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Theorem fsn2 5698
Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)
Hypothesis
Ref Expression
fsn2.1  |-  A  e. 
_V
Assertion
Ref Expression
fsn2  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )

Proof of Theorem fsn2
StepHypRef Expression
1 fsn2.1 . . . . . 6  |-  A  e. 
_V
21snid 3667 . . . . 5  |-  A  e. 
{ A }
3 ffvelrn 5663 . . . . 5  |-  ( ( F : { A }
--> B  /\  A  e. 
{ A } )  ->  ( F `  A )  e.  B
)
42, 3mpan2 652 . . . 4  |-  ( F : { A } --> B  ->  ( F `  A )  e.  B
)
5 ffn 5389 . . . . 5  |-  ( F : { A } --> B  ->  F  Fn  { A } )
6 dffn3 5396 . . . . . . 7  |-  ( F  Fn  { A }  <->  F : { A } --> ran  F )
76biimpi 186 . . . . . 6  |-  ( F  Fn  { A }  ->  F : { A }
--> ran  F )
8 imadmrn 5024 . . . . . . . . 9  |-  ( F
" dom  F )  =  ran  F
9 fndm 5343 . . . . . . . . . 10  |-  ( F  Fn  { A }  ->  dom  F  =  { A } )
109imaeq2d 5012 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  ( F " dom  F )  =  ( F
" { A }
) )
118, 10syl5eqr 2329 . . . . . . . 8  |-  ( F  Fn  { A }  ->  ran  F  =  ( F " { A } ) )
12 fnsnfv 5582 . . . . . . . . 9  |-  ( ( F  Fn  { A }  /\  A  e.  { A } )  ->  { ( F `  A ) }  =  ( F
" { A }
) )
132, 12mpan2 652 . . . . . . . 8  |-  ( F  Fn  { A }  ->  { ( F `  A ) }  =  ( F " { A } ) )
1411, 13eqtr4d 2318 . . . . . . 7  |-  ( F  Fn  { A }  ->  ran  F  =  {
( F `  A
) } )
15 feq3 5377 . . . . . . 7  |-  ( ran 
F  =  { ( F `  A ) }  ->  ( F : { A } --> ran  F  <->  F : { A } --> { ( F `  A ) } ) )
1614, 15syl 15 . . . . . 6  |-  ( F  Fn  { A }  ->  ( F : { A } --> ran  F  <->  F : { A } --> { ( F `  A ) } ) )
177, 16mpbid 201 . . . . 5  |-  ( F  Fn  { A }  ->  F : { A }
--> { ( F `  A ) } )
185, 17syl 15 . . . 4  |-  ( F : { A } --> B  ->  F : { A } --> { ( F `
 A ) } )
194, 18jca 518 . . 3  |-  ( F : { A } --> B  ->  ( ( F `
 A )  e.  B  /\  F : { A } --> { ( F `  A ) } ) )
20 snssi 3759 . . . 4  |-  ( ( F `  A )  e.  B  ->  { ( F `  A ) }  C_  B )
21 fss 5397 . . . . 5  |-  ( ( F : { A }
--> { ( F `  A ) }  /\  { ( F `  A
) }  C_  B
)  ->  F : { A } --> B )
2221ancoms 439 . . . 4  |-  ( ( { ( F `  A ) }  C_  B  /\  F : { A } --> { ( F `
 A ) } )  ->  F : { A } --> B )
2320, 22sylan 457 . . 3  |-  ( ( ( F `  A
)  e.  B  /\  F : { A } --> { ( F `  A ) } )  ->  F : { A } --> B )
2419, 23impbii 180 . 2  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F : { A }
--> { ( F `  A ) } ) )
25 fvex 5539 . . . 4  |-  ( F `
 A )  e. 
_V
261, 25fsn 5696 . . 3  |-  ( F : { A } --> { ( F `  A ) }  <->  F  =  { <. A ,  ( F `  A )
>. } )
2726anbi2i 675 . 2  |-  ( ( ( F `  A
)  e.  B  /\  F : { A } --> { ( F `  A ) } )  <-> 
( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )
2824, 27bitri 240 1  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {csn 3640   <.cop 3643   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255
This theorem is referenced by:  fnressn  5705  fressnfv  5707  mapsnconst  6813  elixpsn  6855  en1  6928  pt1hmeo  17497  cbicp  25166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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