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Theorem List for Metamath Proof Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremweeq2 5101 Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
(𝐴 = 𝐵 → (𝑅 We 𝐴𝑅 We 𝐵))

Theoremwefr 5102 A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)
(𝑅 We 𝐴𝑅 Fr 𝐴)

Theoremweso 5103 A well-ordering is a strict ordering. (Contributed by NM, 16-Mar-1997.)
(𝑅 We 𝐴𝑅 Or 𝐴)

Theoremwecmpep 5104 The elements of an epsilon well-ordering are comparable. (Contributed by NM, 17-May-1994.)
(( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))

Theoremwetrep 5105 An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)
(( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))

Theoremwefrc 5106* A nonempty (possibly proper) subclass of a class well-ordered by E has a minimal element. Special case of Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by NM, 17-Feb-2004.)
(( E We 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)

Theoremwe0 5107 Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
𝑅 We ∅

Theoremwereu 5108* A subset of a well-ordered set has a unique minimal element. (Contributed by NM, 18-Mar-1997.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝑅 We 𝐴 ∧ (𝐵𝑉𝐵𝐴𝐵 ≠ ∅)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)

Theoremwereu2 5109* All nonempty (possibly proper) subclasses of 𝐴, which has a well-founded relation 𝑅, have 𝑅-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 24-Jun-2015.)
(((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)

2.3.10  Relations

Syntaxcxp 5110 Extend the definition of a class to include the Cartesian product.
class (𝐴 × 𝐵)

Syntaxccnv 5111 Extend the definition of a class to include the converse of a class.
class 𝐴

Syntaxcdm 5112 Extend the definition of a class to include the domain of a class.
class dom 𝐴

Syntaxcrn 5113 Extend the definition of a class to include the range of a class.
class ran 𝐴

Syntaxcres 5114 Extend the definition of a class to include the restriction of a class. (Read: The restriction of 𝐴 to 𝐵.)
class (𝐴𝐵)

Syntaxcima 5115 Extend the definition of a class to include the image of a class. (Read: The image of 𝐵 under 𝐴.)
class (𝐴𝐵)

Syntaxccom 5116 Extend the definition of a class to include the composition of two classes. (Read: The composition of 𝐴 and 𝐵.)
class (𝐴𝐵)

Syntaxwrel 5117 Extend the definition of a wff to include the relation predicate. (Read: 𝐴 is a relation.)
wff Rel 𝐴

Definitiondf-xp 5118* Define the Cartesian product of two classes. This is also sometimes called the "cross product" but that term also has other meanings; we intentionally choose a less ambiguous term. Definition 9.11 of [Quine] p. 64. For example, ({1, 5} × {2, 7}) = ({⟨1, 2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5, 7⟩}) (ex-xp 27277). Another example is that the set of rational numbers are defined in df-q 11786 using the Cartesian product (ℤ × ℕ); the left- and right-hand sides of the Cartesian product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994.)
(𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}

Definitiondf-rel 5119 Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5581 and dfrel3 5590. (Contributed by NM, 1-Aug-1994.)
(Rel 𝐴𝐴 ⊆ (V × V))

Definitiondf-cnv 5120* Define the converse of a class. Definition 9.12 of [Quine] p. 64. The converse of a binary relation swaps its arguments, i.e., if 𝐴 ∈ V and 𝐵 ∈ V then (𝐴𝑅𝐵𝐵𝑅𝐴), as proven in brcnv 5303 (see df-br 4652 and df-rel 5119 for more on relations). For example, {⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩} (ex-cnv 27278). We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. Many authors use the postfix superscript "to the minus one." "Converse" is Quine's terminology; some authors call it "inverse," especially when the argument is a function. (Contributed by NM, 4-Jul-1994.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}

