Type | Label | Description |
Statement |
|
Theorem | ofco 6101 |
The composition of a function operation with another function.
(Contributed by Mario Carneiro, 19-Dec-2014.)
|
                         
           |
|
Theorem | offveqb 6102* |
Equivalent expressions for equality with a function operation.
(Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro,
5-Dec-2016.)
|
              
  
       
     
           |
|
Theorem | ofc12 6103 |
Function operation on two constant functions. (Contributed by Mario
Carneiro, 28-Jul-2014.)
|
                              |
|
Theorem | caofref 6104* |
Transfer a reflexive law to the function relation. (Contributed by
Mario Carneiro, 28-Jul-2014.)
|
                    |
|
Theorem | caofinvl 6105* |
Transfer a left inverse law to the function operation. (Contributed
by NM, 22-Oct-2014.)
|
        
       

                      
           |
|
Theorem | caofcom 6106* |
Transfer a commutative law to the function operation. (Contributed by
Mario Carneiro, 26-Jul-2014.)
|
        
       
 
              
       |
|
Theorem | caofrss 6107* |
Transfer a relation subset law to the function relation. (Contributed
by Mario Carneiro, 28-Jul-2014.)
|
        
       
 
       
          |
|
Theorem | caoftrn 6108* |
Transfer a transitivity law to the function relation. (Contributed by
Mario Carneiro, 28-Jul-2014.)
|
        
             
 
           
               |
|
2.6.14 Functions (continued)
|
|
Theorem | resfunexgALT 6109 |
The restriction of a function to a set exists. Compare Proposition 6.17
of [TakeutiZaring] p. 28. This
version has a shorter proof than
resfunexg 5738 but requires ax-pow 4175 and ax-un 4434. (Contributed by NM,
7-Apr-1995.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
       |
|
Theorem | cofunexg 6110 |
Existence of a composition when the first member is a function.
(Contributed by NM, 8-Oct-2007.)
|
    
  |
|
Theorem | cofunex2g 6111 |
Existence of a composition when the second member is one-to-one.
(Contributed by NM, 8-Oct-2007.)
|
 
   
  |
|
Theorem | fnexALT 6112 |
If the domain of a function is a set, the function is a set. Theorem
6.16(1) of [TakeutiZaring] p. 28.
This theorem is derived using the Axiom
of Replacement in the form of funimaexg 5301. This version of fnex 5739
uses
ax-pow 4175 and ax-un 4434, whereas fnex 5739
does not. (Contributed by NM,
14-Aug-1994.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
     |
|
Theorem | funexw 6113 |
Weak version of funex 5740 that holds without ax-coll 4119. If the domain and
codomain of a function exist, so does the function. (Contributed by Rohan
Ridenour, 13-Aug-2023.)
|
     |
|
Theorem | mptexw 6114* |
Weak version of mptex 5743 that holds without ax-coll 4119. If the domain
and codomain of a function given by maps-to notation are sets, the
function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
|
 
  |
|
Theorem | funrnex 6115 |
If the domain of a function exists, so does its range. Part of Theorem
4.15(v) of [Monk1] p. 46. This theorem is
derived using the Axiom of
Replacement in the form of funex 5740. (Contributed by NM, 11-Nov-1995.)
|
     |
|
Theorem | focdmex 6116 |
If the domain of an onto function exists, so does its codomain.
(Contributed by NM, 23-Jul-2004.)
|
         |
|
Theorem | f1dmex 6117 |
If the codomain of a one-to-one function exists, so does its domain. This
can be thought of as a form of the Axiom of Replacement. (Contributed by
NM, 4-Sep-2004.)
|
     

