Type | Label | Description |
Statement |
|
Theorem | f1ovi 5501 |
The identity relation is a one-to-one onto function on the universe.
(Contributed by NM, 16-May-2004.)
|
    |
|
Theorem | f1osn 5502 |
A singleton of an ordered pair is one-to-one onto function.
(Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon,
22-Oct-2011.)
|
              |
|
Theorem | f1osng 5503 |
A singleton of an ordered pair is one-to-one onto function.
(Contributed by Mario Carneiro, 12-Jan-2013.)
|
                  |
|
Theorem | f1sng 5504 |
A singleton of an ordered pair is a one-to-one function. (Contributed
by AV, 17-Apr-2021.)
|
                |
|
Theorem | fsnd 5505 |
A singleton of an ordered pair is a function. (Contributed by AV,
17-Apr-2021.)
|
                  |
|
Theorem | f1oprg 5506 |
An unordered pair of ordered pairs with different elements is a one-to-one
onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
|
    
 
                    
    |
|
Theorem | tz6.12-2 5507* |
Function value when
is not a function. Theorem 6.12(2) of
[TakeutiZaring] p. 27.
(Contributed by NM, 30-Apr-2004.) (Proof
shortened by Mario Carneiro, 31-Aug-2015.)
|
          |
|
Theorem | fveu 5508* |
The value of a function at a unique point. (Contributed by Scott
Fenton, 6-Oct-2017.)
|
   
           |
|
Theorem | brprcneu 5509* |
If is a proper class
and is any class,
then there is no
unique set which is related to through the binary relation .
(Contributed by Scott Fenton, 7-Oct-2017.)
|
      |
|
Theorem | fvprc 5510 |
A function's value at a proper class is the empty set. (Contributed by
NM, 20-May-1998.)
|
    
  |
|
Theorem | fv2 5511* |
Alternate definition of function value. Definition 10.11 of [Quine]
p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew
Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
|
   
      
   |
|
Theorem | dffv3g 5512* |
A definition of function value in terms of iota. (Contributed by Jim
Kingdon, 29-Dec-2018.)
|
    
            |
|
Theorem | dffv4g 5513* |
The previous definition of function value, from before the
operator was introduced. Although based on the idea embodied by
Definition 10.2 of [Quine] p. 65 (see args 4998), this definition
apparently does not appear in the literature. (Contributed by NM,
1-Aug-1994.)
|
    
             |
|
Theorem | elfv 5514* |
Membership in a function value. (Contributed by NM, 30-Apr-2004.)
|
                 |
|
Theorem | fveq1 5515 |
Equality theorem for function value. (Contributed by NM,
29-Dec-1996.)
|
    
      |
|
Theorem | fveq2 5516 |
Equality theorem for function value. (Contributed by NM,
29-Dec-1996.)
|
    
      |
|
Theorem | fveq1i 5517 |
Equality inference for function value. (Contributed by NM,
2-Sep-2003.)
|
   
     |
|
Theorem | fveq1d 5518 |
Equality deduction for function value. (Contributed by NM,
2-Sep-2003.)
|
             |
|
Theorem | fveq2i 5519 |
Equality inference for function value. (Contributed by NM,
28-Jul-1999.)
|
   
     |
|
Theorem | fveq2d 5520 |
Equality deduction for function value. (Contributed by NM,
29-May-1999.)
|
             |
|
Theorem | 2fveq3 5521 |
Equality theorem for nested function values. (Contributed by AV,
14-Aug-2022.)
|
                   |
|
Theorem | fveq12i 5522 |
Equality deduction for function value. (Contributed by FL,
27-Jun-2014.)
|
   
     |
|
Theorem | fveq12d 5523 |
Equality deduction for function value. (Contributed by FL,
22-Dec-2008.)
|
               |
|
Theorem | fveqeq2d 5524 |
Equality deduction for function value. (Contributed by BJ,
30-Aug-2022.)
|
       
   
   |
|
Theorem | fveqeq2 5525 |
Equality deduction for function value. (Contributed by BJ,
31-Aug-2022.)
|
     
