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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | wess 5101 | Subset theorem for the well-ordering predicate. Exercise 4 of [TakeutiZaring] p. 31. (Contributed by NM, 19-Apr-1994.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑅 We 𝐵 → 𝑅 We 𝐴)) | ||
Theorem | weeq1 5102 | Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴)) | ||
Theorem | weeq2 5103 | Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝐴 = 𝐵 → (𝑅 We 𝐴 ↔ 𝑅 We 𝐵)) | ||
Theorem | wefr 5104 | A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.) |
⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | ||
Theorem | weso 5105 | A well-ordering is a strict ordering. (Contributed by NM, 16-Mar-1997.) |
⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | ||
Theorem | wecmpep 5106 | The elements of an epsilon well-ordering are comparable. (Contributed by NM, 17-May-1994.) |
⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | ||
Theorem | wetrep 5107 | An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.) |
⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) | ||
Theorem | wefrc 5108* | 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 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅) | ||
Theorem | we0 5109 | Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.) |
⊢ 𝑅 We ∅ | ||
Theorem | wereu 5110* | 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 ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | ||
Theorem | wereu2 5111* | 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 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | ||
Syntax | cxp 5112 | Extend the definition of a class to include the Cartesian product. |
class (𝐴 × 𝐵) | ||
Syntax | ccnv 5113 | Extend the definition of a class to include the converse of a class. |
class ◡𝐴 | ||
Syntax | cdm 5114 | Extend the definition of a class to include the domain of a class. |
class dom 𝐴 | ||
Syntax | crn 5115 | Extend the definition of a class to include the range of a class. |
class ran 𝐴 | ||
Syntax | cres 5116 | Extend the definition of a class to include the restriction of a class. (Read: The restriction of 𝐴 to 𝐵.) |
class (𝐴 ↾ 𝐵) | ||
Syntax | cima 5117 | Extend the definition of a class to include the image of a class. (Read: The image of 𝐵 under 𝐴.) |
class (𝐴 “ 𝐵) | ||
Syntax | ccom 5118 | Extend the definition of a class to include the composition of two classes. (Read: The composition of 𝐴 and 𝐵.) |
class (𝐴 ∘ 𝐵) | ||
Syntax | wrel 5119 | Extend the definition of a wff to include the relation predicate. (Read: 𝐴 is a relation.) |
wff Rel 𝐴 | ||
Definition | df-xp 5120* | 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 27293). Another example is that the set of rational numbers are defined in df-q 11789 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〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} | ||
Definition | df-rel 5121 | Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5583 and dfrel3 5592. (Contributed by NM, 1-Aug-1994.) |
⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | ||
Definition | df-cnv 5122* | 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 (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴), as proven in brcnv 5305 (see df-br 4654 and df-rel 5121 for more on relations). For example, ◡{〈2, 6〉, 〈3, 9〉} = {〈6, 2〉, 〈9, 3〉} (ex-cnv 27294). 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} | ||
Definition | df-co 5123* | Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, ((exp ∘ cos)‘0) = e (ex-co 27295) because (cos‘0) = 1 (see cos0 14880) and (exp‘1) = e (see df-e 14799). 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.) |
⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} | ||
Definition | df-dm 5124* | 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 27296). Another example is the domain of the complex arctangent, (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) (for proof see atandm 24603). Contrast with range (defined in df-rn 5125). For alternate definitions see dfdm2 5667, dfdm3 5310, and dfdm4 5316. The notation "dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.) |
⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | ||
Definition | df-rn 5125 | Define the range of a class. For example, 𝐹 = {〈2, 6〉, 〈3, 9〉} → ran 𝐹 = {6, 9} (ex-rn 27297). Contrast with domain (defined in df-dm 5124). For alternate definitions, see dfrn2 5311, dfrn3 5312, and dfrn4 5595. 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 | ||
Definition | df-res 5126 | Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, the expression (exp ↾ ℝ) (used in reeff1 14850) means "the exponential function e to the x, but the exponent x must be in the reals" (df-ef 14798 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}) → (𝐹 ↾ 𝐵) = {〈2, 6〉} (ex-res 27298). (Contributed by NM, 2-Aug-1994.) |
⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | ||
Definition | df-ima 5127 | 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}) → (𝐹 “ 𝐵) = {6} (ex-ima 27299). Contrast with restriction (df-res 5126) and range (df-rn 5125). For an alternate definition, see dfima2 5468. (Contributed by NM, 2-Aug-1994.) |
⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | ||
