Type  Label  Description 
Statement 

Definition  dfima 4301 
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 (dfres 4300) and range
(dfrn 4299). For an alternate definition, see dfima2 4613. (Contributed
by NM, 2Aug1994.)



Theorem  xpeq1 4302 
Equality theorem for cross product. (Contributed by NM, 4Jul1994.)



Theorem  xpeq2 4303 
Equality theorem for cross product. (Contributed by NM, 5Jul1994.)



Theorem  elxpi 4304* 
Membership in a cross product. Uses fewer axioms than elxp 4305.
(Contributed by NM, 4Jul1994.)



Theorem  elxp 4305* 
Membership in a cross product. (Contributed by NM, 4Jul1994.)



Theorem  elxp2 4306* 
Membership in a cross product. (Contributed by NM, 23Feb2004.)



Theorem  xpeq12 4307 
Equality theorem for cross product. (Contributed by FL, 31Aug2009.)



Theorem  xpeq1i 4308 
Equality inference for cross product. (Contributed by NM,
21Dec2008.)



Theorem  xpeq2i 4309 
Equality inference for cross product. (Contributed by NM,
21Dec2008.)



Theorem  xpeq12i 4310 
Equality inference for cross product. (Contributed by FL,
31Aug2009.)



Theorem  xpeq1d 4311 
Equality deduction for cross product. (Contributed by Jeff Madsen,
17Jun2010.)



Theorem  xpeq2d 4312 
Equality deduction for cross product. (Contributed by Jeff Madsen,
17Jun2010.)



Theorem  xpeq12d 4313 
Equality deduction for cross product. (Contributed by NM,
8Dec2013.)



Theorem  nfxp 4314 
Boundvariable hypothesis builder for cross product. (Contributed by
NM, 15Sep2003.) (Revised by Mario Carneiro, 15Oct2016.)



Theorem  0nelxp 4315 
The empty set is not a member of a cross product. (Contributed by NM,
2May1996.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  0nelelxp 4316 
A member of a cross product (ordered pair) doesn't contain the empty
set. (Contributed by NM, 15Dec2008.)



Theorem  opelxp 4317 
Ordered pair membership in a cross product. (Contributed by NM,
15Nov1994.) (Proof shortened by Andrew Salmon, 12Aug2011.)
(Revised by Mario Carneiro, 26Apr2015.)



Theorem  brxp 4318 
Binary relation on a cross product. (Contributed by NM,
22Apr2004.)



Theorem  opelxpi 4319 
Ordered pair membership in a cross product (implication). (Contributed by
NM, 28May1995.)



Theorem  opelxp1 4320 
The first member of an ordered pair of classes in a cross product belongs
to first cross product argument. (Contributed by NM, 28May2008.)
(Revised by Mario Carneiro, 26Apr2015.)



Theorem  opelxp2 4321 
The second member of an ordered pair of classes in a cross product belongs
to second cross product argument. (Contributed by Mario Carneiro,
26Apr2015.)



Theorem  otelxp1 4322 
The first member of an ordered triple of classes in a cross product
belongs to first cross product argument. (Contributed by NM,
28May2008.)



Theorem  rabxp 4323* 
Membership in a class builder restricted to a cross product.
(Contributed by NM, 20Feb2014.)



Theorem  brrelex12 4324 
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, 26Apr2015.)



Theorem  brrelex 4325 
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,
18May2004.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  brrelex2 4326 
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, 26Apr2015.)



Theorem  brrelexi 4327 
The first argument of a binary relation exists. (An artifact of our
ordered pair definition.) (Contributed by NM, 4Jun1998.)



Theorem  brrelex2i 4328 
The second argument of a binary relation exists. (An artifact of our
ordered pair definition.) (Contributed by Mario Carneiro,
26Apr2015.)



Theorem  nprrel 4329 
No proper class is related to anything via any relation. (Contributed
by Roy F. Longton, 30Jul2005.)



Theorem  fconstmpt 4330* 
Representation of a constant function using the mapping operation.
(Note that
cannot appear free in .) (Contributed by NM,
12Oct1999.) (Revised by Mario Carneiro, 16Nov2013.)



Theorem  vtoclr 4331* 
Variable to class conversion of transitive relation. (Contributed by
NM, 9Jun1998.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  opelvvg 4332 
Ordered pair membership in the universal class of ordered pairs.
(Contributed by Mario Carneiro, 3May2015.)



Theorem  opelvv 4333 
Ordered pair membership in the universal class of ordered pairs.
(Contributed by NM, 22Aug2013.) (Revised by Mario Carneiro,
26Apr2015.)



Theorem  opthprc 4334 
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, 28Sep2003.)



Theorem  brel 4335 
Two things in a binary relation belong to the relation's domain.
(Contributed by NM, 17May1996.) (Revised by Mario Carneiro,
26Apr2015.)



