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Theorem List for Metamath Proof Explorer - 28701-28800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrexcom4f 28701* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
𝑦𝐴       (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)

Theorem19.9d2rf 28702 A deduction version of one direction of 19.9 2060 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜓)

Theorem19.9d2r 28703* A deduction version of one direction of 19.9 2060 with two variables. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜓)

21.3.2.3  Substitution (without distinct variables) - misc additions

Theoremclelsb3f 28704 Substitution applied to an atomic wff (class version of elsb3 2422). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
𝑦𝐴       ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)

Theoremsbceqbidf 28705 Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))

Theoremsbcies 28706* A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝐴 = (𝐸𝑊)    &   (𝑎 = 𝐴 → (𝜑𝜓))       (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎]𝜓𝜑))

21.3.2.4  Existential "at most one" - misc additions

Theoremmoel 28707* "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.)
(∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦)

Theoremmo5f 28708* Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)
𝑖𝜑    &   𝑗𝜑       (∃*𝑥𝜑 ↔ ∀𝑖𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))

Theoremnmo 28709* Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)
𝑦𝜑       (¬ ∃*𝑥𝜑 ↔ ∀𝑦𝑥(𝜑𝑥𝑦))

Theoremmoimd 28710* "At most one" is preserved through implication (notice wff reversal). (Contributed by Thierry Arnoux, 25-Feb-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜒 → ∃*𝑥𝜓))

TheoremrmoeqALT 28711* Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.) Obsolete version of rmoeq 3372 as of 27-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ∃*𝑥𝐵 𝑥 = 𝐴)

21.3.2.5  Existential uniqueness - misc additions

Theorem2reuswap2 28712* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.)
(∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))

Theoremreuxfr3d 28713* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfr2d 4817. (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 28714* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfrd 4819. (Contributed by Thierry Arnoux, 7-Apr-2017.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))

Theoremrexunirn 28715* Restricted existential quantification over the union of the range of a function. Cf. rexrn 6269 and eluni2 4376. (Contributed by Thierry Arnoux, 19-Sep-2017.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑥𝐴𝐵𝑉)       (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦 ran 𝐹𝜑)

21.3.2.6  Restricted "at most one" - misc additions

TheoremrmoxfrdOLD 28716* 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 28717* 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 28718 "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 → (∃*𝑥𝐵 𝜑 → ∃*𝑥𝐴 𝜑))

Theoremrmo3f 28719* 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 28720* 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 28721* 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))

21.3.3  General Set Theory

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

Theoremrabrab 28722 Abstract builder restricted to another restricted abstract builder. (Contributed by Thierry Arnoux, 30-Aug-2017.)
{𝑥 ∈ {𝑥𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜑𝜓)}

Theoremrabtru 28723 Abtract builder using the constant wff (Contributed by Thierry Arnoux, 4-May-2020.)
𝑥𝐴       {𝑥𝐴 ∣ ⊤} = 𝐴

Theoremrabid2f 28724 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 28725 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 28726* 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 28727* 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 28728* 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})

21.3.3.2  Image Sets

Theoremabrexdomjm 28729* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑦𝐴 → ∃*𝑥𝜑)       (𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝜑} ≼ 𝐴)

Theoremabrexdom2jm 28730* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ≼ 𝐴)

Theoremabrexexd 28731* 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 28732* 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 28733* 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 28734* 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)

21.3.3.3  Set relations and operations - misc additions

Theoremeqri 28735 Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)
𝑥𝐴    &   𝑥𝐵    &   (𝑥𝐴𝑥𝐵)       𝐴 = 𝐵

Theoremrabss3d 28736* Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)
((𝜑 ∧ (𝑥𝐴𝜓)) → 𝑥𝐵)       (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜓})

Theoreminin 28737 Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.)
(𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)

Theoreminindif 28738 See inundif 3998. (Contributed by Thierry Arnoux, 13-Sep-2017.)
((𝐴𝐶) ∩ (𝐴𝐶)) = ∅

Theoremdifeq 28739 Rewriting an equation with class difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)
((𝐴𝐵) = 𝐶 ↔ ((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)))

Theoremindifundif 28740 A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.)
(((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵)) = (𝐴 ∖ (𝐵𝐶))

Theoremelpwincl1 28741 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 28742 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 28743* 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)

21.3.3.4  Unordered pairs

Theoremelpreq 28744 Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)
(𝜑𝑋 ∈ {𝐴, 𝐵})    &   (𝜑𝑌 ∈ {𝐴, 𝐵})    &   (𝜑 → (𝑋 = 𝐴𝑌 = 𝐴))       (𝜑𝑋 = 𝑌)

21.3.3.5  Conditional operator - misc additions

Theoremifeqeqx 28745* An equality theorem tailored for ballotlemsf1o 29902. (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 28746 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 28747 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 28748 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

21.3.3.6  Set union

Theoremuniinn0 28749* Sufficient and necessary condition for a union to intersect with a given set. (Contributed by Thierry Arnoux, 27-Jan-2020.)
(( 𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)