Definitiondf-co 5121* Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, ((exp ∘ cos)‘0) = e (ex-co 27279) because (cos‘0) = 1 (see cos0 14874) and (exp‘1) = e (see df-e 14793). Note that Definition 7 of [Suppes] p. 63 reverses 𝐴 and 𝐵, uses / instead of , and calls the operation "relative product." (Contributed by NM, 4-Jul-1994.)
(𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}

Definitiondf-dm 5122* Define the domain of a class. Definition 3 of [Suppes] p. 59. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → dom 𝐹 = {2, 3} (ex-dm 27280). Another example is the domain of the complex arctangent, (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) (for proof see atandm 24597). Contrast with range (defined in df-rn 5123). For alternate definitions see dfdm2 5665, dfdm3 5308, and dfdm4 5314. The notation "dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.)
dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}

Definitiondf-rn 5123 Define the range of a class. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9} (ex-rn 27281). Contrast with domain (defined in df-dm 5122). For alternate definitions, see dfrn2 5309, dfrn3 5310, and dfrn4 5593. The notation "ran " is used by Enderton; other authors sometimes use script R or script W. (Contributed by NM, 1-Aug-1994.)
ran 𝐴 = dom 𝐴

Definitiondf-res 5124 Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, the expression (exp ↾ ℝ) (used in reeff1 14844) means "the exponential function e to the x, but the exponent x must be in the reals" (df-ef 14792 defines the exponential function, which normally allows the exponent to be a complex number). Another example is that (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} 𝐵 = {1, 2}) → (𝐹𝐵) = {⟨2, 6⟩} (ex-res 27282). (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵) = (𝐴 ∩ (𝐵 × V))

Definitiondf-ima 5125 Define the image of a class (as restricted by another class). Definition 6.6(2) of [TakeutiZaring] p. 24. For example, (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹𝐵) = {6} (ex-ima 27283). Contrast with restriction (df-res 5124) and range (df-rn 5123). For an alternate definition, see dfima2 5466. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵) = ran (𝐴𝐵)

Theoremxpeq1 5126 Equality theorem for Cartesian product. (Contributed by NM, 4-Jul-1994.)
(𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))

Theoremxpeq2 5127 Equality theorem for Cartesian product. (Contributed by NM, 5-Jul-1994.)
(𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))

Theoremelxpi 5128* Membership in a Cartesian product. Uses fewer axioms than elxp 5129. (Contributed by NM, 4-Jul-1994.)
(𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))

Theoremelxp 5129* Membership in a Cartesian product. (Contributed by NM, 4-Jul-1994.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))

Theoremelxp2 5130* Membership in a Cartesian product. (Contributed by NM, 23-Feb-2004.) (Proof shortened by JJ, 13-Aug-2021.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)

Theoremelxp2OLD 5131* Obsolete proof of elxp2 5130 as of 13-Aug-2021. (Contributed by NM, 23-Feb-2004.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)

Theoremxpeq12 5132 Equality theorem for Cartesian product. (Contributed by FL, 31-Aug-2009.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Theoremxpeq1i 5133 Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐴 × 𝐶) = (𝐵 × 𝐶)

Theoremxpeq2i 5134 Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐶 × 𝐴) = (𝐶 × 𝐵)

Theoremxpeq12i 5135 Equality inference for Cartesian product. (Contributed by FL, 31-Aug-2009.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴 × 𝐶) = (𝐵 × 𝐷)

Theoremxpeq1d 5136 Equality deduction for Cartesian product. (Contributed by Jeff Madsen, 17-Jun-2010.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶))

Theoremxpeq2d 5137 Equality deduction for Cartesian product. (Contributed by Jeff Madsen, 17-Jun-2010.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵))

Theoremxpeq12d 5138 Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Theoremsqxpeqd 5139 Equality deduction for a Cartesian square, see Wikipedia "Cartesian product", https://en.wikipedia.org/wiki/Cartesian_product#n-ary_Cartesian_power. (Contributed by AV, 13-Jan-2020.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 × 𝐴) = (𝐵 × 𝐵))

Theoremnfxp 5140 Bound-variable hypothesis builder for Cartesian product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴 × 𝐵)

