P. 47 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4601-4700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	en2lp 4601	No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 27-Nov-2018.)
|- -. ( A e. B /\ B e. A )
Theorem	preleq 4602	Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )
Theorem	opthreg 4603	Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4584 (via the preleq 4602 step). See df-op 3641 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( { A , { A , B } } = { C , { C , D } } <-> ( A = C /\ B = D ) )
Theorem	suc11g 4604	The successor operation behaves like a one-to-one function (assuming the Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)
|- ( ( A e. V /\ B e. W ) -> ( suc A = suc B <-> A = B ) )
Theorem	suc11 4605	The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)
|- ( ( A e. On /\ B e. On ) -> ( suc A = suc B <-> A = B ) )
Theorem	dtruex 4606*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4234 can also be summarized as "at least two sets exist", the difference is that dtruarb 4234 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific y, we can construct a set x which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- E. x -. x = y
Theorem	dtru 4607*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). If we assumed the law of the excluded middle this would be equivalent to dtruex 4606. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- -. A. x x = y
Theorem	eunex 4608	Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- ( E! x ph -> E. x -. ph )
Theorem	ordsoexmid 4609	Weak linearity of ordinals implies the law of the excluded middle (that is, decidability of an arbitrary proposition). (Contributed by Mario Carneiro and Jim Kingdon, 29-Jan-2019.)
|- _E Or On   =>    |- ( ph \/ -. ph )
Theorem	ordsuc 4610	The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon, 20-Jul-2019.)
|- ( Ord A <-> Ord suc A )
Theorem	onsucuni2 4611	A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ( A e. On /\ A = suc B ) -> suc U. A = A )
Theorem	0elsucexmid 4612*	If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)
|- A. x e. On (/) e. suc x   =>    |- ( ph \/ -. ph )
Theorem	nlimsucg 4613	A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A e. V -> -. Lim suc A )
Theorem	ordpwsucss 4614	The collection of ordinals in the power class of an ordinal is a superset of its successor. We can think of ( ~P A i^i On ) as another possible definition of successor, which would be equivalent to df-suc 4417 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if A e. On then both U. suc A = A (onunisuci 4478) and U. { x e. On | x C_ A } = A (onuniss2 4559). Constructively ( ~P A i^i On ) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4617). (Contributed by Jim Kingdon, 21-Jul-2019.)
|- ( Ord A -> suc A C_ ( ~P A i^i On ) )
Theorem	onnmin 4615	No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) (Constructive proof by Mario Carneiro and Jim Kingdon, 21-Jul-2019.)
|- ( ( A C_ On /\ B e. A ) -> -. B e. |^| A )
Theorem	ssnel 4616	Relationship between subset and elementhood. In the context of ordinals this can be seen as an ordering law. (Contributed by Jim Kingdon, 22-Jul-2019.)
|- ( A C_ B -> -. B e. A )
Theorem	ordpwsucexmid 4617*	The subset in ordpwsucss 4614 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.)
|- A. x e. On suc x = ( ~P x i^i On )   =>    |- ( ph \/ -. ph )
Theorem	ordtri2or2exmid 4618*	Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
|- A. x e. On A. y e. On ( x C_ y \/ y C_ x )   =>    |- ( ph \/ -. ph )
Theorem	ontri2orexmidim 4619*	Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4618. (Contributed by Jim Kingdon, 26-Aug-2024.)
|- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> DECID ph )
Theorem	onintexmid 4620*	If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
|- ( ( y C_ On /\ E. x x e. y ) -> |^| y e. y )   =>    |- ( ph \/ -. ph )
Theorem	zfregfr 4621	The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
|- _E Fr A
Theorem	ordfr 4622	Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
|- ( Ord A -> _E Fr A )
Theorem	ordwe 4623	Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
|- ( Ord A -> _E We A )
Theorem	wetriext 4624*	A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
|- ( ph -> R We A )   &    |- ( ph -> A e. V )   &    |- ( ph -> A. a e. A A. b e. A ( a R b \/ a = b \/ b R a ) )   &    |- ( ph -> B e. A )   &    |- ( ph -> C e. A )   &    |- ( ph -> A. z e. A ( z R B <-> z R C ) )   =>    |- ( ph -> B = C )
Theorem	wessep 4625	A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.)