  |
|
Theorem | abrexex 6118* |
Existence of a class abstraction of existentially restricted sets.
is normally a free-variable parameter in the class expression
substituted for , which can be thought of as    . This
simple-looking theorem is actually quite powerful and appears to involve
the Axiom of Replacement in an intrinsic way, as can be seen by tracing
back through the path mptexg 5742, funex 5740, fnex 5739, resfunexg 5738, and
funimaexg 5301. See also abrexex2 6125. (Contributed by NM, 16-Oct-2003.)
(Proof shortened by Mario Carneiro, 31-Aug-2015.)
|
 
  |
|
Theorem | abrexexg 6119* |
Existence of a class abstraction of existentially restricted sets.
is normally a free-variable parameter in . The antecedent assures
us that is a
set. (Contributed by NM, 3-Nov-2003.)
|
  
   |
|
Theorem | iunexg 6120* |
The existence of an indexed union. is normally a free-variable
parameter in .
(Contributed by NM, 23-Mar-2006.)
|
    
  |
|
Theorem | abrexex2g 6121* |
Existence of an existentially restricted class abstraction.
(Contributed by Jeff Madsen, 2-Sep-2009.)
|
    
      |
|
Theorem | opabex3d 6122* |
Existence of an ordered pair abstraction, deduction version.
(Contributed by Alexander van der Vekens, 19-Oct-2017.)
|
                  |
|
Theorem | opabex3 6123* |
Existence of an ordered pair abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
|

         
 |
|
Theorem | iunex 6124* |
The existence of an indexed union. is normally a free-variable
parameter in the class expression substituted for , which can be
read informally as    . (Contributed by NM, 13-Oct-2003.)
|

 |
|
Theorem | abrexex2 6125* |
Existence of an existentially restricted class abstraction. is
normally has free-variable parameters and . See also
abrexex 6118. (Contributed by NM, 12-Sep-2004.)
|
      |
|
Theorem | abexssex 6126* |
Existence of a class abstraction with an existentially quantified
expression. Both and can be
free in .
(Contributed
by NM, 29-Jul-2006.)
|
       
 |
|
Theorem | abexex 6127* |
A condition where a class builder continues to exist after its wff is
existentially quantified. (Contributed by NM, 4-Mar-2007.)
|
         |
|
Theorem | oprabexd 6128* |
Existence of an operator abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
|
     
             
 
       |
|
Theorem | oprabex 6129* |
Existence of an operation class abstraction. (Contributed by NM,
19-Oct-2004.)
|
            
 
    |
|
Theorem | oprabex3 6130* |
Existence of an operation class abstraction (special case).
(Contributed by NM, 19-Oct-2004.)
|
          
               
        |
|
Theorem | oprabrexex2 6131* |
Existence of an existentially restricted operation abstraction.
(Contributed by Jeff Madsen, 11-Jun-2010.)
|
   
  
        
  |
|
Theorem | ab2rexex 6132* |
Existence of a class abstraction of existentially restricted sets.
Variables and
are normally
free-variable parameters in the
class expression substituted for , which can be thought of as
    . See comments for abrexex 6118. (Contributed by NM,
20-Sep-2011.)
|
 
 
 |
|
Theorem | ab2rexex2 6133* |
Existence of an existentially restricted class abstraction.
normally has free-variable parameters , , and .
Compare abrexex2 6125. (Contributed by NM, 20-Sep-2011.)
|
 
  
  |
|
Theorem | xpexgALT 6134 |
The cross product of two sets is a set. Proposition 6.2 of
[TakeutiZaring] p. 23. This
version is proven using Replacement; see
xpexg 4741 for a version that uses the Power Set axiom
instead.
(Contributed by Mario Carneiro, 20-May-2013.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
    
  |
|
Theorem | offval3 6135* |
General value of      with no assumptions on functionality
of and . (Contributed by Stefan
O'Rear, 24-Jan-2015.)
|
                          |
|
Theorem | offres 6136 |
Pointwise combination commutes with restriction. (Contributed by Stefan
O'Rear, 24-Jan-2015.)
|
                
