MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnsnb Structured version   Unicode version

Theorem fnsnb 6081
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 5690 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  ( F  |`  { A } )  =  F )
2 fnfun 5678 . . . . . . . . . 10  |-  ( F  Fn  { A }  ->  Fun  F )
3 funressn 6075 . . . . . . . . . 10  |-  ( Fun 
F  ->  ( F  |` 
{ A } ) 
C_  { <. A , 
( F `  A
) >. } )
42, 3syl 16 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  ( F  |`  { A } )  C_  { <. A ,  ( F `  A ) >. } )
51, 4eqsstr3d 3539 . . . . . . . 8  |-  ( F  Fn  { A }  ->  F  C_  { <. A , 
( F `  A
) >. } )
65sseld 3503 . . . . . . 7  |-  ( F  Fn  { A }  ->  ( x  e.  F  ->  x  e.  { <. A ,  ( F `  A ) >. } ) )
7 elsni 4052 . . . . . . 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 5591 . . . . . . . . 9  |-  ( F  Fn  { A }  <->  ( Fun  F  /\  dom  F  =  { A }
) )
10 fnsnb.1 . . . . . . . . . . . 12  |-  A  e. 
_V
1110snid 4055 . . . . . . . . . . 11  |-  A  e. 
{ A }
12 eleq2 2540 . . . . . . . . . . 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 5994 . . . . . . . 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 2539 . . . . . . 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 4041 . . . . . 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 1695 . . 3  |-  ( F  Fn  { A }  ->  A. x ( x  e.  F  <->  x  e.  {
<. A ,  ( F `
 A ) >. } ) )
25 dfcleq 2460 . . 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 5876 . . . 4  |-  ( F `
 A )  e. 
_V
2810, 27fnsn 5641 . . 3  |-  { <. A ,  ( F `  A ) >. }  Fn  { A }
29 fneq1 5669 . . 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 1377    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   {csn 4027   <.cop 4033   dom cdm 4999    |` cres 5001   Fun wfun 5582    Fn wfn 5583   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596
This theorem is referenced by:  fnprb  6120
  Copyright terms: Public domain W3C validator