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Theorem fsn2 5911
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 3843 . . . . 5  |-  A  e. 
{ A }
3 ffvelrn 5871 . . . . 5  |-  ( ( F : { A }
--> B  /\  A  e. 
{ A } )  ->  ( F `  A )  e.  B
)
42, 3mpan2 654 . . . 4  |-  ( F : { A } --> B  ->  ( F `  A )  e.  B
)
5 ffn 5594 . . . . 5  |-  ( F : { A } --> B  ->  F  Fn  { A } )
6 dffn3 5601 . . . . . . 7  |-  ( F  Fn  { A }  <->  F : { A } --> ran  F )
76biimpi 188 . . . . . 6  |-  ( F  Fn  { A }  ->  F : { A }
--> ran  F )
8 imadmrn 5218 . . . . . . . . 9  |-  ( F
" dom  F )  =  ran  F
9 fndm 5547 . . . . . . . . . 10  |-  ( F  Fn  { A }  ->  dom  F  =  { A } )
109imaeq2d 5206 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  ( F " dom  F )  =  ( F
" { A }
) )
118, 10syl5eqr 2484 . . . . . . . 8  |-  ( F  Fn  { A }  ->  ran  F  =  ( F " { A } ) )
12 fnsnfv 5789 . . . . . . . . 9  |-  ( ( F  Fn  { A }  /\  A  e.  { A } )  ->  { ( F `  A ) }  =  ( F
" { A }
) )
132, 12mpan2 654 . . . . . . . 8  |-  ( F  Fn  { A }  ->  { ( F `  A ) }  =  ( F " { A } ) )
1411, 13eqtr4d 2473 . . . . . . 7  |-  ( F  Fn  { A }  ->  ran  F  =  {
( F `  A
) } )
15 feq3 5581 . . . . . . 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 203 . . . . 5  |-  ( F  Fn  { A }  ->  F : { A }
--> { ( F `  A ) } )
185, 17syl 16 . . . 4  |-  ( F : { A } --> B  ->  F : { A } --> { ( F `
 A ) } )
194, 18jca 520 . . 3  |-  ( F : { A } --> B  ->  ( ( F `
 A )  e.  B  /\  F : { A } --> { ( F `  A ) } ) )
20 snssi 3944 . . . 4  |-  ( ( F `  A )  e.  B  ->  { ( F `  A ) }  C_  B )
21 fss 5602 . . . . 5  |-  ( ( F : { A }
--> { ( F `  A ) }  /\  { ( F `  A
) }  C_  B
)  ->  F : { A } --> B )
2221ancoms 441 . . . 4  |-  ( ( { ( F `  A ) }  C_  B  /\  F : { A } --> { ( F `
 A ) } )  ->  F : { A } --> B )
2320, 22sylan 459 . . 3  |-  ( ( ( F `  A
)  e.  B  /\  F : { A } --> { ( F `  A ) } )  ->  F : { A } --> B )
2419, 23impbii 182 . 2  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F : { A }
--> { ( F `  A ) } ) )
25 fvex 5745 . . . 4  |-  ( F `
 A )  e. 
_V
261, 25fsn 5909 . . 3  |-  ( F : { A } --> { ( F `  A ) }  <->  F  =  { <. A ,  ( F `  A )
>. } )
2726anbi2i 677 . 2  |-  ( ( ( F `  A
)  e.  B  /\  F : { A } --> { ( F `  A ) } )  <-> 
( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )
2824, 27bitri 242 1  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   {csn 3816   <.cop 3819   dom cdm 4881   ran crn 4882   "cima 4884    Fn wfn 5452   -->wf 5453   ` cfv 5457
This theorem is referenced by:  fnressn  5921  fressnfv  5923  mapsnconst  7062  elixpsn  7104  en1  7177  pt1hmeo  17843  0spth  21576  wrdlen1  28199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465
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