    |
|
Theorem | ofmres 6137* |
Equivalent expressions for a restriction of the function operation map.
Unlike   which is a proper class,   
  can
be a set by ofmresex 6138, allowing it to be used as a function or
structure argument. By ofmresval 6094, the restricted operation map
values are the same as the original values, allowing theorems for
  to be reused. (Contributed by NM, 20-Oct-2014.)
|
    
 
       |
|
Theorem | ofmresex 6138 |
Existence of a restriction of the function operation map. (Contributed
by NM, 20-Oct-2014.)
|
            |
|
2.6.15 First and second members of an ordered
pair
|
|
Syntax | c1st 6139 |
Extend the definition of a class to include the first member an ordered
pair function.
|
 |
|
Syntax | c2nd 6140 |
Extend the definition of a class to include the second member an ordered
pair function.
|
 |
|
Definition | df-1st 6141 |
Define a function that extracts the first member, or abscissa, of an
ordered pair. Theorem op1st 6147 proves that it does this. For example,
(  3 , 4 ) = 3 . Equivalent to Definition
5.13 (i) of
[Monk1] p. 52 (compare op1sta 5111 and op1stb 4479). The notation is the same
as Monk's. (Contributed by NM, 9-Oct-2004.)
|
      |
|
Definition | df-2nd 6142 |
Define a function that extracts the second member, or ordinate, of an
ordered pair. Theorem op2nd 6148 proves that it does this. For example,
   3 , 4 ) = 4 . Equivalent to Definition 5.13 (ii)
of [Monk1] p. 52 (compare op2nda 5114 and op2ndb 5113). The notation is the
same as Monk's. (Contributed by NM, 9-Oct-2004.)
|
      |
|
Theorem | 1stvalg 6143 |
The value of the function that extracts the first member of an ordered
pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro,
8-Sep-2013.)
|
    
     |
|
Theorem | 2ndvalg 6144 |
The value of the function that extracts the second member of an ordered
pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro,
8-Sep-2013.)
|
    
     |
|
Theorem | 1st0 6145 |
The value of the first-member function at the empty set. (Contributed by
NM, 23-Apr-2007.)
|
     |
|
Theorem | 2nd0 6146 |
The value of the second-member function at the empty set. (Contributed by
NM, 23-Apr-2007.)
|
     |
|
Theorem | op1st 6147 |
Extract the first member of an ordered pair. (Contributed by NM,
5-Oct-2004.)
|
        |
|
Theorem | op2nd 6148 |
Extract the second member of an ordered pair. (Contributed by NM,
5-Oct-2004.)
|
        |
|
Theorem | op1std 6149 |
Extract the first member of an ordered pair. (Contributed by Mario
Carneiro, 31-Aug-2015.)
|
          |
|
Theorem | op2ndd 6150 |
Extract the second member of an ordered pair. (Contributed by Mario
Carneiro, 31-Aug-2015.)
|
          |
|
Theorem | op1stg 6151 |
Extract the first member of an ordered pair. (Contributed by NM,
19-Jul-2005.)
|
            |
|
Theorem | op2ndg 6152 |
Extract the second member of an ordered pair. (Contributed by NM,
19-Jul-2005.)
|
            |
|
Theorem | ot1stg 6153 |
Extract the first member of an ordered triple. (Due to infrequent
usage, it isn't worthwhile at this point to define special extractors
for triples, so we reuse the ordered pair extractors for ot1stg 6153,
ot2ndg 6154, ot3rdgg 6155.) (Contributed by NM, 3-Apr-2015.) (Revised
by
Mario Carneiro, 2-May-2015.)
|
                 |
|
Theorem | ot2ndg 6154 |
Extract the second member of an ordered triple. (See ot1stg 6153 comment.)
(Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro,
2-May-2015.)
|
                 |
|
Theorem | ot3rdgg 6155 |
Extract the third member of an ordered triple. (See ot1stg 6153 comment.)
(Contributed by NM, 3-Apr-2015.)
|
        