Theorem  brab2a 4336* 
Ordered pair membership in an ordered pair class abstraction.
(Contributed by Mario Carneiro, 9Nov2015.)



Theorem  elxp3 4337* 
Membership in a cross product. (Contributed by NM, 5Mar1995.)



Theorem  opeliunxp 4338 
Membership in a union of cross products. (Contributed by Mario
Carneiro, 29Dec2014.) (Revised by Mario Carneiro, 1Jan2017.)



Theorem  xpundi 4339 
Distributive law for cross product over union. Theorem 103 of [Suppes]
p. 52. (Contributed by NM, 12Aug2004.)



Theorem  xpundir 4340 
Distributive law for cross product over union. Similar to Theorem 103
of [Suppes] p. 52. (Contributed by NM,
30Sep2002.)



Theorem  xpiundi 4341* 
Distributive law for cross product over indexed union. (Contributed by
Mario Carneiro, 27Apr2014.)



Theorem  xpiundir 4342* 
Distributive law for cross product over indexed union. (Contributed by
Mario Carneiro, 27Apr2014.)



Theorem  iunxpconst 4343* 
Membership in a union of cross products when the second factor is
constant. (Contributed by Mario Carneiro, 29Dec2014.)



Theorem  xpun 4344 
The cross product of two unions. (Contributed by NM, 12Aug2004.)



Theorem  elvv 4345* 
Membership in universal class of ordered pairs. (Contributed by NM,
4Jul1994.)



Theorem  elvvv 4346* 
Membership in universal class of ordered triples. (Contributed by NM,
17Dec2008.)



Theorem  elvvuni 4347 
An ordered pair contains its union. (Contributed by NM,
16Sep2006.)



Theorem  mosubopt 4348* 
"At most one" remains true inside ordered pair quantification.
(Contributed by NM, 28Aug2007.)



Theorem  mosubop 4349* 
"At most one" remains true inside ordered pair quantification.
(Contributed by NM, 28May1995.)



Theorem  brinxp2 4350 
Intersection of binary relation with cross product. (Contributed by NM,
3Mar2007.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  brinxp 4351 
Intersection of binary relation with cross product. (Contributed by NM,
9Mar1997.)



Theorem  poinxp 4352 
Intersection of partial order with cross product of its field.
(Contributed by Mario Carneiro, 10Jul2014.)



Theorem  soinxp 4353 
Intersection of linear order with cross product of its field.
(Contributed by Mario Carneiro, 10Jul2014.)



Theorem  seinxp 4354 
Intersection of setlike relation with cross product of its field.
(Contributed by Mario Carneiro, 22Jun2015.)

Se Se 

Theorem  posng 4355 
Partial ordering of a singleton. (Contributed by Jim Kingdon,
5Dec2018.)



Theorem  sosng 4356 
Strict linear ordering on a singleton. (Contributed by Jim Kingdon,
5Dec2018.)



Theorem  opabssxp 4357* 
An abstraction relation is a subset of a related cross product.
(Contributed by NM, 16Jul1995.)



Theorem  brab2ga 4358* 
The law of concretion for a binary relation. See brab2a 4336 for alternate
proof. TODO: should one of them be deleted? (Contributed by Mario
Carneiro, 28Apr2015.) (Proof modification is discouraged.)



Theorem  optocl 4359* 
Implicit substitution of class for ordered pair. (Contributed by NM,
5Mar1995.)



Theorem  2optocl 4360* 
Implicit substitution of classes for ordered pairs. (Contributed by NM,
12Mar1995.)



Theorem  3optocl 4361* 
Implicit substitution of classes for ordered pairs. (Contributed by NM,
12Mar1995.)



Theorem  opbrop 4362* 
Ordered pair membership in a relation. Special case. (Contributed by
NM, 5Aug1995.)



Theorem  0xp 4363 
The cross product with the empty set is empty. Part of Theorem 3.13(ii)
of [Monk1] p. 37. (Contributed by NM,
4Jul1994.)



Theorem  csbxpg 4364 
Distribute proper substitution through the cross product of two
classes. (Contributed by Alan Sare, 10Nov2012.)



Theorem  releq 4365 
Equality theorem for the relation predicate. (Contributed by NM,
1Aug1994.)



Theorem  releqi 4366 
Equality inference for the relation predicate. (Contributed by NM,
8Dec2006.)



Theorem  releqd 4367 
Equality deduction for the relation predicate. (Contributed by NM,
8Mar2014.)



Theorem  nfrel 4368 
Boundvariable hypothesis builder for a relation. (Contributed by NM,
31Jan2004.) (Revised by Mario Carneiro, 15Oct2016.)



Theorem  sbcrel 4369 
Distribute proper substitution through a relation predicate. (Contributed
by Alexander van der Vekens, 23Jul2017.)



