Type | Label | Description |
Statement |
|
Theorem | funimaexglem 5301 |
Lemma for funimaexg 5302. It constitutes the interesting part of
funimaexg 5302, in which
. (Contributed by Jim
Kingdon,
27-Dec-2018.)
|
 
    
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|
Theorem | funimaexg 5302 |
Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284.
Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM,
10-Sep-2006.)
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|
Theorem | funimaex 5303 |
The image of a set under any function is also a set. Equivalent of
Axiom of Replacement. Axiom 39(vi) of [Quine] p. 284. Compare Exercise
9 of [TakeutiZaring] p. 29.
(Contributed by NM, 17-Nov-2002.)
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|
Theorem | isarep1 5304* |
Part of a study of the Axiom of Replacement used by the Isabelle prover.
The object PrimReplace is apparently the image of the function encoded
by     i.e. the class          .
If so, we can prove Isabelle's "Axiom of Replacement"
conclusion without
using the Axiom of Replacement, for which I (N. Megill) currently have
no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by
Mario Carneiro, 4-Dec-2016.)
|
         
 
 ![] ]](rbrack.gif)   |
|
Theorem | isarep2 5305* |
Part of a study of the Axiom of Replacement used by the Isabelle prover.
In Isabelle, the sethood of PrimReplace is apparently postulated
implicitly by its type signature " i, i, i
=> o
=> i", which automatically asserts that it is a set without
using any
axioms. To prove that it is a set in Metamath, we need the hypotheses
of Isabelle's "Axiom of Replacement" as well as the Axiom of
Replacement
in the form funimaex 5303. (Contributed by NM, 26-Oct-2006.)
|
         ![] ]](rbrack.gif)              |
|
Theorem | fneq1 5306 |
Equality theorem for function predicate with domain. (Contributed by NM,
1-Aug-1994.)
|
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|
Theorem | fneq2 5307 |
Equality theorem for function predicate with domain. (Contributed by NM,
1-Aug-1994.)
|
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|
Theorem | fneq1d 5308 |
Equality deduction for function predicate with domain. (Contributed by
Paul Chapman, 22-Jun-2011.)
|
   
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|
Theorem | fneq2d 5309 |
Equality deduction for function predicate with domain. (Contributed by
Paul Chapman, 22-Jun-2011.)
|
   
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|
Theorem | fneq12d 5310 |
Equality deduction for function predicate with domain. (Contributed by
NM, 26-Jun-2011.)
|
     
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|
Theorem | fneq12 5311 |
Equality theorem for function predicate with domain. (Contributed by
Thierry Arnoux, 31-Jan-2017.)
|
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|
Theorem | fneq1i 5312 |
Equality inference for function predicate with domain. (Contributed by
Paul Chapman, 22-Jun-2011.)
|
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|
Theorem | fneq2i 5313 |
Equality inference for function predicate with domain. (Contributed by
NM, 4-Sep-2011.)
|
   |
|
Theorem | nffn 5314 |
Bound-variable hypothesis builder for a function with domain.
(Contributed by NM, 30-Jan-2004.)
|
      |
|
Theorem | fnfun 5315 |
A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
|
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|
Theorem | fnrel 5316 |
A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
|
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|
Theorem | fndm 5317 |
The domain of a function. (Contributed by NM, 2-Aug-1994.)
|
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|
Theorem | funfni 5318 |
Inference to convert a function and domain antecedent. (Contributed by
NM, 22-Apr-2004.)
|
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|
Theorem | fndmu 5319 |
A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
|
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|
Theorem | fnbr 5320 |
The first argument of binary relation on a function belongs to the
function's domain. (Contributed by NM, 7-May-2004.)
|
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|
Theorem | fnop 5321 |
The first argument of an ordered pair in a function belongs to the
function's domain. (Contributed by NM, 8-Aug-1994.)
|
     
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|
Theorem | fneu 5322* |
There is exactly one value of a function. (Contributed by NM,
22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|
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|
Theorem | fneu2 5323* |
There is exactly one value of a function. (Contributed by NM,
7-Nov-1995.)
|
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|
Theorem | fnun 5324 |
The union of two functions with disjoint domains. (Contributed by NM,
22-Sep-2004.)
|
      
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|
Theorem | fnunsn 5325 |
Extension of a function with a new ordered pair. (Contributed by NM,
28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
|
                   
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|
Theorem | fnco 5326 |
Composition of two functions. (Contributed by NM, 22-May-2006.)
|
    
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|
Theorem | fnresdm 5327 |
A function does not change when restricted to its domain. (Contributed by
NM, 5-Sep-2004.)
|
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|
Theorem | fnresdisj 5328 |
A function restricted to a class disjoint with its domain is empty.
(Contributed by NM, 23-Sep-2004.)
|
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|
Theorem | 2elresin 5329 |
Membership in two functions restricted by each other's domain.
(Contributed by NM, 8-Aug-1994.)
|
                   
