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Theorem fnsnb 5910
Description: A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Revised to add reverse implication. (Revised by NM, 29-Dec-2018.)
Hypothesis
Ref Expression
fnsnb.1  |-  A  e. 
_V
Assertion
Ref Expression
fnsnb  |-  ( F  Fn  { A }  <->  F  =  { <. A , 
( F `  A
) >. } )

Proof of Theorem fnsnb
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fnresdm 5532 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  ( F  |`  { A } )  =  F )
2 fnfun 5520 . . . . . . . . . 10  |-  ( F  Fn  { A }  ->  Fun  F )
3 funressn 5907 . . . . . . . . . 10  |-  ( Fun 
F  ->  ( F  |` 
{ A } ) 
C_  { <. A , 
( F `  A
) >. } )
42, 3syl 16 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  ( F  |`  { A } )  C_  { <. A ,  ( F `  A ) >. } )
51, 4eqsstr3d 3403 . . . . . . . 8  |-  ( F  Fn  { A }  ->  F  C_  { <. A , 
( F `  A
) >. } )
65sseld 3367 . . . . . . 7  |-  ( F  Fn  { A }  ->  ( x  e.  F  ->  x  e.  { <. A ,  ( F `  A ) >. } ) )
7 elsni 3914 . . . . . . 7  |-  ( x  e.  { <. A , 
( F `  A
) >. }  ->  x  =  <. A ,  ( F `  A )
>. )
86, 7syl6 33 . . . . . 6  |-  ( F  Fn  { A }  ->  ( x  e.  F  ->  x  =  <. A , 
( F `  A
) >. ) )
9 df-fn 5433 . . . . . . . . 9  |-  ( F  Fn  { A }  <->  ( Fun  F  /\  dom  F  =  { A }
) )
10 fnsnb.1 . . . . . . . . . . . 12  |-  A  e. 
_V
1110snid 3917 . . . . . . . . . . 11  |-  A  e. 
{ A }
12 eleq2 2504 . . . . . . . . . . 11  |-  ( dom 
F  =  { A }  ->  ( A  e. 
dom  F  <->  A  e.  { A } ) )
1311, 12mpbiri 233 . . . . . . . . . 10  |-  ( dom 
F  =  { A }  ->  A  e.  dom  F )
1413anim2i 569 . . . . . . . . 9  |-  ( ( Fun  F  /\  dom  F  =  { A }
)  ->  ( Fun  F  /\  A  e.  dom  F ) )
159, 14sylbi 195 . . . . . . . 8  |-  ( F  Fn  { A }  ->  ( Fun  F  /\  A  e.  dom  F ) )
16 funfvop 5827 . . . . . . . 8  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
1715, 16syl 16 . . . . . . 7  |-  ( F  Fn  { A }  -> 
<. A ,  ( F `
 A ) >.  e.  F )
18 eleq1 2503 . . . . . . 7  |-  ( x  =  <. A ,  ( F `  A )
>.  ->  ( x  e.  F  <->  <. A ,  ( F `  A )
>.  e.  F ) )
1917, 18syl5ibrcom 222 . . . . . 6  |-  ( F  Fn  { A }  ->  ( x  =  <. A ,  ( F `  A ) >.  ->  x  e.  F ) )
208, 19impbid 191 . . . . 5  |-  ( F  Fn  { A }  ->  ( x  e.  F  <->  x  =  <. A ,  ( F `  A )
>. ) )
21 elsn 3903 . . . . . 6  |-  ( x  e.  { <. A , 
( F `  A
) >. }  <->  x  =  <. A ,  ( F `
 A ) >.
)
2221bibi2i 313 . . . . 5  |-  ( ( x  e.  F  <->  x  e.  {
<. A ,  ( F `
 A ) >. } )  <->  ( x  e.  F  <->  x  =  <. A ,  ( F `  A ) >. )
)
2320, 22sylibr 212 . . . 4  |-  ( F  Fn  { A }  ->  ( x  e.  F  <->  x  e.  { <. A , 
( F `  A
) >. } ) )
2423alrimiv 1685 . . 3  |-  ( F  Fn  { A }  ->  A. x ( x  e.  F  <->  x  e.  {
<. A ,  ( F `
 A ) >. } ) )
25 dfcleq 2437 . . 3  |-  ( F  =  { <. A , 
( F `  A
) >. }  <->  A. x
( x  e.  F  <->  x  e.  { <. A , 
( F `  A
) >. } ) )
2624, 25sylibr 212 . 2  |-  ( F  Fn  { A }  ->  F  =  { <. A ,  ( F `  A ) >. } )
27 fvex 5713 . . . 4  |-  ( F `
 A )  e. 
_V
2810, 27fnsn 5483 . . 3  |-  { <. A ,  ( F `  A ) >. }  Fn  { A }
29 fneq1 5511 . . 3  |-  ( F  =  { <. A , 
( F `  A
) >. }  ->  ( F  Fn  { A } 
<->  { <. A ,  ( F `  A )
>. }  Fn  { A } ) )
3028, 29mpbiri 233 . 2  |-  ( F  =  { <. A , 
( F `  A
) >. }  ->  F  Fn  { A } )
3126, 30impbii 188 1  |-  ( F  Fn  { A }  <->  F  =  { <. A , 
( F `  A
) >. } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369    e. wcel 1756   _Vcvv 2984    C_ wss 3340   {csn 3889   <.cop 3895   dom cdm 4852    |` cres 4854   Fun wfun 5424    Fn wfn 5425   ` cfv 5430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438
This theorem is referenced by:  fnprb  5948
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