|- ( ( _E We A /\ B C_ A ) -> _E We B )
Theorem	reg3exmidlemwe 4626*	Lemma for reg3exmid 4627. Our counterexample A satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- _E We A
Theorem	reg3exmid 4627*	If any inhabited set satisfying df-wetr 4380 for _E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
|- ( ( _E We z /\ E. w w e. z ) -> E. x e. z A. y e. z x C_ y )   =>    |- ( ph \/ -. ph )
Theorem	dcextest 4628*	If it is decidable whether { x | ph } is a set, then -. ph is decidable (where x does not occur in ph). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition -. ph is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.)
|- DECID { x | ph } e. _V   =>    |- DECID -. ph
2.5.3  Transfinite induction
Theorem	tfi 4629*	The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if A is a class of ordinal numbers with the property that every ordinal number included in A also belongs to A, then every ordinal number is in A. (Contributed by NM, 18-Feb-2004.)
|- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> A = On )
Theorem	tfis 4630*	Transfinite Induction Schema. If all ordinal numbers less than a given number x have a property (induction hypothesis), then all ordinal numbers have the property (conclusion). Exercise 25 of [Enderton] p. 200. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 20-Nov-2016.)
|- ( x e. On -> ( A. y e. x [ y / x ] ph -> ph ) )   =>    |- ( x e. On -> ph )
Theorem	tfis2f 4631*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   &    |- ( x e. On -> ( A. y e. x ps -> ph ) )   =>    |- ( x e. On -> ph )
Theorem	tfis2 4632*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x e. On -> ( A. y e. x ps -> ph ) )   =>    |- ( x e. On -> ph )
Theorem	tfis3 4633*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = A -> ( ph <-> ch ) )   &    |- ( x e. On -> ( A. y e. x ps -> ph ) )   =>    |- ( A e. On -> ch )
Theorem	tfisi 4634*	A transfinite induction scheme in "implicit" form where the induction is done on an object derived from the object of interest. (Contributed by Stefan O'Rear, 24-Aug-2015.)
|- ( ph -> A e. V )   &    |- ( ph -> T e. On )   &    |- ( ( ph /\ ( R e. On /\ R C_ T ) /\ A. y ( S e. R -> ch ) ) -> ps )   &    |- ( x = y -> ( ps <-> ch ) )   &    |- ( x = A -> ( ps <-> th ) )   &    |- ( x = y -> R = S )   &    |- ( x = A -> R = T )   =>    |- ( ph -> th )
2.6  IZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity
Axiom	ax-iinf 4635*	Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.)
|- E. x ( (/) e. x /\ A. y ( y e. x -> suc y e. x ) )
Theorem	zfinf2 4636*	A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.)
|- E. x ( (/) e. x /\ A. y e. x suc y e. x )
2.6.2  The natural numbers
Syntax	com 4637	Extend class notation to include the class of natural numbers.
class om
Definition	df-iom 4638*	Define the class of natural numbers as the smallest inductive set, which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. Note: the natural numbers om are a subset of the ordinal numbers df-on 4414. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 9036) with analogous properties and operations, but they will be different sets. We are unable to use the terms finite ordinal and natural number interchangeably, as shown at exmidonfin 7301. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4639 instead for naming consistency with set.mm. (New usage is discouraged.)
|- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
Theorem	dfom3 4639*	Alias for df-iom 4638. Use it instead of df-iom 4638 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.)
|- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
Theorem	omex 4640	The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.)
|- om e. _V
2.6.3  Peano's postulates
Theorem	peano1 4641	Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. (Contributed by NM, 15-May-1994.)
|- (/) e. om
Theorem	peano2 4642	The successor of any natural number is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
|- ( A e. om -> suc A e. om )
Theorem	peano3 4643	The successor of any natural number is not zero. One of Peano's five postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
|- ( A e. om -> suc A =/= (/) )
Theorem	peano4 4644	Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's five postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.)
|- ( ( A e. om /\ B e. om ) -> ( suc A = suc B <-> A = B ) )
Theorem	peano5 4645*	The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as Theorem findes 4650. (Contributed by NM, 18-Feb-2004.)
|- ( ( (/) e. A /\ A. x e. om ( x e. A -> suc x e. A ) ) -> om C_ A )
2.6.4  Finite induction (for finite ordinals)
Theorem	find 4646*	The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that A is a set of natural numbers, zero belongs to A, and given any member of A the member's successor also belongs to A. The conclusion is that every natural number is in A. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A )   =>    |- A = om
Theorem	finds 4647*	Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = suc y -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( y e. om -> ( ch -> th ) )   =>    |- ( A e. om -> ta )
Theorem	finds2 4648*	Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = suc y -> ( ph <-> th ) )   &    |- ( ta -> ps )   &    |- ( y e. om -> ( ta -> ( ch -> th ) ) )   =>    |- ( x e. om -> ( ta -> ph ) )
Theorem	finds1 4649*	Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = suc y -> ( ph <-> th ) )   &    |- ps   &    |- ( y e. om -> ( ch -> th ) )   =>    |- ( x e. om -> ph )
Theorem	findes 4650	Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.)