Theoremuniin1 28750* 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 28751* 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 28752 Express a class difference using unions and class complements. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝐴𝐶 → (𝐴𝐵) = (𝐶 ∖ ((𝐶𝐴) ∪ 𝐵)))

Theorempwuniss 28753 Condition for a class union to be a subset. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)

Theoremelpwunicl 28754 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)

21.3.3.7  Indexed union - misc additions

Theoremcbviunf 28755* 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 28756 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 28757 Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4509 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) (Revised by Thierry Arnoux, 2-May-2020.)
𝑥𝐶        𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)

Theoremiunxsngf 28758* 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 28759* Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)
(∀𝑦𝐶𝑥𝐴 𝑦𝐵𝐶 𝑥𝐴 𝐵)

Theoremssiun2sf 28760 Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)
𝑥𝐴    &   𝑥𝐶    &   𝑥𝐷    &   (𝑥 = 𝐶𝐵 = 𝐷)       (𝐶𝐴𝐷 𝑥𝐴 𝐵)

Theoremiuninc 28761* 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 28762* 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 28763* The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 28762. (Contributed by Thierry Arnoux, 4-Sep-2016.)
𝑂 ∈ V    &   𝐴 ⊆ 𝒫 𝑂       (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))

Theoremiunrdx 28764* Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)
(𝜑𝐹:𝐴onto𝐶)    &   ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)       (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)

Theoremiunpreima 28765* Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.)
(Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))

Theoremdisjnf 28766* 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 28767* 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 28768 A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))

Theoremdisjeq1f 28769 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))

Theoremdisjdifprg 28770* A trivial partition into a subset and its complement. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝐴𝑉𝐵𝑊) → Disj 𝑥 ∈ {(𝐵𝐴), 𝐴}𝑥)

Theoremdisjdifprg2 28771* 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 28772* 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 28773* 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 28774* 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 28775* 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 28776* 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 28777* 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 28778* 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 28779* A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)
((Fun 𝐹Disj 𝑥𝐴 𝐵) → Disj 𝑥𝐴 (𝐹𝐵))

Theoremdisjrnmpt 28780* 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 28781 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 28782 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 28783* 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 28784* Rewrite a countable union as a disjoint union. Cf. iundisj 23123. (Contributed by Thierry Arnoux, 31-Dec-2016.)
𝑘𝐴    &   𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)        𝑛 ∈ ℕ 𝐴 = 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)

Theoremiundisj2f 28785* A disjoint union is disjoint. Cf. iundisj2 23124. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑘𝐴    &   𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)       Disj 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)

Theoremdisjrdx 28786* 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 28787* 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 28788* A variant of disjex 28787, applicable for more generic families. (Contributed by Thierry Arnoux, 4-Oct-2016.)
(𝑥 = 𝑦𝐴 = 𝐵)       ((∃𝑧(𝑧𝐴𝑧𝐵) → 𝑥 = 𝑦) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))

Theoremdisjunsn 28789* Append an element to a disjoint collection. Similar to ralunsn 4360, gsumunsn 18182, etc. (Contributed by Thierry Arnoux, 28-Mar-2018.)
(𝑥 = 𝑀𝐵 = 𝐶)       ((𝑀𝑉 ∧ ¬ 𝑀𝐴) → (Disj 𝑥 ∈ (𝐴 ∪ {𝑀})𝐵 ↔ (Disj 𝑥𝐴 𝐵 ∧ ( 𝑥𝐴 𝐵𝐶) = ∅)))

Theoremdisjun0 28790* Adding the empty element preserves disjointness. (Contributed by Thierry Arnoux, 30-May-2020.)
(Disj 𝑥𝐴 𝑥Disj 𝑥 ∈ (𝐴 ∪ {∅})𝑥)

Theoremdisjiunel 28791* 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 28792* 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) = ∅)

21.3.4  Relations and Functions

Theoremxpdisjres 28793 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 28794 Ordered pair elementhood outside of the diagonal. (Contributed by Thierry Arnoux, 1-Jan-2020.)
(Rel 𝐴 → (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ I ) ↔ (⟨𝑋, 𝑌⟩ ∈ 𝐴𝑋𝑌)))

Theoremdifres 28795 Case when class difference in unaffected by restriction. (Contributed by Thierry Arnoux, 1-Jan-2020.)
(𝐴 ⊆ (𝐵 × V) → (𝐴 ∖ (𝐶𝐵)) = (𝐴𝐶))

Theoremimadifxp 28796 Image of the difference with a Cartesian product. (Contributed by Thierry Arnoux, 13-Dec-2017.)
(𝐶𝐴 → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))

Theoremrelfi 28797 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 28798 Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 28799. (Contributed by Thierry Arnoux, 3-Jan-2020.)
(𝐹 Fn 𝑋 → (𝐹𝐹) Er 𝑋)

Theoremfcoinvbr 28799 Binary relation for the equivalence relation from fcoinver 28798. (Contributed by Thierry Arnoux, 3-Jan-2020.)
= (𝐹𝐹)       ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))

Theorembrabgaf 28800* 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))

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