Theorem | xpeq1 5128 | Equality theorem for Cartesian product. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | ||
Theorem | xpeq2 5129 | Equality theorem for Cartesian product. (Contributed by NM, 5-Jul-1994.) |
⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | ||
Theorem | elxpi 5130* | Membership in a Cartesian product. Uses fewer axioms than elxp 5131. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | ||
Theorem | elxp 5131* | Membership in a Cartesian product. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | ||
Theorem | elxp2 5132* | Membership in a Cartesian product. (Contributed by NM, 23-Feb-2004.) (Proof shortened by JJ, 13-Aug-2021.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉) | ||
Theorem | elxp2OLD 5133* | Obsolete proof of elxp2 5132 as of 13-Aug-2021. (Contributed by NM, 23-Feb-2004.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉) | ||
Theorem | xpeq12 5134 | Equality theorem for Cartesian product. (Contributed by FL, 31-Aug-2009.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | ||
Theorem | xpeq1i 5135 | Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐶) | ||
Theorem | xpeq2i 5136 | Equality inference for Cartesian product. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 × 𝐴) = (𝐶 × 𝐵) | ||
Theorem | xpeq12i 5137 | Equality inference for Cartesian product. (Contributed by FL, 31-Aug-2009.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷) | ||
Theorem | xpeq1d 5138 | Equality deduction for Cartesian product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | ||
Theorem | xpeq2d 5139 | Equality deduction for Cartesian product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | ||
Theorem | xpeq12d 5140 | Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | ||
Theorem | sqxpeqd 5141 | 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.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 × 𝐴) = (𝐵 × 𝐵)) | ||
Theorem | nfxp 5142 | Bound-variable hypothesis builder for Cartesian product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 × 𝐵) | ||
Theorem | 0nelxp 5143 | 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) | ||
Theorem | 0nelxpOLD 5144 | Obsolete proof of 0nelxp 5143 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) | ||
Theorem | 0nelelxp 5145 | A member of a Cartesian product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.) |
⊢ (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶) | ||
Theorem | opelxp 5146 | 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) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | ||
Theorem | brxp 5147 | Binary relation on a Cartesian product. (Contributed by NM, 22-Apr-2004.) |
⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | ||
Theorem | opelxpi 5148 | Ordered pair membership in a Cartesian product (implication). (Contributed by NM, 28-May-1995.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | ||
Theorem | opelxpd 5149 | Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | ||
Theorem | opelxp1 5150 | 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) → 𝐴 ∈ 𝐶) | ||
Theorem | opelxp2 5151 | 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) → 𝐵 ∈ 𝐷) | ||
Theorem | otelxp1 5152 | 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) → 𝐴 ∈ 𝑅) | ||
Theorem | otel3xp 5153 | An ordered triple is an element of a doubled Cartesian product. (Contributed by Alexander van der Vekens, 26-Feb-2018.) |
⊢ ((𝑇 = 〈𝐴, 𝐵, 𝐶〉 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍)) | ||
Theorem | rabxp 5154* | Membership in a class builder restricted to a Cartesian product. (Contributed by NM, 20-Feb-2014.) |
⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ∧ 𝜓)} | ||
Theorem | brrelex12 5155 | 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 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | brrelex 5156 | 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 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V) | ||
Theorem | brrelex2 5157 | 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 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V) | ||
Theorem | brrelexi 5158 | The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.) |
⊢ Rel 𝑅 ⇒ ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V) | ||
Theorem | brrelex2i 5159 | The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ Rel 𝑅 ⇒ ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ V) | ||
Theorem | nprrel12 5160 | Proper classes are not related via any relation. (Contributed by AV, 29-Oct-2021.) |
⊢ Rel 𝑅 ⇒ ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 𝐴𝑅𝐵) | ||
Theorem | nprrel 5161 | No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.) |
⊢ Rel 𝑅 & ⊢ ¬ 𝐴 ∈ V ⇒ ⊢ ¬ 𝐴𝑅𝐵 | ||
Theorem | 0nelrel 5162 | A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.) |
⊢ (Rel 𝑅 → ∅ ∉ 𝑅) | ||
Theorem | fconstmpt 5163* | 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}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | vtoclr 5164* | Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ Rel 𝑅 & ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ⇒ ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) | ||
Theorem | opelvvg 5165 | Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (V × V)) | ||