Theorem0nelxp 5141 The empty set is not a member of a Cartesian product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof shortened by JJ, 13-Aug-2021.)
¬ ∅ ∈ (𝐴 × 𝐵)

Theorem0nelxpOLD 5142 Obsolete proof of 0nelxp 5141 as of 13-Aug-2021. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
¬ ∅ ∈ (𝐴 × 𝐵)

Theorem0nelelxp 5143 A member of a Cartesian product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)
(𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶)

Theoremopelxp 5144 Ordered pair membership in a Cartesian product. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
(⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷))

Theorembrxp 5145 Binary relation on a Cartesian product. (Contributed by NM, 22-Apr-2004.)
(𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐷))

Theoremopelxpi 5146 Ordered pair membership in a Cartesian product (implication). (Contributed by NM, 28-May-1995.)
((𝐴𝐶𝐵𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))

Theoremopelxpd 5147 Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))

Theoremopelxp1 5148 The first member of an ordered pair of classes in a Cartesian product belongs to first Cartesian product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
(⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐴𝐶)

Theoremopelxp2 5149 The second member of an ordered pair of classes in a Cartesian product belongs to second Cartesian product argument. (Contributed by Mario Carneiro, 26-Apr-2015.)
(⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐵𝐷)

Theoremotelxp1 5150 The first member of an ordered triple of classes in a Cartesian product belongs to first Cartesian product argument. (Contributed by NM, 28-May-2008.)
(⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑅 × 𝑆) × 𝑇) → 𝐴𝑅)

Theoremotel3xp 5151 An ordered triple is an element of a doubled Cartesian product. (Contributed by Alexander van der Vekens, 26-Feb-2018.)
((𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ ∧ (𝐴𝑋𝐵𝑌𝐶𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍))

Theoremrabxp 5152* Membership in a class builder restricted to a Cartesian product. (Contributed by NM, 20-Feb-2014.)
(𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))       {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵𝜓)}

Theorembrrelex12 5153 A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theorembrrelex 5154 A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)

Theorembrrelex2 5155 A true binary relation on a relation implies the second argument is a set. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)

Theorembrrelexi 5156 The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)
Rel 𝑅       (𝐴𝑅𝐵𝐴 ∈ V)

Theorembrrelex2i 5157 The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐵 ∈ V)

Theoremnprrel12 5158 Proper classes are not related via any relation. (Contributed by AV, 29-Oct-2021.)
Rel 𝑅       (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 𝐴𝑅𝐵)

Theoremnprrel 5159 No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)
Rel 𝑅    &    ¬ 𝐴 ∈ V        ¬ 𝐴𝑅𝐵

Theorem0nelrel 5160 A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.)
(Rel 𝑅 → ∅ ∉ 𝑅)

Theoremfconstmpt 5161* Representation of a constant function using the mapping operation. (Note that 𝑥 cannot appear free in 𝐵.) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.)
(𝐴 × {𝐵}) = (𝑥𝐴𝐵)

Theoremvtoclr 5162* Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Rel 𝑅    &   ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)       ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Theoremopelvvg 5163 Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)
((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))

Theoremopelvv 5164 Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐴, 𝐵⟩ ∈ (V × V)

Theoremopthprc 5165 Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes." (Contributed by NM, 28-Sep-2003.)
(((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theorembrel 5166 Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝑅 ⊆ (𝐶 × 𝐷)       (𝐴𝑅𝐵 → (𝐴𝐶𝐵𝐷))

Theoremelxp3 5167* Membership in a Cartesian product. (Contributed by NM, 5-Mar-1995.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))

Theoremopeliunxp 5168 Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
(⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐶𝐵))

Theoremxpundi 5169 Distributive law for Cartesian product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
(𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶))

Theoremxpundir 5170 Distributive law for Cartesian product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))

Theoremxpiundi 5171* Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
(𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵)