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|
Theorem | fnssresb 5330 |
Restriction of a function with a subclass of its domain. (Contributed by
NM, 10-Oct-2007.)
|
   
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|
Theorem | fnssres 5331 |
Restriction of a function with a subclass of its domain. (Contributed by
NM, 2-Aug-1994.)
|
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|
Theorem | fnresin1 5332 |
Restriction of a function's domain with an intersection. (Contributed by
NM, 9-Aug-1994.)
|
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|
Theorem | fnresin2 5333 |
Restriction of a function's domain with an intersection. (Contributed by
NM, 9-Aug-1994.)
|
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|
Theorem | fnres 5334* |
An equivalence for functionality of a restriction. Compare dffun8 5246.
(Contributed by Mario Carneiro, 20-May-2015.)
|
    
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|
Theorem | fnresi 5335 |
Functionality and domain of restricted identity. (Contributed by NM,
27-Aug-2004.)
|
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|
Theorem | fnima 5336 |
The image of a function's domain is its range. (Contributed by NM,
4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|
    
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|
Theorem | fn0 5337 |
A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.)
(Proof shortened by Andrew Salmon, 17-Sep-2011.)
|

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|
Theorem | fnimadisj 5338 |
A class that is disjoint with the domain of a function has an empty image
under the function. (Contributed by FL, 24-Jan-2007.)
|
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|
Theorem | fnimaeq0 5339 |
Images under a function never map nonempty sets to empty sets.
(Contributed by Stefan O'Rear, 21-Jan-2015.)
|
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|
Theorem | dfmpt3 5340 |
Alternate definition for the maps-to notation df-mpt 4068. (Contributed
by Mario Carneiro, 30-Dec-2016.)
|
 
        |
|
Theorem | fnopabg 5341* |
Functionality and domain of an ordered-pair class abstraction.
(Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro,
4-Dec-2016.)
|
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|
Theorem | fnopab 5342* |
Functionality and domain of an ordered-pair class abstraction.
(Contributed by NM, 5-Mar-1996.)
|
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|
Theorem | mptfng 5343* |
The maps-to notation defines a function with domain. (Contributed by
Scott Fenton, 21-Mar-2011.)
|
   
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|
Theorem | fnmpt 5344* |
The maps-to notation defines a function with domain. (Contributed by
NM, 9-Apr-2013.)
|
   
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|
Theorem | mpt0 5345 |
A mapping operation with empty domain. (Contributed by Mario Carneiro,
28-Dec-2014.)
|
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|
Theorem | fnmpti 5346* |
Functionality and domain of an ordered-pair class abstraction.
(Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro,
31-Aug-2015.)
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|
Theorem | dmmpti 5347* |
Domain of an ordered-pair class abstraction that specifies a function.
(Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro,
31-Aug-2015.)
|
 
 |
|
Theorem | dmmptd 5348* |
The domain of the mapping operation, deduction form. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
|
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|
Theorem | mptun 5349 |
Union of mappings which are mutually compatible. (Contributed by Mario
Carneiro, 31-Aug-2015.)
|
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|
Theorem | feq1 5350 |
Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|
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|
Theorem | feq2 5351 |
Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|
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|
Theorem | feq3 5352 |
Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
|
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|
Theorem | feq23 5353 |
Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
|
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Theorem | feq1d 5354 |
Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
|
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Theorem | feq2d 5355 |
Equality deduction for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
|
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|
Theorem | feq3d 5356 |
Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
|
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|
Theorem | feq12d 5357 |
Equality deduction for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq123d 5358 |
Equality deduction for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
|
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|
Theorem | feq123 5359 |
Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
|
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|
Theorem | feq1i 5360 |
Equality inference for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
|
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|
Theorem | feq2i 5361 |
Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
|
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|
Theorem | feq23i 5362 |
Equality inference for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
|
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|
Theorem | feq23d 5363 |
Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
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Theorem | nff 5364 |
Bound-variable hypothesis builder for a mapping. (Contributed by NM,
29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
|
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|
Theorem | sbcfng 5365* |
Distribute proper substitution through the function predicate with a
domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
|
    ![]. ].](_drbrack.gif)   ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)    |
|
Theorem | sbcfg 5366* |
Distribute proper substitution through the function predicate with
domain and codomain. (Contributed by Alexander van der Vekens,
15-Jul-2018.)
|
    ![]. ].](_drbrack.gif)       ![]_ ]_](_urbrack.gif)     ![]_ ]_](_urbrack.gif)     ![]_ ]_](_urbrack.gif)    |
|
Theorem | ffn 5367 |
A mapping is a function. (Contributed by NM, 2-Aug-1994.)
|
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Theorem | ffnd 5368 |
A mapping is a function with domain, deduction form. (Contributed by
Glauco Siliprandi, 17-Aug-2020.)
|
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Theorem | dffn2 5369 |
Any function is a mapping into . (Contributed by NM, 31-Oct-1995.)
(Proof shortened by Andrew Salmon, 17-Sep-2011.)
|
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Theorem | ffun 5370 |
A mapping is a function. (Contributed by NM, 3-Aug-1994.)
|
    