Theorem  relss 4370 
Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58.
(Contributed by NM, 15Aug1994.)



Theorem  ssrel 4371* 
A subclass relationship depends only on a relation's ordered pairs.
Theorem 3.2(i) of [Monk1] p. 33.
(Contributed by NM, 2Aug1994.)
(Proof shortened by Andrew Salmon, 27Aug2011.)



Theorem  eqrel 4372* 
Extensionality principle for relations. Theorem 3.2(ii) of [Monk1]
p. 33. (Contributed by NM, 2Aug1994.)



Theorem  ssrel2 4373* 
A subclass relationship depends only on a relation's ordered pairs.
This version of ssrel 4371 is restricted to the relation's domain.
(Contributed by Thierry Arnoux, 25Jan2018.)



Theorem  relssi 4374* 
Inference from subclass principle for relations. (Contributed by NM,
31Mar1998.)



Theorem  relssdv 4375* 
Deduction from subclass principle for relations. (Contributed by NM,
11Sep2004.)



Theorem  eqrelriv 4376* 
Inference from extensionality principle for relations. (Contributed by
FL, 15Oct2012.)



Theorem  eqrelriiv 4377* 
Inference from extensionality principle for relations. (Contributed by
NM, 17Mar1995.)



Theorem  eqbrriv 4378* 
Inference from extensionality principle for relations. (Contributed by
NM, 12Dec2006.)



Theorem  eqrelrdv 4379* 
Deduce equality of relations from equivalence of membership.
(Contributed by Rodolfo Medina, 10Oct2010.)



Theorem  eqbrrdv 4380* 
Deduction from extensionality principle for relations. (Contributed by
Mario Carneiro, 3Jan2017.)



Theorem  eqbrrdiv 4381* 
Deduction from extensionality principle for relations. (Contributed by
Rodolfo Medina, 10Oct2010.)



Theorem  eqrelrdv2 4382* 
A version of eqrelrdv 4379. (Contributed by Rodolfo Medina,
10Oct2010.)



Theorem  ssrelrel 4383* 
A subclass relationship determined by ordered triples. Use relrelss 4787
to express the antecedent in terms of the relation predicate.
(Contributed by NM, 17Dec2008.) (Proof shortened by Andrew Salmon,
27Aug2011.)



Theorem  eqrelrel 4384* 
Extensionality principle for ordered triples, analogous to eqrel 4372.
Use relrelss 4787 to express the antecedent in terms of the
relation
predicate. (Contributed by NM, 17Dec2008.)



Theorem  elrel 4385* 
A member of a relation is an ordered pair. (Contributed by NM,
17Sep2006.)



Theorem  relsn 4386 
A singleton is a relation iff it is an ordered pair. (Contributed by
NM, 24Sep2013.)



Theorem  relsnop 4387 
A singleton of an ordered pair is a relation. (Contributed by NM,
17May1998.) (Revised by Mario Carneiro, 26Apr2015.)



Theorem  xpss12 4388 
Subset theorem for cross product. Generalization of Theorem 101 of
[Suppes] p. 52. (Contributed by NM,
26Aug1995.) (Proof shortened by
Andrew Salmon, 27Aug2011.)



Theorem  xpss 4389 
A cross product is included in the ordered pair universe. Exercise 3 of
[TakeutiZaring] p. 25.
(Contributed by NM, 2Aug1994.)



Theorem  relxp 4390 
A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37.
(Contributed by NM, 2Aug1994.)



Theorem  xpss1 4391 
Subset relation for cross product. (Contributed by Jeff Hankins,
30Aug2009.)



Theorem  xpss2 4392 
Subset relation for cross product. (Contributed by Jeff Hankins,
30Aug2009.)



Theorem  xpsspw 4393 
A cross product is included in the power of the power of the union of
its arguments. (Contributed by NM, 13Sep2006.)



Theorem  unixpss 4394 
The double class union of a cross product is included in the union of its
arguments. (Contributed by NM, 16Sep2006.)



Theorem  xpexg 4395 
The cross product of two sets is a set. Proposition 6.2 of
[TakeutiZaring] p. 23. (Contributed
by NM, 14Aug1994.)



Theorem  xpex 4396 
The cross product of two sets is a set. Proposition 6.2 of
[TakeutiZaring] p. 23.
(Contributed by NM, 14Aug1994.)



Theorem  relun 4397 
The union of two relations is a relation. Compare Exercise 5 of
[TakeutiZaring] p. 25. (Contributed
by NM, 12Aug1994.)



Theorem  relin1 4398 
The intersection with a relation is a relation. (Contributed by NM,
16Aug1994.)



Theorem  relin2 4399 
The intersection with a relation is a relation. (Contributed by NM,
17Jan2006.)



Theorem  reldif 4400 
A difference cutting down a relation is a relation. (Contributed by NM,
31Mar1998.)