       |
|
Theorem | nffv 5526 |
Bound-variable hypothesis builder for function value. (Contributed by
NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
|
           |
|
Theorem | nffvmpt1 5527* |
Bound-variable hypothesis builder for mapping, special case.
(Contributed by Mario Carneiro, 25-Dec-2016.)
|
         |
|
Theorem | nffvd 5528 |
Deduction version of bound-variable hypothesis builder nffv 5526.
(Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro,
15-Oct-2016.)
|
                 |
|
Theorem | funfveu 5529* |
A function has one value given an argument in its domain. (Contributed
by Jim Kingdon, 29-Dec-2018.)
|
        |
|
Theorem | fvss 5530* |
The value of a function is a subset of if every element that could
be a candidate for the value is a subset of . (Contributed by
Mario Carneiro, 24-May-2019.)
|
             |
|
Theorem | fvssunirng 5531 |
The result of a function value is always a subset of the union of the
range, if the input is a set. (Contributed by Stefan O'Rear,
2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
|
        |
|
Theorem | relfvssunirn 5532 |
The result of a function value is always a subset of the union of the
range, even if it is invalid and thus empty. (Contributed by Stefan
O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
|
        |
|
Theorem | funfvex 5533 |
The value of a function exists. A special case of Corollary 6.13 of
[TakeutiZaring] p. 27.
(Contributed by Jim Kingdon, 29-Dec-2018.)
|
      
  |
|
Theorem | relrnfvex 5534 |
If a function has a set range, then the function value exists
unconditional on the domain. (Contributed by Mario Carneiro,
24-May-2019.)
|
      
  |
|
Theorem | fvexg 5535 |
Evaluating a set function at a set exists. (Contributed by Mario
Carneiro and Jim Kingdon, 28-May-2019.)
|
      
  |
|
Theorem | fvex 5536 |
Evaluating a set function at a set exists. (Contributed by Mario
Carneiro and Jim Kingdon, 28-May-2019.)
|
   
 |
|
Theorem | sefvex 5537 |
If a function is set-like, then the function value exists if the input
does. (Contributed by Mario Carneiro, 24-May-2019.)
|
   Se     
  |
|
Theorem | fvifdc 5538 |
Move a conditional outside of a function. (Contributed by Jim Kingdon,
1-Jan-2022.)
|
DECID                         |
|
Theorem | fv3 5539* |
Alternate definition of the value of a function. Definition 6.11 of
[TakeutiZaring] p. 26.
(Contributed by NM, 30-Apr-2004.) (Revised by
Mario Carneiro, 31-Aug-2015.)
|
   
        
     |
|
Theorem | fvres 5540 |
The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
|
             |
|
Theorem | fvresd 5541 |
The value of a restricted function, deduction version of fvres 5540.
(Contributed by Glauco Siliprandi, 8-Apr-2021.)
|
               |
|
Theorem | funssfv 5542 |
The value of a member of the domain of a subclass of a function.
(Contributed by NM, 15-Aug-1994.)
|
 
    
      |
|
Theorem | tz6.12-1 5543* |
Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed
by NM, 30-Apr-2004.)
|
              |
|
Theorem | tz6.12 5544* |
Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed
by NM, 10-Jul-1994.)
|
                 |
|
Theorem | tz6.12f 5545* |
Function value, using bound-variable hypotheses instead of distinct
variable conditions. (Contributed by NM, 30-Aug-1999.)
|
                   |
|
Theorem | tz6.12c 5546* |
Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by
NM, 30-Apr-2004.)
|
        
     |
|
Theorem | ndmfvg 5547 |
The value of a class outside its domain is the empty set. (Contributed
by Jim Kingdon, 15-Jan-2019.)
|
 
       |
|
Theorem | relelfvdm 5548 |
If a function value has a member, the argument belongs to the domain.
(Contributed by Jim Kingdon, 22-Jan-2019.)
|
         |
|
Theorem | nfvres 5549 |
The value of a non-member of a restriction is the empty set.
(Contributed by NM, 13-Nov-1995.)
|
      
  |
|
Theorem | nfunsn 5550 |
If the restriction of a class to a singleton is not a function, its
value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof
shortened by Andrew Salmon, 22-Oct-2011.)
|
    