|- [. (/) / x ]. ph   &    |- ( x e. om -> ( ph -> [. suc x / x ]. ph ) )   =>    |- ( x e. om -> ph )
2.6.5  The Natural Numbers (continued)
Theorem	nn0suc 4651*	A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.)
|- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )
Theorem	elomssom 4652	A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4653. (Revised by BJ, 7-Aug-2024.)
|- ( A e. om -> A C_ om )
Theorem	elnn 4653	A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
|- ( ( A e. B /\ B e. om ) -> A e. om )
Theorem	ordom 4654	Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
|- Ord om
Theorem	omelon2 4655	Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
|- ( om e. _V -> om e. On )
Theorem	omelon 4656	Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
|- om e. On
Theorem	nnon 4657	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
|- ( A e. om -> A e. On )
Theorem	nnoni 4658	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
|- A e. om   =>    |- A e. On
Theorem	nnord 4659	A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
|- ( A e. om -> Ord A )
Theorem	omsson 4660	Omega is a subset of On. (Contributed by NM, 13-Jun-1994.)
|- om C_ On
Theorem	limom 4661	Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. (Contributed by NM, 26-Mar-1995.) (Proof rewritten by Jim Kingdon, 5-Jan-2019.)
|- Lim om
Theorem	peano2b 4662	A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
|- ( A e. om <-> suc A e. om )
Theorem	nnsuc 4663*	A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
|- ( ( A e. om /\ A =/= (/) ) -> E. x e. om A = suc x )
Theorem	nnsucpred 4664	The successor of the precedessor of a nonzero natural number. (Contributed by Jim Kingdon, 31-Jul-2022.)
|- ( ( A e. om /\ A =/= (/) ) -> suc U. A = A )
Theorem	nndceq0 4665	A natural number is either zero or nonzero. Decidable equality for natural numbers is a special case of the law of the excluded middle which holds in most constructive set theories including ours. (Contributed by Jim Kingdon, 5-Jan-2019.)
|- ( A e. om -> DECID A = (/) )
Theorem	0elnn 4666	A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.)
|- ( A e. om -> ( A = (/) \/ (/) e. A ) )
Theorem	nn0eln0 4667	A natural number is nonempty iff it contains the empty set. Although in constructive mathematics it is generally more natural to work with inhabited sets and ignore the whole concept of nonempty sets, in the specific case of natural numbers this theorem may be helpful in converting proofs which were written assuming excluded middle. (Contributed by Jim Kingdon, 28-Aug-2019.)
|- ( A e. om -> ( (/) e. A <-> A =/= (/) ) )
Theorem	nnregexmid 4668*	If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4582 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6584 or nntri3or 6578), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.)
|- ( ( x C_ om /\ E. y y e. x ) -> E. y ( y e. x /\ A. z ( z e. y -> -. z e. x ) ) )   =>    |- ( ph \/ -. ph )
Theorem	omsinds 4669*	Strong (or "total") induction principle over om. (Contributed by Scott Fenton, 17-Jul-2015.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = A -> ( ph <-> ch ) )   &    |- ( x e. om -> ( A. y e. x ps -> ph ) )   =>    |- ( A e. om -> ch )
Theorem	nnpredcl 4670	The predecessor of a natural number is a natural number. This theorem is most interesting when the natural number is a successor (as seen in theorems like onsucuni2 4611) but also holds when it is (/) by uni0 3876. (Contributed by Jim Kingdon, 31-Jul-2022.)
|- ( A e. om -> U. A e. om )
Theorem	nnpredlt 4671	The predecessor (see nnpredcl 4670) of a nonzero natural number is less than (see df-iord 4412) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
|- ( ( A e. om /\ A =/= (/) ) -> U. A e. A )
2.6.6  Relations
Syntax	cxp 4672	Extend the definition of a class to include the cross product.
class ( A X. B )
Syntax	ccnv 4673	Extend the definition of a class to include the converse of a class.
class `' A
Syntax	cdm 4674	Extend the definition of a class to include the domain of a class.
class dom A
Syntax	crn 4675	Extend the definition of a class to include the range of a class.