    |
|
Theorem | 1stval2 6156 |
Alternate value of the function that extracts the first member of an
ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by
NM, 18-Aug-2006.)
|
           |
|
Theorem | 2ndval2 6157 |
Alternate value of the function that extracts the second member of an
ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by
NM, 18-Aug-2006.)
|
               |
|
Theorem | fo1st 6158 |
The function
maps the universe onto the universe. (Contributed
by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
|
     |
|
Theorem | fo2nd 6159 |
The function
maps the universe onto the universe. (Contributed
by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
|
     |
|
Theorem | f1stres 6160 |
Mapping of a restriction of the (first member of an ordered
pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario
Carneiro, 8-Sep-2013.)
|
           |
|
Theorem | f2ndres 6161 |
Mapping of a restriction of the (second member of an ordered
pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario
Carneiro, 8-Sep-2013.)
|
           |
|
Theorem | fo1stresm 6162* |
Onto mapping of a restriction of the (first member of an ordered
pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
|
              |
|
Theorem | fo2ndresm 6163* |
Onto mapping of a restriction of the (second member of an
ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
|
              |
|
Theorem | 1stcof 6164 |
Composition of the first member function with another function.
(Contributed by NM, 12-Oct-2007.)
|
     
         |
|
Theorem | 2ndcof 6165 |
Composition of the second member function with another function.
(Contributed by FL, 15-Oct-2012.)
|
     
         |
|
Theorem | xp1st 6166 |
Location of the first element of a Cartesian product. (Contributed by
Jeff Madsen, 2-Sep-2009.)
|
         |
|
Theorem | xp2nd 6167 |
Location of the second element of a Cartesian product. (Contributed by
Jeff Madsen, 2-Sep-2009.)
|
         |
|
Theorem | 1stexg 6168 |
Existence of the first member of a set. (Contributed by Jim Kingdon,
26-Jan-2019.)
|
    
  |
|
Theorem | 2ndexg 6169 |
Existence of the first member of a set. (Contributed by Jim Kingdon,
26-Jan-2019.)
|
    
  |
|
Theorem | elxp6 6170 |
Membership in a cross product. This version requires no quantifiers or
dummy variables. See also elxp4 5117. (Contributed by NM, 9-Oct-2004.)
|
         
                  |
|
Theorem | elxp7 6171 |
Membership in a cross product. This version requires no quantifiers or
dummy variables. See also elxp4 5117. (Contributed by NM, 19-Aug-2006.)
|
          
        |
|
Theorem | oprssdmm 6172* |
Domain of closure of an operation. (Contributed by Jim Kingdon,
23-Oct-2023.)
|
   
  
        
   
  |
|
Theorem | eqopi 6173 |
Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.)
(Revised by Mario Carneiro, 23-Feb-2014.)
|
  
         
 
     |
|
Theorem | xp2 6174* |
Representation of cross product based on ordered pair component
functions. (Contributed by NM, 16-Sep-2006.)
|
 
  
            |
|
Theorem | unielxp 6175 |
The membership relation for a cross product is inherited by union.
(Contributed by NM, 16-Sep-2006.)
|
   
     |
|
Theorem | 1st2nd2 6176 |
Reconstruction of a member of a cross product in terms of its ordered pair
components. (Contributed by NM, 20-Oct-2013.)
|
  
             |
|
Theorem | xpopth 6177 |
An ordered pair theorem for members of cross products. (Contributed by
NM, 20-Jun-2007.)
|
  