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Theorem | ffund 5371 |
A mapping is a function, deduction version. (Contributed by Glauco
Siliprandi, 3-Mar-2021.)
|
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Theorem | frel 5372 |
A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
|
    
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Theorem | fdm 5373 |
The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
|
    
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Theorem | fdmd 5374 |
Deduction form of fdm 5373. The domain of a mapping. (Contributed by
Glauco Siliprandi, 26-Jun-2021.)
|
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|
Theorem | fdmi 5375 |
The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
|
   
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Theorem | frn 5376 |
The range of a mapping. (Contributed by NM, 3-Aug-1994.)
|
    
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|
Theorem | frnd 5377 |
Deduction form of frn 5376. The range of a mapping. (Contributed by
Glauco Siliprandi, 26-Jun-2021.)
|
      
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|
Theorem | dffn3 5378 |
A function maps to its range. (Contributed by NM, 1-Sep-1999.)
|
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Theorem | fss 5379 |
Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.)
(Proof shortened by Andrew Salmon, 17-Sep-2011.)
|
      
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|
Theorem | fssd 5380 |
Expanding the codomain of a mapping, deduction form. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
|
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Theorem | fssdmd 5381 |
Expressing that a class is a subclass of the domain of a function
expressed in maps-to notation, deduction form. (Contributed by AV,
21-Aug-2022.)
|
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|
Theorem | fssdm 5382 |
Expressing that a class is a subclass of the domain of a function
expressed in maps-to notation, semi-deduction form. (Contributed by AV,
21-Aug-2022.)
|
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|
Theorem | fco 5383 |
Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
|
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|
Theorem | fco2 5384 |
Functionality of a composition with weakened out of domain condition on
the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
|
            
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|
Theorem | fssxp 5385 |
A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.)
(Proof shortened by Andrew Salmon, 17-Sep-2011.)
|
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Theorem | fex2 5386 |
A function with bounded domain and codomain is a set. This version is
proven without the Axiom of Replacement. (Contributed by Mario Carneiro,
24-Jun-2015.)
|
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|
Theorem | funssxp 5387 |
Two ways of specifying a partial function from to .
(Contributed by NM, 13-Nov-2007.)
|
 
           |
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Theorem | ffdm 5388 |
A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
|
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Theorem | opelf 5389 |
The members of an ordered pair element of a mapping belong to the
mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.)
(Revised by Mario Carneiro, 26-Apr-2015.)
|
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Theorem | fun 5390 |
The union of two functions with disjoint domains. (Contributed by NM,
22-Sep-2004.)
|
               
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|
Theorem | fun2 5391 |
The union of two functions with disjoint domains. (Contributed by Mario
Carneiro, 12-Mar-2015.)
|
               
         |
|
Theorem | fnfco 5392 |
Composition of two functions. (Contributed by NM, 22-May-2006.)
|
      
    |
|
Theorem | fssres 5393 |
Restriction of a function with a subclass of its domain. (Contributed by
NM, 23-Sep-2004.)
|
      
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|
Theorem | fssresd 5394 |
Restriction of a function with a subclass of its domain, deduction form.
(Contributed by Glauco Siliprandi, 11-Dec-2019.)
|
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Theorem | fssres2 5395 |
Restriction of a restricted function with a subclass of its domain.
(Contributed by NM, 21-Jul-2005.)
|
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Theorem | fresin 5396 |
An identity for the mapping relationship under restriction. (Contributed
by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro,
26-May-2016.)
|
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Theorem | resasplitss 5397 |
If two functions agree on their common domain, their union contains a
union of three functions with pairwise disjoint domains. If we assumed
the law of the excluded middle, this would be equality rather than subset.
(Contributed by Jim Kingdon, 28-Dec-2018.)
|
          
                    |
|
Theorem | fcoi1 5398 |
Composition of a mapping and restricted identity. (Contributed by NM,
13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|
     
    |
|
Theorem | fcoi2 5399 |
Composition of restricted identity and a mapping. (Contributed by NM,
13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|
       
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|
Theorem | feu 5400* |
There is exactly one value of a function in its codomain. (Contributed
by NM, 10-Dec-2003.)
|
      
      |