Type | Label | Description |
Statement |
|
Theorem | fimacnvdisj 5401 |
The preimage of a class disjoint with a mapping's codomain is empty.
(Contributed by FL, 24-Jan-2007.)
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|
Theorem | fintm 5402* |
Function into an intersection. (Contributed by Jim Kingdon,
28-Dec-2018.)
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|
Theorem | fin 5403 |
Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
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|
Theorem | fabexg 5404* |
Existence of a set of functions. (Contributed by Paul Chapman,
25-Feb-2008.)
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|
Theorem | fabex 5405* |
Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
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        |
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Theorem | dmfex 5406 |
If a mapping is a set, its domain is a set. (Contributed by NM,
27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|
      
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|
Theorem | f0 5407 |
The empty function. (Contributed by NM, 14-Aug-1999.)
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|
Theorem | f00 5408 |
A class is a function with empty codomain iff it and its domain are empty.
(Contributed by NM, 10-Dec-2003.)
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|
Theorem | f0bi 5409 |
A function with empty domain is empty. (Contributed by Alexander van der
Vekens, 30-Jun-2018.)
|
    
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|
Theorem | f0dom0 5410 |
A function is empty iff it has an empty domain. (Contributed by AV,
10-Feb-2019.)
|
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|
Theorem | f0rn0 5411* |
If there is no element in the range of a function, its domain must be
empty. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
|
      

  |
|
Theorem | fconst 5412 |
A cross product with a singleton is a constant function. (Contributed
by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon,
17-Sep-2011.)
|
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|
Theorem | fconstg 5413 |
A cross product with a singleton is a constant function. (Contributed
by NM, 19-Oct-2004.)
|
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|
Theorem | fnconstg 5414 |
A cross product with a singleton is a constant function. (Contributed by
NM, 24-Jul-2014.)
|
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|
Theorem | fconst6g 5415 |
Constant function with loose range. (Contributed by Stefan O'Rear,
1-Feb-2015.)
|
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|
Theorem | fconst6 5416 |
A constant function as a mapping. (Contributed by Jeff Madsen,
30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
|
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|
Theorem | f1eq1 5417 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
|
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|
Theorem | f1eq2 5418 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
|
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|
Theorem | f1eq3 5419 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
|
             |
|
Theorem | nff1 5420 |
Bound-variable hypothesis builder for a one-to-one function.
(Contributed by NM, 16-May-2004.)
|
            |
|
Theorem | dff12 5421* |
Alternate definition of a one-to-one function. (Contributed by NM,
31-Dec-1996.)
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|
Theorem | f1f 5422 |
A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
|
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|
Theorem | f1rn 5423 |
The range of a one-to-one mapping. (Contributed by BJ, 6-Jul-2022.)
|
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|
Theorem | f1fn 5424 |
A one-to-one mapping is a function on its domain. (Contributed by NM,
8-Mar-2014.)
|
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|
Theorem | f1fun 5425 |
A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
|
       |
|
Theorem | f1rel 5426 |
A one-to-one onto mapping is a relation. (Contributed by NM,
8-Mar-2014.)
|
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|
Theorem | f1dm 5427 |
The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
|
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|
Theorem | f1ss 5428 |
A function that is one-to-one is also one-to-one on some superset of its
range. (Contributed by Mario Carneiro, 12-Jan-2013.)
|
     

      |
|
Theorem | f1ssr 5429 |
Combine a one-to-one function with a restriction on the domain.
(Contributed by Stefan O'Rear, 20-Feb-2015.)
|
     
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|
Theorem | f1ff1 5430 |
If a function is one-to-one from to and is
also a function
from to , then it is a one-to-one
function from to
. (Contributed
by BJ, 4-Jul-2022.)
|
     
           |
|
Theorem | f1ssres 5431 |
A function that is one-to-one is also one-to-one on any subclass of its
domain. (Contributed by Mario Carneiro, 17-Jan-2015.)
|
     

        |
|
Theorem | f1resf1 5432 |
The restriction of an injective function is injective. (Contributed by
AV, 28-Jun-2022.)
|
        
              |
|
Theorem | f1cnvcnv 5433 |
Two ways to express that a set (not necessarily a function) is
one-to-one. Each side is equivalent to Definition 6.4(3) of
[TakeutiZaring] p. 24, who use the
notation "Un2 (A)" for one-to-one.
We
do not introduce a separate notation since we rarely use it. (Contributed
by NM, 13-Aug-2004.)
|
        
     |
|
Theorem | f1co 5434 |
Composition of one-to-one functions. Exercise 30 of [TakeutiZaring]
p. 25. (Contributed by NM, 28-May-1998.)
|
     