      |
|
Theorem | 0fv 5551 |
Function value of the empty set. (Contributed by Stefan O'Rear,
26-Nov-2014.)
|
     |
|
Theorem | csbfv12g 5552 |
Move class substitution in and out of a function value. (Contributed by
NM, 11-Nov-2005.)
|
   ![]_ ]_](_urbrack.gif)    
   ![]_ ]_](_urbrack.gif)     ![]_ ]_](_urbrack.gif)    |
|
Theorem | csbfv2g 5553* |
Move class substitution in and out of a function value. (Contributed by
NM, 10-Nov-2005.)
|
   ![]_ ]_](_urbrack.gif)    
     ![]_ ]_](_urbrack.gif)    |
|
Theorem | csbfvg 5554* |
Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
|
   ![]_ ]_](_urbrack.gif)    
      |
|
Theorem | funbrfv 5555 |
The second argument of a binary relation on a function is the function's
value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro,
28-Apr-2015.)
|
           |
|
Theorem | funopfv 5556 |
The second element in an ordered pair member of a function is the
function's value. (Contributed by NM, 19-Jul-1996.)
|
    
       |
|
Theorem | fnbrfvb 5557 |
Equivalence of function value and binary relation. (Contributed by NM,
19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
|
             |
|
Theorem | fnopfvb 5558 |
Equivalence of function value and ordered pair membership. (Contributed
by NM, 7-Nov-1995.)
|
              |
|
Theorem | funbrfvb 5559 |
Equivalence of function value and binary relation. (Contributed by NM,
26-Mar-2006.)
|
             |
|
Theorem | funopfvb 5560 |
Equivalence of function value and ordered pair membership. Theorem
4.3(ii) of [Monk1] p. 42. (Contributed by
NM, 26-Jan-1997.)
|
              |
|
Theorem | funbrfv2b 5561 |
Function value in terms of a binary relation. (Contributed by Mario
Carneiro, 19-Mar-2014.)
|
    
        |
|
Theorem | dffn5im 5562* |
Representation of a function in terms of its values. The converse holds
given the law of the excluded middle; as it is we have most of the
converse via funmpt 5255 and dmmptss 5126. (Contributed by Jim Kingdon,
31-Dec-2018.)
|
         |
|
Theorem | fnrnfv 5563* |
The range of a function expressed as a collection of the function's
values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario
Carneiro, 31-Aug-2015.)
|
          |
|
Theorem | fvelrnb 5564* |
A member of a function's range is a value of the function. (Contributed
by NM, 31-Oct-1995.)
|
  
       |
|
Theorem | dfimafn 5565* |
Alternate definition of the image of a function. (Contributed by Raph
Levien, 20-Nov-2006.)
|
 
    
         |
|
Theorem | dfimafn2 5566* |
Alternate definition of the image of a function as an indexed union of
singletons of function values. (Contributed by Raph Levien,
20-Nov-2006.)
|
 
    
         |
|
Theorem | funimass4 5567* |
Membership relation for the values of a function whose image is a
subclass. (Contributed by Raph Levien, 20-Nov-2006.)
|
 
     
        |
|
Theorem | fvelima 5568* |
Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42.
(Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon,
22-Oct-2011.)
|
              |
|
Theorem | foelcdmi 5569* |
A member of a surjective function's codomain is a value of the function.
(Contributed by Thierry Arnoux, 23-Jan-2020.)
|
     

    
  |
|
Theorem | feqmptd 5570* |
Deduction form of dffn5im 5562. (Contributed by Mario Carneiro,
8-Jan-2015.)
|
               |
|
Theorem | feqresmpt 5571* |
Express a restricted function as a mapping. (Contributed by Mario
Carneiro, 18-May-2016.)
|
                   |
|
Theorem | dffn5imf 5572* |
Representation of a function in terms of its values. (Contributed by
Jim Kingdon, 31-Dec-2018.)
|
           |
|
Theorem | fvelimab 5573* |
Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof
shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy,
17-Dec-2011.)
|
        
       |
|
Theorem | fvi 5574 |
The value of the identity function. (Contributed by NM, 1-May-2004.)
(Revised by Mario Carneiro, 28-Apr-2015.)
|
  
  |
|
Theorem | fniinfv 5575* |
The indexed intersection of a function's values is the intersection of
its range. (Contributed by NM, 20-Oct-2005.)
|
 
