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Theorem List for Metamath Proof Explorer - 28501-28600   *Has distinct variable group(s)
TypeLabelDescription
Statement

20.3.2.5  Existential uniqueness - misc additions

Theorem2reuswap2 28501* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.)
(∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))

Theoremreuxfr3d 28502* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfr2d 4716. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃*𝑦𝐶 𝑥 = 𝐴)       (𝜑 → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐶 𝜓))

Theoremreuxfr4d 28503* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfrd 4718. (Contributed by Thierry Arnoux, 7-Apr-2017.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))

Theoremrexunirn 28504* Restricted existential quantification over the union of the range of a function. Cf. rexrn 6152 and eluni2 4274. (Contributed by Thierry Arnoux, 19-Sep-2017.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑥𝐴𝐵𝑉)       (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦 ran 𝐹𝜑)

20.3.2.6  Restricted "at most one" - misc additions

TheoremrmoxfrdOLD 28505* Transfer "at most one" restricted quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Thierry Arnoux, 7-Apr-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥(𝑥𝐵𝜓) ↔ ∃*𝑦(𝑦𝐶𝜒)))

Theoremrmoxfrd 28506* Transfer "at most one" restricted quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐵 𝜓 ↔ ∃*𝑦𝐶 𝜒))

Theoremssrmo 28507 "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 → (∃*𝑥𝐵 𝜑 → ∃*𝑥𝐴 𝜑))

Theoremrmo3f 28508* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑       (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Theoremrmo4fOLD 28509* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝐴    &   𝑦𝐴    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))

Theoremrmo4f 28510* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
𝑥𝐴    &   𝑦𝐴    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑𝜓) → 𝑥 = 𝑦))

20.3.3  General Set Theory

20.3.3.1  Class abstractions (a.k.a. class builders)

Theoremrabrab 28511 Abstract builder restricted to another restricted abstract builder. (Contributed by Thierry Arnoux, 30-Aug-2017.)
{𝑥 ∈ {𝑥𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜑𝜓)}

Theoremrabtru 28512 Abtract builder using the constant wff (Contributed by Thierry Arnoux, 4-May-2020.)
𝑥𝐴       {𝑥𝐴 ∣ ⊤} = 𝐴

Theoremrabid2f 28513 An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
𝑥𝐴       (𝐴 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)

TheoremrabexgfGS 28514 Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017.)
𝑥𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Theoremrabsnel 28515* Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018.)
𝐵 ∈ V       ({𝑥𝐴𝜑} = {𝐵} → 𝐵𝐴)

Theoremforesf1o 28516* From a surjective function, *choose* a subset of the domain, such that the restricted function is bijective. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝐴𝑉𝐹:𝐴onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)

Theoremrabfodom 28517* Domination relation for restricted abstract class builders, based on a surjective function. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝜑𝑥𝐴𝑦 = (𝐹𝑥)) → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴onto𝐵)       (𝜑 → {𝑦𝐵𝜒} ≼ {𝑥𝐴𝜓})

20.3.3.2  Image Sets

Theoremabrexdomjm 28518* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑦𝐴 → ∃*𝑥𝜑)       (𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝜑} ≼ 𝐴)

Theoremabrexdom2jm 28519* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ≼ 𝐴)

Theoremabrexexd 28520* Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.)
𝑥𝐴    &   (𝜑𝐴 ∈ V)       (𝜑 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)

Theoremelabreximd 28521* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑥𝜑    &   𝑥𝜒    &   (𝐴 = 𝐵 → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐶) → 𝜓)       ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)

Theoremelabreximdv 28522* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
(𝐴 = 𝐵 → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐶) → 𝜓)       ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)

Theoremabrexss 28523* A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.)
𝑥𝐶       (∀𝑥𝐴 𝐵𝐶 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶)

20.3.3.3  Set relations and operations - misc additions

Theoremeqri 28524 Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)
𝑥𝐴    &   𝑥𝐵    &   (𝑥𝐴𝑥𝐵)       𝐴 = 𝐵

Theoremrabss3d 28525* Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)
((𝜑 ∧ (𝑥𝐴𝜓)) → 𝑥𝐵)       (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜓})

Theoreminin 28526 Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.)
(𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)

Theoreminindif 28527 See inundif 3901. (Contributed by Thierry Arnoux, 13-Sep-2017.)
((𝐴𝐶) ∩ (𝐴𝐶)) = ∅

Theoremdifeq 28528 Rewriting an equation with class difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)
((𝐴𝐵) = 𝐶 ↔ ((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)))

Theoremindifundif 28529 A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.)
(((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵)) = (𝐴 ∖ (𝐵𝐶))

Theoremelpwincl1 28530 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 28531 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 28532* 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

Theoremelpreq 28533 Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)
(𝜑𝑋 ∈ {𝐴, 𝐵})    &   (𝜑𝑌 ∈ {𝐴, 𝐵})    &   (𝜑 → (𝑋 = 𝐴𝑌 = 𝐴))       (𝜑𝑋 = 𝑌)

20.3.3.5  Conditional operator - misc additions

Theoremifeqeqx 28534* An equality theorem tailored for ballotlemsf1o 29710. (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 28535 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 28536 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 28537 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

20.3.3.6  Set union

Theoremuniinn0 28538* Sufficient and necessary condition for a union to intersect with a given set. (Contributed by Thierry Arnoux, 27-Jan-2020.)
(( 𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)