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|
Theorem | foeq1 5435 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|
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|
Theorem | foeq2 5436 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|
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|
Theorem | foeq3 5437 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|
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|
Theorem | nffo 5438 |
Bound-variable hypothesis builder for an onto function. (Contributed by
NM, 16-May-2004.)
|
            |
|
Theorem | fof 5439 |
An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
|
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|
Theorem | fofun 5440 |
An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
|
    
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|
Theorem | fofn 5441 |
An onto mapping is a function on its domain. (Contributed by NM,
16-Dec-2008.)
|
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|
Theorem | forn 5442 |
The codomain of an onto function is its range. (Contributed by NM,
3-Aug-1994.)
|
    
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|
Theorem | dffo2 5443 |
Alternate definition of an onto function. (Contributed by NM,
22-Mar-2006.)
|
         
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|
Theorem | foima 5444 |
The image of the domain of an onto function. (Contributed by NM,
29-Nov-2002.)
|
        
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|
Theorem | dffn4 5445 |
A function maps onto its range. (Contributed by NM, 10-May-1998.)
|
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|
Theorem | funforn 5446 |
A function maps its domain onto its range. (Contributed by NM,
23-Jul-2004.)
|
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|
Theorem | fodmrnu 5447 |
An onto function has unique domain and range. (Contributed by NM,
5-Nov-2006.)
|
     
     
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|
Theorem | fores 5448 |
Restriction of a function. (Contributed by NM, 4-Mar-1997.)
|
 
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Theorem | foco 5449 |
Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
|
     
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|
Theorem | f1oeq1 5450 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
|
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|
Theorem | f1oeq2 5451 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
|
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|
Theorem | f1oeq3 5452 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
|
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|
Theorem | f1oeq23 5453 |
Equality theorem for one-to-one onto functions. (Contributed by FL,
14-Jul-2012.)
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|
Theorem | f1eq123d 5454 |
Equality deduction for one-to-one functions. (Contributed by Mario
Carneiro, 27-Jan-2017.)
|
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|
Theorem | foeq123d 5455 |
Equality deduction for onto functions. (Contributed by Mario Carneiro,
27-Jan-2017.)
|
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|
Theorem | f1oeq123d 5456 |
Equality deduction for one-to-one onto functions. (Contributed by Mario
Carneiro, 27-Jan-2017.)
|
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|
Theorem | f1oeq1d 5457 |
Equality deduction for one-to-one onto functions. (Contributed by
Glauco Siliprandi, 17-Aug-2020.)
|
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|
Theorem | f1oeq2d 5458 |
Equality deduction for one-to-one onto functions. (Contributed by
Glauco Siliprandi, 17-Aug-2020.)
|
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|
Theorem | f1oeq3d 5459 |
Equality deduction for one-to-one onto functions. (Contributed by
Glauco Siliprandi, 17-Aug-2020.)
|
               |
|
Theorem | nff1o 5460 |
Bound-variable hypothesis builder for a one-to-one onto function.
(Contributed by NM, 16-May-2004.)
|
            |
|
Theorem | f1of1 5461 |
A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM,
12-Dec-2003.)
|
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|
Theorem | f1of 5462 |
A one-to-one onto mapping is a mapping. (Contributed by NM,
12-Dec-2003.)
|
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|
Theorem | f1ofn 5463 |
A one-to-one onto mapping is function on its domain. (Contributed by NM,
12-Dec-2003.)
|
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|
Theorem | f1ofun 5464 |
A one-to-one onto mapping is a function. (Contributed by NM,
12-Dec-2003.)
|
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|
Theorem | f1orel 5465 |
A one-to-one onto mapping is a relation. (Contributed by NM,
13-Dec-2003.)
|
       |
|
Theorem | f1odm 5466 |
The domain of a one-to-one onto mapping. (Contributed by NM,
8-Mar-2014.)
|
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|
Theorem | dff1o2 5467 |
Alternate definition of one-to-one onto function. (Contributed by NM,
10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|
      