                 
    
   |
|
Theorem | eqop 6178 |
Two ways to express equality with an ordered pair. (Contributed by NM,
3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
|
   
           
    |
|
Theorem | eqop2 6179 |
Two ways to express equality with an ordered pair. (Contributed by NM,
25-Feb-2014.)
|
           
        |
|
Theorem | op1steq 6180* |
Two ways of expressing that an element is the first member of an ordered
pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro,
23-Feb-2014.)
|
       
       |
|
Theorem | 2nd1st 6181 |
Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
|
            
       |
|
Theorem | 1st2nd 6182 |
Reconstruction of a member of a relation in terms of its ordered pair
components. (Contributed by NM, 29-Aug-2006.)
|
                |
|
Theorem | 1stdm 6183 |
The first ordered pair component of a member of a relation belongs to the
domain of the relation. (Contributed by NM, 17-Sep-2006.)
|
      
  |
|
Theorem | 2ndrn 6184 |
The second ordered pair component of a member of a relation belongs to the
range of the relation. (Contributed by NM, 17-Sep-2006.)
|
      
  |
|
Theorem | 1st2ndbr 6185 |
Express an element of a relation as a relationship between first and
second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
|
               |
|
Theorem | releldm2 6186* |
Two ways of expressing membership in the domain of a relation.
(Contributed by NM, 22-Sep-2013.)
|
          |
|
Theorem | reldm 6187* |
An expression for the domain of a relation. (Contributed by NM,
22-Sep-2013.)
|
 
       |
|
Theorem | sbcopeq1a 6188 |
Equality theorem for substitution of a class for an ordered pair (analog
of sbceq1a 2973 that avoids the existential quantifiers of copsexg 4245).
(Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro,
31-Aug-2015.)
|
           ![]. ].](_drbrack.gif)       ![]. ].](_drbrack.gif)
   |
|
Theorem | csbopeq1a 6189 |
Equality theorem for substitution of a class for an ordered pair
  
in (analog of csbeq1a 3067). (Contributed by NM,
19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
|
          ![]_ ]_](_urbrack.gif)       ![]_ ]_](_urbrack.gif)
  |
|
Theorem | dfopab2 6190* |
A way to define an ordered-pair class abstraction without using
existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by
Mario Carneiro, 31-Aug-2015.)
|
     
      
 ![]. ].](_drbrack.gif)       ![]. ].](_drbrack.gif)   |
|
Theorem | dfoprab3s 6191* |
A way to define an operation class abstraction without using existential
quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario
Carneiro, 31-Aug-2015.)
|
              
      ![]. ].](_drbrack.gif)       ![]. ].](_drbrack.gif)    |
|
Theorem | dfoprab3 6192* |
Operation class abstraction expressed without existential quantifiers.
(Contributed by NM, 16-Dec-2008.)
|
             
 
         |
|
Theorem | dfoprab4 6193* |
Operation class abstraction expressed without existential quantifiers.
(Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro,
31-Aug-2015.)
|
             
 
             |
|
Theorem | dfoprab4f 6194* |
Operation class abstraction expressed without existential quantifiers.
(Unnecessary distinct variable restrictions were removed by David
Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by
Mario Carneiro, 31-Aug-2015.)
|
    
            
 
             |
|
Theorem | dfxp3 6195* |
Define the cross product of three classes. Compare df-xp 4633.
(Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro,
3-Nov-2015.)
|
  
        
   |
|
Theorem | elopabi 6196* |
A consequence of membership in an ordered-pair class abstraction, using
ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
|
                     
  |
|
Theorem | eloprabi 6197* |
A consequence of membership in an operation class abstraction, using
ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by
David Abernethy, 19-Jun-2012.)
|
                    
                      |
|
Theorem | mpomptsx 6198* |
Express a two-argument function as a one-argument function, or
vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
|
 

            ![]_ ]_](_urbrack.gif)       ![]_ ]_](_urbrack.gif)   |
|
Theorem | mpompts 6199* |
Express a two-argument function as a one-argument function, or
vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
|
 

         ![]_ ]_](_urbrack.gif)       ![]_ ]_](_urbrack.gif)   |
|
Theorem | dmmpossx 6200* |
The domain of a mapping is a subset of its base class. (Contributed by
Mario Carneiro, 9-Feb-2015.)
|
  
      |