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Theorem fsn2 5875
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 3809 . . . . 5  |-  A  e. 
{ A }
3 ffvelrn 5835 . . . . 5  |-  ( ( F : { A }
--> B  /\  A  e. 
{ A } )  ->  ( F `  A )  e.  B
)
42, 3mpan2 653 . . . 4  |-  ( F : { A } --> B  ->  ( F `  A )  e.  B
)
5 ffn 5558 . . . . 5  |-  ( F : { A } --> B  ->  F  Fn  { A } )
6 dffn3 5565 . . . . . . 7  |-  ( F  Fn  { A }  <->  F : { A } --> ran  F )
76biimpi 187 . . . . . 6  |-  ( F  Fn  { A }  ->  F : { A }
--> ran  F )
8 imadmrn 5182 . . . . . . . . 9  |-  ( F
" dom  F )  =  ran  F
9 fndm 5511 . . . . . . . . . 10  |-  ( F  Fn  { A }  ->  dom  F  =  { A } )
109imaeq2d 5170 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  ( F " dom  F )  =  ( F
" { A }
) )
118, 10syl5eqr 2458 . . . . . . . 8  |-  ( F  Fn  { A }  ->  ran  F  =  ( F " { A } ) )
12 fnsnfv 5753 . . . . . . . . 9  |-  ( ( F  Fn  { A }  /\  A  e.  { A } )  ->  { ( F `  A ) }  =  ( F
" { A }
) )
132, 12mpan2 653 . . . . . . . 8  |-  ( F  Fn  { A }  ->  { ( F `  A ) }  =  ( F " { A } ) )
1411, 13eqtr4d 2447 . . . . . . 7  |-  ( F  Fn  { A }  ->  ran  F  =  {
( F `  A
) } )
15 feq3 5545 . . . . . . 7  |-  ( ran 
F  =  { ( F `  A ) }  ->  ( F : { A } --> ran  F  <->  F : { A } --> { ( F `  A ) } ) )
1614, 15syl 16 . . . . . 6  |-  ( F  Fn  { A }  ->  ( F : { A } --> ran  F  <->  F : { A } --> { ( F `  A ) } ) )
177, 16mpbid 202 . . . . 5  |-  ( F  Fn  { A }  ->  F : { A }
--> { ( F `  A ) } )
185, 17syl 16 . . . 4  |-  ( F : { A } --> B  ->  F : { A } --> { ( F `
 A ) } )
194, 18jca 519 . . 3  |-  ( F : { A } --> B  ->  ( ( F `
 A )  e.  B  /\  F : { A } --> { ( F `  A ) } ) )
20 snssi 3910 . . . 4  |-  ( ( F `  A )  e.  B  ->  { ( F `  A ) }  C_  B )
21 fss 5566 . . . . 5  |-  ( ( F : { A }
--> { ( F `  A ) }  /\  { ( F `  A
) }  C_  B
)  ->  F : { A } --> B )
2221ancoms 440 . . . 4  |-  ( ( { ( F `  A ) }  C_  B  /\  F : { A } --> { ( F `
 A ) } )  ->  F : { A } --> B )
2320, 22sylan 458 . . 3  |-  ( ( ( F `  A
)  e.  B  /\  F : { A } --> { ( F `  A ) } )  ->  F : { A } --> B )
2419, 23impbii 181 . 2  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F : { A }
--> { ( F `  A ) } ) )
25 fvex 5709 . . . 4  |-  ( F `
 A )  e. 
_V
261, 25fsn 5873 . . 3  |-  ( F : { A } --> { ( F `  A ) }  <->  F  =  { <. A ,  ( F `  A )
>. } )
2726anbi2i 676 . 2  |-  ( ( ( F `  A
)  e.  B  /\  F : { A } --> { ( F `  A ) } )  <-> 
( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )
2824, 27bitri 241 1  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924    C_ wss 3288   {csn 3782   <.cop 3785   dom cdm 4845   ran crn 4846   "cima 4848    Fn wfn 5416   -->wf 5417   ` cfv 5421
This theorem is referenced by:  fnressn  5885  fressnfv  5887  mapsnconst  7026  elixpsn  7068  en1  7141  pt1hmeo  17799  0spth  21532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429
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