Theorem | opelvv 5166 | 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) | ||
Theorem | opthprc 5167 | 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 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | brel 5168 | 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 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | ||
Theorem | elxp3 5169* | Membership in a Cartesian product. (Contributed by NM, 5-Mar-1995.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶))) | ||
Theorem | opeliunxp 5170 | Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.) |
⊢ (〈𝑥, 𝐶〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)) | ||
Theorem | xpundi 5171 | Distributive law for Cartesian product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.) |
⊢ (𝐴 × (𝐵 ∪ 𝐶)) = ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶)) | ||
Theorem | xpundir 5172 | Distributive law for Cartesian product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.) |
⊢ ((𝐴 ∪ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) | ||
Theorem | xpiundi 5173* | Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.) |
⊢ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) | ||
Theorem | xpiundir 5174* | Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.) |
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) | ||
Theorem | iunxpconst 5175* | Membership in a union of Cartesian products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) | ||
Theorem | xpun 5176 | The Cartesian product of two unions. (Contributed by NM, 12-Aug-2004.) |
⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷))) | ||
Theorem | elvv 5177* | Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | ||
Theorem | elvvv 5178* | Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.) |
⊢ (𝐴 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) | ||
Theorem | elvvuni 5179 | An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.) |
⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴) | ||
Theorem | brinxp2 5180 | 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 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵)) | ||
Theorem | brinxp 5181 | Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵)) | ||
Theorem | poinxp 5182 | 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) | ||
Theorem | soinxp 5183 | 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) | ||
Theorem | frinxp 5184 | 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) | ||
Theorem | seinxp 5185 | 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) | ||
Theorem | weinxp 5186 | 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) | ||
Theorem | posn 5187 | Partial ordering of a singleton. (Contributed by NM, 27-Apr-2009.) (Revised by Mario Carneiro, 23-Apr-2015.) |
⊢ (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴)) | ||
Theorem | sosn 5188 | Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
⊢ (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴)) | ||
Theorem | frsn 5189 | 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)) | ||
Theorem | wesn 5190 | Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
⊢ (Rel 𝑅 → (𝑅 We {𝐴} ↔ ¬ 𝐴𝑅𝐴)) | ||
Theorem | elopaelxp 5191* | 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)) | ||
Theorem | bropaex12 5192* | 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)) | ||
Theorem | opabssxp 5193* | An abstraction relation is a subset of a related Cartesian product. (Contributed by NM, 16-Jul-1995.) |
⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵) | ||
Theorem | brab2a 5194* | 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 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ⇒ ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝜓)) | ||
Theorem | optocl 5195* | Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.) |
⊢ 𝐷 = (𝐵 × 𝐶) & ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝐷 → 𝜓) | ||
Theorem | 2optocl 5196* | Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.) |
⊢ 𝑅 = (𝐶 × 𝐷) & ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (〈𝑧, 𝑤〉 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) → 𝜑) ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒) | ||
Theorem | 3optocl 5197* | Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.) |
⊢ 𝑅 = (𝐷 × 𝐹) & ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (〈𝑧, 𝑤〉 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (〈𝑣, 𝑢〉 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐹) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐹) ∧ (𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹)) → 𝜑) ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑅) → 𝜃) | ||
Theorem | opbrop 5198* | Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.) |
⊢ (((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝜑 ↔ 𝜓)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ 𝜑))} ⇒ ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (〈𝐴, 𝐵〉𝑅〈𝐶, 𝐷〉 ↔ 𝜓)) | ||
Theorem | 0xp 5199 | 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) = ∅ | ||
Theorem | csbxp 5200 | 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 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) |
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