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

Syntaxwwe 5101 Extend wff notation to include the well-ordering predicate. Read: ' 𝑅 well-orders 𝐴.'
wff 𝑅 We 𝐴

Definitiondf-fr 5102* Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 5108 and dffr3 5533. (Contributed by NM, 3-Apr-1994.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))

Definitiondf-se 5103* Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)
(𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)

Definitiondf-we 5104 Define the well-ordering predicate. For an alternate definition, see dfwe2 7023. (Contributed by NM, 3-Apr-1994.)
(𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))

Theoremfri 5105* Property of well-founded relation (one direction of definition). (Contributed by NM, 18-Mar-1997.)
(((𝐵𝐶𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)

Theoremseex 5106* The 𝑅-preimage of an element of the base set in a set-like relation is a set. (Contributed by Mario Carneiro, 19-Nov-2014.)
((𝑅 Se 𝐴𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)

Theoremexse 5107 Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
(𝐴𝑉𝑅 Se 𝐴)

Theoremdffr2 5108* Alternate definition of well-founded relation. Similar to Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Mario Carneiro, 23-Jun-2015.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))

Theoremfrc 5109* Property of well-founded relation (one direction of definition using class variables). (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 19-Nov-2014.)
𝐵 ∈ V       ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} = ∅)

Theoremfrss 5110 Subset theorem for the well-founded predicate. Exercise 1 of [TakeutiZaring] p. 31. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))

Theoremsess1 5111 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝑅𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))

Theoremsess2 5112 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))

Theoremfreq1 5113 Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
(𝑅 = 𝑆 → (𝑅 Fr 𝐴𝑆 Fr 𝐴))

Theoremfreq2 5114 Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
(𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))

Theoremseeq1 5115 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))

Theoremseeq2 5116 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))

Theoremnffr 5117 Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝑅    &   𝑥𝐴       𝑥 𝑅 Fr 𝐴

Theoremnfse 5118 Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝑅    &   𝑥𝐴       𝑥 𝑅 Se 𝐴

Theoremnfwe 5119 Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝑅    &   𝑥𝐴       𝑥 𝑅 We 𝐴

Theoremfrirr 5120 A well-founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 2-Jan-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
((𝑅 Fr 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)

Theoremfr2nr 5121 A well-founded relation has no 2-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))

Theoremfr0 5122 Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)
𝑅 Fr ∅

Theoremfrminex 5123* If an element of a well-founded set satisfies a property 𝜑, then there is a minimal element that satisfies 𝜑. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
𝐴 ∈ V    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝑅 Fr 𝐴 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 (𝜓 → ¬ 𝑦𝑅𝑥))))

Theoremefrirr 5124 Irreflexivity of the epsilon relation: a class founded by epsilon is not a member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
( E Fr 𝐴 → ¬ 𝐴𝐴)

Theoremefrn2lp 5125 A set founded by epsilon contains no 2-cycle loops. (Contributed by NM, 19-Apr-1994.)
(( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝐶𝐶𝐵))

Theoremepse 5126 The epsilon relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the epsilon relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.)
E Se 𝐴

Theoremtz7.2 5127 Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent E Fr 𝐴. (Contributed by NM, 4-May-1994.)
((Tr 𝐴 ∧ E Fr 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))

Theoremdfepfr 5128* An alternate way of saying that the epsilon relation is well-founded. (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)
( E Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥𝑦) = ∅))

Theoremepfrc 5129* A subset of an epsilon-founded class has a minimal element. (Contributed by NM, 17-Feb-2004.) (Revised by David Abernethy, 22-Feb-2011.)
𝐵 ∈ V       (( E Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)

Theoremwess 5130 Subset theorem for the well-ordering predicate. Exercise 4 of [TakeutiZaring] p. 31. (Contributed by NM, 19-Apr-1994.)
(𝐴𝐵 → (𝑅 We 𝐵𝑅 We 𝐴))

Theoremweeq1 5131 Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)
(𝑅 = 𝑆 → (𝑅 We 𝐴𝑆 We 𝐴))

Theoremweeq2 5132 Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
(𝐴 = 𝐵 → (𝑅 We 𝐴𝑅 We 𝐵))

Theoremwefr 5133 A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)
(𝑅 We 𝐴𝑅 Fr 𝐴)

Theoremweso 5134 A well-ordering is a strict ordering. (Contributed by NM, 16-Mar-1997.)
(𝑅 We 𝐴𝑅 Or 𝐴)

Theoremwecmpep 5135 The elements of an epsilon well-ordering are comparable. (Contributed by NM, 17-May-1994.)
(( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))

Theoremwetrep 5136 An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)
(( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))

Theoremwefrc 5137* 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 5138 Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
𝑅 We ∅

Theoremwereu 5139* 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 5140* 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 5141 Extend the definition of a class to include the Cartesian product.
class (𝐴 × 𝐵)

Syntaxccnv 5142 Extend the definition of a class to include the converse of a class.
class 𝐴

Syntaxcdm 5143 Extend the definition of a class to include the domain of a class.
class dom 𝐴

Syntaxcrn 5144 Extend the definition of a class to include the range of a class.
class ran 𝐴

Syntaxcres 5145 Extend the definition of a class to include the restriction of a class. (Read: The restriction of 𝐴 to 𝐵.)
class (𝐴𝐵)

