. (Contributed by Thierry Arnoux, 4-Oct-2016.)
(𝑥 = 𝑦𝐴 = 𝐵)       ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝑥 = 𝑦) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
 
Theoremdisjunsn 30273* Append an element to a disjoint collection. Similar to ralunsn 4818, gsumunsn 19011, etc. (Contributed by Thierry Arnoux, 28-Mar-2018.)
(𝑥 = 𝑀𝐵 = 𝐶)       ((𝑀𝑉 ∧ ¬ 𝑀𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {𝑀})𝐵 ↔ (Disj 𝑥𝐴 𝐵 ∧ ( 𝑥𝐴 𝐵𝐶) = ∅)))
 
Theoremdisjun0 30274* Adding the empty element preserves disjointness. (Contributed by Thierry Arnoux, 30-May-2020.)
(Disj 𝑥𝐴 𝑥Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)
 
Theoremdisjiunel 30275* A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.)
(𝜑Disj 𝑥𝐴 𝐵)    &   (𝑥 = 𝑌𝐵 = 𝐷)    &   (𝜑𝐸𝐴)    &   (𝜑𝑌 ∈ (𝐴𝐸))       (𝜑 → ( 𝑥𝐸 𝐵𝐷) = ∅)
 
Theoremdisjuniel 30276* A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.)
(𝜑Disj 𝑥𝐴 𝑥)    &   (𝜑𝐵𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))       (𝜑 → ( 𝐵𝐶) = ∅)
 
20.3.4  Relations and Functions
 
20.3.4.1  Relations - misc additions
 
Theoremxpdisjres 30277 Restriction of a constant function (or other Cartesian product) outside of its domain. (Contributed by Thierry Arnoux, 25-Jan-2017.)
((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) ↾ 𝐶) = ∅)
 
Theoremopeldifid 30278 Ordered pair elementhood outside of the diagonal. (Contributed by Thierry Arnoux, 1-Jan-2020.)
(Rel 𝐴 → (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ I ) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴𝑋𝑌)))
 
Theoremdifres 30279 Case when class difference in unaffected by restriction. (Contributed by Thierry Arnoux, 1-Jan-2020.)
(𝐴 ⊆ (𝐵 × V) → (𝐴 ∖ (𝐶𝐵)) = (𝐴𝐶))
 
Theoremimadifxp 30280 Image of the difference with a Cartesian product. (Contributed by Thierry Arnoux, 13-Dec-2017.)
(𝐶𝐴 → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
 
Theoremrelfi 30281 A relation (set) is finite if and only if both its domain and range are finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)
(Rel 𝐴 → (𝐴 ∈ Fin ↔ (dom 𝐴 ∈ Fin ∧ ran 𝐴 ∈ Fin)))
 
Theoremreldisjun 30282 Split a relation into two disjoint parts based on its domain. (Contributed by Thierry Arnoux, 9-Oct-2023.)
((Rel 𝑅 ∧ dom 𝑅 = (𝐴𝐵) ∧ (𝐴𝐵) = ∅) → 𝑅 = ((𝑅𝐴) ∪ (𝑅𝐵)))
 
Theorem0res 30283 Restriction of the empty function. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(∅ ↾ 𝐴) = ∅
 
Theoremfunresdm1 30284 Restriction of a disjoint union to the domain of the first term. (Contributed by Thierry Arnoux, 9-Dec-2021.)
((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴𝐵) ↾ dom 𝐴) = 𝐴)
 
Theoremfnunres1 30285 Restriction of a disjoint union to the domain of the first function. (Contributed by Thierry Arnoux, 9-Dec-2021.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)
 
Theoremfcoinver 30286 Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 30287. (Contributed by Thierry Arnoux, 3-Jan-2020.)
(𝐹 Fn 𝑋 → (𝐹𝐹) Er 𝑋)
 
Theoremfcoinvbr 30287 Binary relation for the equivalence relation from fcoinver 30286. (Contributed by Thierry Arnoux, 3-Jan-2020.)
= (𝐹𝐹)       ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))
 
Theorembrabgaf 30288* The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) (Revised by Thierry Arnoux, 17-May-2020.)
𝑥𝜓    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝜓))
 
Theorembrelg 30289 Two things in a binary relation belong to the relation's domain. (Contributed by Thierry Arnoux, 29-Aug-2017.)
((𝑅 ⊆ (𝐶 × 𝐷) ∧ 𝐴𝑅𝐵) → (𝐴𝐶𝐵𝐷))
 
Theorembr8d 30290* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by Thierry Arnoux, 21-Mar-2019.)
(𝑎 = 𝐴 → (𝜓𝜒))    &   (𝑏 = 𝐵 → (𝜒𝜃))    &   (𝑐 = 𝐶 → (𝜃𝜏))    &   (𝑑 = 𝐷 → (𝜏𝜂))    &   (𝑒 = 𝐸 → (𝜂𝜁))    &   (𝑓 = 𝐹 → (𝜁𝜎))    &   (𝑔 = 𝐺 → (𝜎𝜌))    &   ( = 𝐻 → (𝜌𝜇))    &   (𝜑𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃𝑒𝑃𝑓𝑃𝑔𝑃𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ⟩⟩ ∧ 𝜓)})    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐺𝑃)    &   (𝜑𝐻𝑃)       (𝜑 → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜇))
 
Theoremopabdm 30291* Domain of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑})
 
Theoremopabrn 30292* Range of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ran 𝑅 = {𝑦 ∣ ∃𝑥𝜑})
 
Theoremopabssi 30293* Sufficient condition for a collection of ordered pairs to be a subclass of a relation. (Contributed by Peter Mazsa, 21-Oct-2019.) (Revised by Thierry Arnoux, 18-Feb-2022.)
(𝜑 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)       {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ 𝐴
 
Theoremopabid2ss 30294* One direction of opabid2 5694 which holds without a Rel 𝐴 requirement. (Contributed by Thierry Arnoux, 18-Feb-2022.)
{⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} ⊆ 𝐴
 
Theoremssrelf 30295* A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Thierry Arnoux, 6-Nov-2017.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝐴    &   𝑦𝐴    &   𝑥𝐵    &   𝑦𝐵       (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
 
Theoremeqrelrd2 30296* A version of eqrelrdv2 5662 with explicit non-free declarations. (Contributed by Thierry Arnoux, 28-Aug-2017.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝐴    &   𝑦𝐴    &   𝑥𝐵    &   𝑦𝐵    &   (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
 
Theoremerbr3b 30297 Biconditional for equivalent elements. (Contributed by Thierry Arnoux, 6-Jan-2020.)
((𝑅 Er 𝑋𝐴𝑅𝐵) → (𝐴𝑅𝐶𝐵𝑅𝐶))
 
Theoremiunsnima 30298 Image of a singleton by an indexed union involving that singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)       ((𝜑𝑥𝐴) → ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑥}) = 𝐵)
 
20.3.4.2  Functions - misc additions
 
Theoremac6sf2 30299* Alternate version of ac6 9891 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.) (Revised by Thierry Arnoux, 17-May-2020.)
𝑦𝐵    &   𝑦𝜓    &   𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
 
Theoremfnresin 30300 Restriction of a function with a subclass of its domain. (Contributed by Thierry Arnoux, 10-Oct-2017.)
(𝐹 Fn 𝐴 → (𝐹𝐵) Fn (𝐴𝐵))
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