    
  |
|
Theorem | fnsnfv 5576 |
Singleton of function value. (Contributed by NM, 22-May-1998.)
|
                 |
|
Theorem | fnimapr 5577 |
The image of a pair under a function. (Contributed by Jeff Madsen,
6-Jan-2011.)
|
       
               |
|
Theorem | ssimaex 5578* |
The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
|
 
               |
|
Theorem | ssimaexg 5579* |
The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
|
 
               |
|
Theorem | funfvdm 5580 |
A simplified expression for the value of a function when we know it's a
function. (Contributed by Jim Kingdon, 1-Jan-2019.)
|
      
         |
|
Theorem | funfvdm2 5581* |
The value of a function. Definition of function value in [Enderton]
p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)
|
      
       |
|
Theorem | funfvdm2f 5582 |
The value of a function. Version of funfvdm2 5581 using a bound-variable
hypotheses instead of distinct variable conditions. (Contributed by Jim
Kingdon, 1-Jan-2019.)
|
     
            |
|
Theorem | fvun1 5583 |
The value of a union when the argument is in the first domain.
(Contributed by Scott Fenton, 29-Jun-2013.)
|
            
      |
|
Theorem | fvun2 5584 |
The value of a union when the argument is in the second domain.
(Contributed by Scott Fenton, 29-Jun-2013.)
|
            
      |
|
Theorem | dmfco 5585 |
Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
|
         
   |
|
Theorem | fvco2 5586 |
Value of a function composition. Similar to second part of Theorem 3H
of [Enderton] p. 47. (Contributed by
NM, 9-Oct-2004.) (Proof shortened
by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear,
16-Oct-2014.)
|
        
          |
|
Theorem | fvco 5587 |
Value of a function composition. Similar to Exercise 5 of [TakeutiZaring]
p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario
Carneiro, 26-Dec-2014.)
|
        
          |
|
Theorem | fvco3 5588 |
Value of a function composition. (Contributed by NM, 3-Jan-2004.)
(Revised by Mario Carneiro, 26-Dec-2014.)
|
      
                |
|
Theorem | fvco4 5589 |
Value of a composition. (Contributed by BJ, 7-Jul-2022.)
|
       

 
     
          |
|
Theorem | fvopab3g 5590* |
Value of a function given by ordered-pair class abstraction.
(Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro,
28-Apr-2015.)
|
        
                     |
|
Theorem | fvopab3ig 5591* |
Value of a function given by ordered-pair class abstraction.
(Contributed by NM, 23-Oct-1999.)
|
        
                     |
|
Theorem | fvmptss2 5592* |
A mapping always evaluates to a subset of the substituted expression in
the mapping, even if this is a proper class, or we are out of the
domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by
Mario Carneiro, 3-Jul-2019.)
|
 
     
 |
|
Theorem | fvmptg 5593* |
Value of a function given in maps-to notation. (Contributed by NM,
2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
|
 
           |
|
Theorem | fvmpt 5594* |
Value of a function given in maps-to notation. (Contributed by NM,
17-Aug-2011.)
|
 
      
  |
|
Theorem | fvmpts 5595* |
Value of a function given in maps-to notation, using explicit class
substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by
Mario Carneiro, 31-Aug-2015.)
|
      ![]_ ]_](_urbrack.gif)     
  ![]_ ]_](_urbrack.gif)   |
|
Theorem | fvmpt3 5596* |
Value of a function given in maps-to notation, with a slightly
different sethood condition. (Contributed by Stefan O'Rear,
30-Jan-2015.)
|
 
  
     
  |
|
Theorem | fvmpt3i 5597* |
Value of a function given in maps-to notation, with a slightly different
sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
|
 
      
  |
|
Theorem | fvmptd 5598* |
Deduction version of fvmpt 5594. (Contributed by Scott Fenton,
18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
|
     
             |
|
Theorem | mptrcl 5599* |
Reverse closure for a mapping: If the function value of a mapping has a
member, the argument belongs to the base class of the mapping.
(Contributed by AV, 4-Apr-2020.) (Revised by Jim Kingdon,
27-Mar-2023.)
|
  
   
  |
|
Theorem | fvmpt2 5600* |
Value of a function given by the maps-to notation. (Contributed by FL,
21-Jun-2010.)
|
    
      |