Syntaxcima 5146 Extend the definition of a class to include the image of a class. (Read: The image of 𝐵 under 𝐴.)
class (𝐴𝐵)

Syntaxccom 5147 Extend the definition of a class to include the composition of two classes. (Read: The composition of 𝐴 and 𝐵.)
class (𝐴𝐵)

Syntaxwrel 5148 Extend the definition of a wff to include the relation predicate. (Read: 𝐴 is a relation.)
wff Rel 𝐴

Definitiondf-xp 5149* 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 27423). Another example is that the set of rational numbers are defined in df-q 11827 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 5150 Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5618 and dfrel3 5627. (Contributed by NM, 1-Aug-1994.)
(Rel 𝐴𝐴 ⊆ (V × V))

Definitiondf-cnv 5151* 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 5337 (see df-br 4686 and df-rel 5150 for more on relations). For example, {⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩} (ex-cnv 27424). 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 5152* Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, ((exp ∘ cos)‘0) = e (ex-co 27425) because (cos‘0) = 1 (see cos0 14924) and (exp‘1) = e (see df-e 14843). 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 5153* 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 27426). Another example is the domain of the complex arctangent, (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) (for proof see atandm 24648). Contrast with range (defined in df-rn 5154). For alternate definitions see dfdm2 5705, dfdm3 5342, and dfdm4 5348. 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 5154 Define the range of a class. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9} (ex-rn 27427). Contrast with domain (defined in df-dm 5153). For alternate definitions, see dfrn2 5343, dfrn3 5344, and dfrn4 5630. 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 5155 Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, the expression (exp ↾ ℝ) (used in reeff1 14894) means "the exponential function e to the x, but the exponent x must be in the reals" (df-ef 14842 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 27428). (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵) = (𝐴 ∩ (𝐵 × V))

Definitiondf-ima 5156 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 27429). Contrast with restriction (df-res 5155) and range (df-rn 5154). For an alternate definition, see dfima2 5503. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵) = ran (𝐴𝐵)

Theoremxpeq1 5157 Equality theorem for Cartesian product. (Contributed by NM, 4-Jul-1994.)
(𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))

Theoremxpss12 5158 Subset theorem for Cartesian product. Generalization of Theorem 101 of [Suppes] p. 52. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐷))

Theoremxpss 5159 A Cartesian product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
(𝐴 × 𝐵) ⊆ (V × V)

Theoremrelxp 5160 A Cartesian product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)
Rel (𝐴 × 𝐵)

Theoremxpss1 5161 Subset relation for Cartesian product. (Contributed by Jeff Hankins, 30-Aug-2009.)
(𝐴𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))

Theoremxpss2 5162 Subset relation for Cartesian product. (Contributed by Jeff Hankins, 30-Aug-2009.)
(𝐴𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵))

Theoremxpeq2 5163 Equality theorem for Cartesian product. (Contributed by NM, 5-Jul-1994.)
(𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))

Theoremelxpi 5164* Membership in a Cartesian product. Uses fewer axioms than elxp 5165. (Contributed by NM, 4-Jul-1994.)
(𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))

Theoremelxp 5165* Membership in a Cartesian product. (Contributed by NM, 4-Jul-1994.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))

Theoremelxp2 5166* 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 5167* Obsolete proof of elxp2 5166 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 5168 Equality theorem for Cartesian product. (Contributed by FL, 31-Aug-2009.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Theoremxpeq1i 5169 Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐴 × 𝐶) = (𝐵 × 𝐶)

Theoremxpeq2i 5170 Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐶 × 𝐴) = (𝐶 × 𝐵)

Theoremxpeq12i 5171 Equality inference for Cartesian product. (Contributed by FL, 31-Aug-2009.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴 × 𝐶) = (𝐵 × 𝐷)

Theoremxpeq1d 5172 Equality deduction for Cartesian product. (Contributed by Jeff Madsen, 17-Jun-2010.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶))

Theoremxpeq2d 5173 Equality deduction for Cartesian product. (Contributed by Jeff Madsen, 17-Jun-2010.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵))

Theoremxpeq12d 5174 Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷))

Theoremsqxpeqd 5175 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 5176 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 5177 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 5178 Obsolete proof of 0nelxp 5177 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 5179 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 5180 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 5181 Binary relation on a Cartesian product. (Contributed by NM, 22-Apr-2004.)
(𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐷))

Theoremopelxpi 5182 Ordered pair membership in a Cartesian product (implication). (Contributed by NM, 28-May-1995.)
((𝐴𝐶𝐵𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))

Theoremopelxpd 5183 Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))

Theoremopelxp1 5184 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 5185 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 5186 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 5187 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 5188* 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 5189 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 5190 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 5191 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 5192 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 5193 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 5194 Proper classes are not related via any relation. (Contributed by AV, 29-Oct-2021.)
Rel 𝑅       (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 𝐴𝑅𝐵)

Theoremnprrel 5195 No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)
Rel 𝑅    &    ¬ 𝐴 ∈ V        ¬ 𝐴𝑅𝐵

Theorem0nelrel 5196 A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.)
(Rel 𝑅 → ∅ ∉ 𝑅)

Theoremfconstmpt 5197* 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 5198* 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 5199 Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)
((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))

Theoremopelvv 5200 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)

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