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|
Theorem | dff1o3 5468 |
Alternate definition of one-to-one onto function. (Contributed by NM,
25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|
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|
Theorem | f1ofo 5469 |
A one-to-one onto function is an onto function. (Contributed by NM,
28-Apr-2004.)
|
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|
Theorem | dff1o4 5470 |
Alternate definition of one-to-one onto function. (Contributed by NM,
25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|
          |
|
Theorem | dff1o5 5471 |
Alternate definition of one-to-one onto function. (Contributed by NM,
10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|
             |
|
Theorem | f1orn 5472 |
A one-to-one function maps onto its range. (Contributed by NM,
13-Aug-2004.)
|
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|
Theorem | f1f1orn 5473 |
A one-to-one function maps one-to-one onto its range. (Contributed by NM,
4-Sep-2004.)
|
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|
Theorem | f1oabexg 5474* |
The class of all 1-1-onto functions mapping one set to another is a set.
(Contributed by Paul Chapman, 25-Feb-2008.)
|
             |
|
Theorem | f1ocnv 5475 |
The converse of a one-to-one onto function is also one-to-one onto.
(Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon,
22-Oct-2011.)
|
            |
|
Theorem | f1ocnvb 5476 |
A relation is a one-to-one onto function iff its converse is a one-to-one
onto function with domain and codomain/range interchanged. (Contributed
by NM, 8-Dec-2003.)
|
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|
Theorem | f1ores 5477 |
The restriction of a one-to-one function maps one-to-one onto the image.
(Contributed by NM, 25-Mar-1998.)
|
     

            |
|
Theorem | f1orescnv 5478 |
The converse of a one-to-one-onto restricted function. (Contributed by
Paul Chapman, 21-Apr-2008.)
|
           
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|
Theorem | f1imacnv 5479 |
Preimage of an image. (Contributed by NM, 30-Sep-2004.)
|
     

           |
|
Theorem | foimacnv 5480 |
A reverse version of f1imacnv 5479. (Contributed by Jeff Hankins,
16-Jul-2009.)
|
     

           |
|
Theorem | foun 5481 |
The union of two onto functions with disjoint domains is an onto function.
(Contributed by Mario Carneiro, 22-Jun-2016.)
|
               
           |
|
Theorem | f1oun 5482 |
The union of two one-to-one onto functions with disjoint domains and
ranges. (Contributed by NM, 26-Mar-1998.)
|
           
  
   
        
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|
Theorem | fun11iun 5483* |
The union of a chain (with respect to inclusion) of one-to-one functions
is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.)
(Revised by Mario Carneiro, 24-Jun-2015.)
|
 
      
   
   
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Theorem | resdif 5484 |
The restriction of a one-to-one onto function to a difference maps onto
the difference of the images. (Contributed by Paul Chapman,
11-Apr-2009.)
|
         
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|
Theorem | f1oco 5485 |
Composition of one-to-one onto functions. (Contributed by NM,
19-Mar-1998.)
|
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|
Theorem | f1cnv 5486 |
The converse of an injective function is bijective. (Contributed by FL,
11-Nov-2011.)
|
            |
|
Theorem | funcocnv2 5487 |
Composition with the converse. (Contributed by Jeff Madsen,
2-Sep-2009.)
|
   
   |
|
Theorem | fococnv2 5488 |
The composition of an onto function and its converse. (Contributed by
Stefan O'Rear, 12-Feb-2015.)
|
       
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|
Theorem | f1ococnv2 5489 |
The composition of a one-to-one onto function and its converse equals the
identity relation restricted to the function's range. (Contributed by NM,
13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.)
|
       
   |
|
Theorem | f1cocnv2 5490 |
Composition of an injective function with its converse. (Contributed by
FL, 11-Nov-2011.)
|
       
   |
|
Theorem | f1ococnv1 5491 |
The composition of a one-to-one onto function's converse and itself equals
the identity relation restricted to the function's domain. (Contributed
by NM, 13-Dec-2003.)
|
       
   |
|
Theorem | f1cocnv1 5492 |
Composition of an injective function with its converse. (Contributed by
FL, 11-Nov-2011.)
|
       
   |
|
Theorem | funcoeqres 5493 |
Express a constraint on a composition as a constraint on the composand.
(Contributed by Stefan O'Rear, 7-Mar-2015.)
|
   
  
     |
|
Theorem | ffoss 5494* |
Relationship between a mapping and an onto mapping. Figure 38 of
[Enderton] p. 145. (Contributed by NM,
10-May-1998.)
|
               |
|
Theorem | f11o 5495* |
Relationship between one-to-one and one-to-one onto function.
(Contributed by NM, 4-Apr-1998.)
|
    
          |
|
Theorem | f10 5496 |
The empty set maps one-to-one into any class. (Contributed by NM,
7-Apr-1998.)
|
     |
|
Theorem | f1o00 5497 |
One-to-one onto mapping of the empty set. (Contributed by NM,
15-Apr-1998.)
|
         |
|
Theorem | fo00 5498 |
Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
|
         |
|
Theorem | f1o0 5499 |
One-to-one onto mapping of the empty set. (Contributed by NM,
10-Sep-2004.)
|
     |
|
Theorem | f1oi 5500 |
A restriction of the identity relation is a one-to-one onto function.
(Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon,
22-Oct-2011.)
|
      |