class ran A
Syntax	cres 4676	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 4677	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 4678	Extend the definition of a class to include the composition of two classes. (Read: The composition of A and B.)
class ( A o. B )
Syntax	wrel 4679	Extend the definition of a wff to include the relation predicate. (Read: A is a relation.)
wff Rel A
Definition	df-xp 4680*	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 } X. { 2 , 7 } ) = ( { <. 1 , 2 >. , <. 1 , 7 >. } u. { <. 5 , 2 >. , <. 5 , 7 >. } ). Another example is that the set of rational numbers is defined using the Cartesian product as ( ZZ X. NN ); 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 X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }
Definition	df-rel 4681	Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5132 and dfrel3 5139. (Contributed by NM, 1-Aug-1994.)
|- ( Rel A <-> A C_ ( _V X. _V ) )
Definition	df-cnv 4682*	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 e. _V and B e. _V then ( A `' R B <-> B R A ), as proven in brcnv 4860 (see df-br 4044 and df-rel 4681 for more on relations). For example, `' { <. 2 , 6 >. , <. 3 , 9 >. } = { <. 6 , 2 >. , <. 9 , 3 >. }. We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. "Converse" is Quine's terminology. Some authors use a "minus one" exponent and call it "inverse", especially when the argument is a function, although this is not in general a genuine inverse. (Contributed by NM, 4-Jul-1994.)
|- `' A = { <. x , y >. | y A x }
Definition	df-co 4683*	Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. Note that Definition 7 of [Suppes] p. 63 reverses A and B, uses a slash instead of o., and calls the operation "relative product". (Contributed by NM, 4-Jul-1994.)
|- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) }
Definition	df-dm 4684*	Define the domain of a class. Definition 3 of [Suppes] p. 59. For example, F = { <. 2 , 6 >., <. 3 , 9 >. } -> dom F = { 2 , 3 } . Contrast with range (defined in df-rn 4685). For alternate definitions see dfdm2 5216, dfdm3 4864, and dfdm4 4869. The notation " dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.)
|- dom A = { x | E. y x A y }
Definition	df-rn 4685	Define the range of a class. For example, F = { <. 2 , 6 >., <. 3 , 9 >. } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4684). For alternate definitions, see dfrn2 4865, dfrn3 4866, and dfrn4 5142. 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 4686	Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, ( F = { <. 2 , 6 >. , <. 3 , 9 >. } /\ B = { 1 , 2 } ) -> ( F |` B ) = { <. 2 , 6 >. }. We do not introduce a special syntax for the corestriction of a class: it will be expressed either as the intersection ( A i^i ( _V X. B ) ) or as the converse of the restricted converse. (Contributed by NM, 2-Aug-1994.)
|- ( A |` B ) = ( A i^i ( B X. _V ) )
Definition	df-ima 4687	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 } . Contrast with restriction (df-res 4686) and range (df-rn 4685). For an alternate definition, see dfima2 5023. (Contributed by NM, 2-Aug-1994.)
|- ( A " B ) = ran ( A |` B )
Theorem	xpeq1 4688	Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
|- ( A = B -> ( A X. C ) = ( B X. C ) )
Theorem	xpeq2 4689	Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
|- ( A = B -> ( C X. A ) = ( C X. B ) )
Theorem	elxpi 4690*	Membership in a cross product. Uses fewer axioms than elxp 4691. (Contributed by NM, 4-Jul-1994.)
|- ( A e. ( B X. C ) -> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )
Theorem	elxp 4691*	Membership in a cross product. (Contributed by NM, 4-Jul-1994.)
|- ( A e. ( B X. C ) <-> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )
Theorem	elxp2 4692*	Membership in a cross product. (Contributed by NM, 23-Feb-2004.)
|- ( A e. ( B X. C ) <-> E. x e. B E. y e. C A = <. x , y >. )
Theorem	xpeq12 4693	Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
|- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )
Theorem	xpeq1i 4694	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( A X. C ) = ( B X. C )
Theorem	xpeq2i 4695	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( C X. A ) = ( C X. B )
Theorem	xpeq12i 4696	Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
|- A = B   &    |- C = D   =>    |- ( A X. C ) = ( B X. D )
Theorem	xpeq1d 4697	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A X. C ) = ( B X. C ) )
Theorem	xpeq2d 4698	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C X. A ) = ( C X. B ) )
Theorem	xpeq12d 4699	Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A X. C ) = ( B X. D ) )
Theorem	sqxpeqd 4700	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.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A X. A ) = ( B X. B ) )