Theoremxpiundir 5172* Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
( 𝑥𝐴 𝐵 × 𝐶) = 𝑥𝐴 (𝐵 × 𝐶)

Theoremiunxpconst 5173* Membership in a union of Cartesian products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)

Theoremxpun 5174 The Cartesian product of two unions. (Contributed by NM, 12-Aug-2004.)
((𝐴𝐵) × (𝐶𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))

Theoremelvv 5175* Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)
(𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)

Theoremelvvv 5176* Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)
(𝐴 ∈ ((V × V) × V) ↔ ∃𝑥𝑦𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)

Theoremelvvuni 5177 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
(𝐴 ∈ (V × V) → 𝐴𝐴)

Theorembrinxp2 5178 Intersection of binary relation with Cartesian product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝐶𝐵𝐷𝐴𝑅𝐵))

Theorembrinxp 5179 Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.)
((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵))

Theorempoinxp 5180 Intersection of partial order with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
(𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)

Theoremsoinxp 5181 Intersection of total order with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
(𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)

Theoremfrinxp 5182 Intersection of well-founded relation with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
(𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴)

Theoremseinxp 5183 Intersection of set-like relation with Cartesian product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
(𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)

Theoremweinxp 5184 Intersection of well-ordering with Cartesian product of its field. (Contributed by NM, 9-Mar-1997.) (Revised by Mario Carneiro, 10-Jul-2014.)
(𝑅 We 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) We 𝐴)

Theoremposn 5185 Partial ordering of a singleton. (Contributed by NM, 27-Apr-2009.) (Revised by Mario Carneiro, 23-Apr-2015.)
(Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Theoremsosn 5186 Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Theoremfrsn 5187 Founded relation on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 23-Apr-2015.)
(Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Theoremwesn 5188 Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(Rel 𝑅 → (𝑅 We {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Theoremelopaelxp 5189* Membership in an ordered pair class builder implies membership in a Cartesian product. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
(𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))

Theorembropaex12 5190* Two classes related by an ordered pair class builder are sets. (Contributed by AV, 21-Jan-2020.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}       (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theoremopabssxp 5191* An abstraction relation is a subset of a related Cartesian product. (Contributed by NM, 16-Jul-1995.)
{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)

Theorembrab2a 5192* The law of concretion for a binary relation. Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 28-Apr-2015.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)}       (𝐴𝑅𝐵 ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓))

Theoremoptocl 5193* Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)
𝐷 = (𝐵 × 𝐶)    &   (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))    &   ((𝑥𝐵𝑦𝐶) → 𝜑)       (𝐴𝐷𝜓)

Theorem2optocl 5194* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
𝑅 = (𝐶 × 𝐷)    &   (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))    &   (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓𝜒))    &   (((𝑥𝐶𝑦𝐷) ∧ (𝑧𝐶𝑤𝐷)) → 𝜑)       ((𝐴𝑅𝐵𝑅) → 𝜒)

Theorem3optocl 5195* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
𝑅 = (𝐷 × 𝐹)    &   (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))    &   (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓𝜒))    &   (⟨𝑣, 𝑢⟩ = 𝐶 → (𝜒𝜃))    &   (((𝑥𝐷𝑦𝐹) ∧ (𝑧𝐷𝑤𝐹) ∧ (𝑣𝐷𝑢𝐹)) → 𝜑)       ((𝐴𝑅𝐵𝑅𝐶𝑅) → 𝜃)

Theoremopbrop 5196* Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)
(((𝑧 = 𝐴𝑤 = 𝐵) ∧ (𝑣 = 𝐶𝑢 = 𝐷)) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))}       (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ 𝜓))

Theorem0xp 5197 The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.)
(∅ × 𝐴) = ∅

Theoremcsbxp 5198 Distribute proper substitution through the Cartesian product of two classes. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 23-Aug-2018.)
𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶)

Theoremreleq 5199 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))

Theoremreleqi 5200 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
𝐴 = 𝐵       (Rel 𝐴 ↔ Rel 𝐵)

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