Theoremuniin1 28539* 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 28540* 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 28541 Express a class difference using unions and class complements. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝐴𝐶 → (𝐴𝐵) = (𝐶 ∖ ((𝐶𝐴) ∪ 𝐵)))

Theorempwuniss 28542 Condition for a class union to be a subset. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)

Theoremelpwunicl 28543 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 28544* 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 28545 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 28546 Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4407 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) (Revised by Thierry Arnoux, 2-May-2020.)
𝑥𝐶        𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)

Theoremiunxsngf 28547* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry Arnoux, 2-May-2020.)
𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)

Theoremssiun3 28548* Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)
(∀𝑦𝐶𝑥𝐴 𝑦𝐵𝐶 𝑥𝐴 𝐵)

Theoremssiun2sf 28549 Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)
𝑥𝐴    &   𝑥𝐶    &   𝑥𝐷    &   (𝑥 = 𝐶𝐵 = 𝐷)       (𝐶𝐴𝐷 𝑥𝐴 𝐵)

Theoremiuninc 28550* 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 28551* 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 28552* The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 28551. (Contributed by Thierry Arnoux, 4-Sep-2016.)
𝑂 ∈ V    &   𝐴 ⊆ 𝒫 𝑂       (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))

Theoremiunrdx 28553* Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)
(𝜑𝐹:𝐴onto𝐶)    &   ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)       (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)

Theoremiunpreima 28554* Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.)
(Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))

Theoremdisjnf 28555* 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 Masza.) (Contributed by Thierry Arnoux, 26-Jul-2018.)
(Disj 𝑥𝐴 𝐵 ↔ (𝐵 = ∅ ∨ ∃*𝑥 𝑥𝐴))

Theoremcbvdisjf 28556* 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 28557 A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))

Theoremdisjeq1f 28558 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))

Theoremdisjdifprg 28559* A trivial partition into a subset and its complement. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝐴𝑉𝐵𝑊) → Disj 𝑥 ∈ {(𝐵𝐴), 𝐴}𝑥)

Theoremdisjdifprg2 28560* 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 28561* 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 28562* 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 28563* 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 28564* 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 28565* 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 28566* 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 28567* 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 28568* A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)
((Fun 𝐹Disj 𝑥𝐴 𝐵) → Disj 𝑥𝐴 (𝐹𝐵))

Theoremdisjrnmpt 28569* 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 28570 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 28571 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 28572* 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 28573* Rewrite a countable union as a disjoint union. Cf. iundisj 22999. (Contributed by Thierry Arnoux, 31-Dec-2016.)
𝑘𝐴    &   𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)        𝑛 ∈ ℕ 𝐴 = 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)

Theoremiundisj2f 28574* A disjoint union is disjoint. Cf. iundisj2 23000. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑘𝐴    &   𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)       Disj 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)

Theoremdisjrdx 28575* 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 28576* 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 28577* A variant of disjex 28576, applicable for more generic families. (Contributed by Thierry Arnoux, 4-Oct-2016.)
(𝑥 = 𝑦𝐴 = 𝐵)       ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝑥 = 𝑦) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))

Theoremdisjunsn 28578* Append an element to a disjoint collection. Similar to ralunsn 4258, gsumunsn 18087, etc. (Contributed by Thierry Arnoux, 28-Mar-2018.)
(𝑥 = 𝑀𝐵 = 𝐶)       ((𝑀𝑉 ∧ ¬ 𝑀𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {𝑀})𝐵 ↔ (Disj 𝑥𝐴 𝐵 ∧ ( 𝑥𝐴 𝐵𝐶) = ∅)))

Theoremdisjun0 28579* Adding the empty element preserves disjointness. (Contributed by Thierry Arnoux, 30-May-2020.)
(Disj 𝑥𝐴 𝑥Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)

Theoremdisjiunel 28580* 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 28581* 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

Theoremxpdisjres 28582 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 28583 Ordered pair elementhood outside of the diagonal. (Contributed by Thierry Arnoux, 1-Jan-2020.)
(Rel 𝐴 → (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ I ) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴𝑋𝑌)))

Theoremdifres 28584 Case when class difference in unaffected by restriction. (Contributed by Thierry Arnoux, 1-Jan-2020.)
(𝐴 ⊆ (𝐵 × V) → (𝐴 ∖ (𝐶𝐵)) = (𝐴𝐶))

Theoremimadifxp 28585 Image of the difference with a Cartesian product. (Contributed by Thierry Arnoux, 13-Dec-2017.)
(𝐶𝐴 → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))

Theoremrelfi 28586 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)))

Theoremfcoinver 28587 Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 28588. (Contributed by Thierry Arnoux, 3-Jan-2020.)
(𝐹 Fn 𝑋 → (𝐹𝐹) Er 𝑋)

Theoremfcoinvbr 28588 Binary relation for the equivalence relation from fcoinver 28587. (Contributed by Thierry Arnoux, 3-Jan-2020.)
= (𝐹𝐹)       ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))

Theorembrabgaf 28589* 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 28590 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 28591* 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 28592* Domain of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑})

Theoremopabrn 28593* Range of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ran 𝑅 = {𝑦 ∣ ∃𝑥𝜑})

Theoremssrelf 28594* 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 28595* A version of eqrelrdv2 5035 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 28596 Biconditional for equivalent elements. (Contributed by Thierry Arnoux, 6-Jan-2020.)
((𝑅 Er 𝑋𝐴𝑅𝐵) → (𝐴𝑅𝐶𝐵𝑅𝐶))

Theoremiunsnima 28597 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)