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Theorem fsn2 5562
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 ffn 5242 . . 3  |-  ( F : { A } --> B  ->  F  Fn  { A } )
2 fsn2.1 . . . . 5  |-  A  e. 
_V
32snid 3526 . . . 4  |-  A  e. 
{ A }
4 funfvex 5406 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
54funfni 5193 . . . 4  |-  ( ( F  Fn  { A }  /\  A  e.  { A } )  ->  ( F `  A )  e.  _V )
63, 5mpan2 421 . . 3  |-  ( F  Fn  { A }  ->  ( F `  A
)  e.  _V )
71, 6syl 14 . 2  |-  ( F : { A } --> B  ->  ( F `  A )  e.  _V )
8 elex 2671 . . 3  |-  ( ( F `  A )  e.  B  ->  ( F `  A )  e.  _V )
98adantr 274 . 2  |-  ( ( ( F `  A
)  e.  B  /\  F  =  { <. A , 
( F `  A
) >. } )  -> 
( F `  A
)  e.  _V )
10 ffvelrn 5521 . . . . . 6  |-  ( ( F : { A }
--> B  /\  A  e. 
{ A } )  ->  ( F `  A )  e.  B
)
113, 10mpan2 421 . . . . 5  |-  ( F : { A } --> B  ->  ( F `  A )  e.  B
)
12 dffn3 5253 . . . . . . . 8  |-  ( F  Fn  { A }  <->  F : { A } --> ran  F )
1312biimpi 119 . . . . . . 7  |-  ( F  Fn  { A }  ->  F : { A }
--> ran  F )
14 imadmrn 4861 . . . . . . . . . 10  |-  ( F
" dom  F )  =  ran  F
15 fndm 5192 . . . . . . . . . . 11  |-  ( F  Fn  { A }  ->  dom  F  =  { A } )
1615imaeq2d 4851 . . . . . . . . . 10  |-  ( F  Fn  { A }  ->  ( F " dom  F )  =  ( F
" { A }
) )
1714, 16syl5eqr 2164 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  ran  F  =  ( F " { A } ) )
18 fnsnfv 5448 . . . . . . . . . 10  |-  ( ( F  Fn  { A }  /\  A  e.  { A } )  ->  { ( F `  A ) }  =  ( F
" { A }
) )
193, 18mpan2 421 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  { ( F `  A ) }  =  ( F " { A } ) )
2017, 19eqtr4d 2153 . . . . . . . 8  |-  ( F  Fn  { A }  ->  ran  F  =  {
( F `  A
) } )
21 feq3 5227 . . . . . . . 8  |-  ( ran 
F  =  { ( F `  A ) }  ->  ( F : { A } --> ran  F  <->  F : { A } --> { ( F `  A ) } ) )
2220, 21syl 14 . . . . . . 7  |-  ( F  Fn  { A }  ->  ( F : { A } --> ran  F  <->  F : { A } --> { ( F `  A ) } ) )
2313, 22mpbid 146 . . . . . 6  |-  ( F  Fn  { A }  ->  F : { A }
--> { ( F `  A ) } )
241, 23syl 14 . . . . 5  |-  ( F : { A } --> B  ->  F : { A } --> { ( F `
 A ) } )
2511, 24jca 304 . . . 4  |-  ( F : { A } --> B  ->  ( ( F `
 A )  e.  B  /\  F : { A } --> { ( F `  A ) } ) )
26 snssi 3634 . . . . 5  |-  ( ( F `  A )  e.  B  ->  { ( F `  A ) }  C_  B )
27 fss 5254 . . . . . 6  |-  ( ( F : { A }
--> { ( F `  A ) }  /\  { ( F `  A
) }  C_  B
)  ->  F : { A } --> B )
2827ancoms 266 . . . . 5  |-  ( ( { ( F `  A ) }  C_  B  /\  F : { A } --> { ( F `
 A ) } )  ->  F : { A } --> B )
2926, 28sylan 281 . . . 4  |-  ( ( ( F `  A
)  e.  B  /\  F : { A } --> { ( F `  A ) } )  ->  F : { A } --> B )
3025, 29impbii 125 . . 3  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F : { A }
--> { ( F `  A ) } ) )
31 fsng 5561 . . . . 5  |-  ( ( A  e.  _V  /\  ( F `  A )  e.  _V )  -> 
( F : { A } --> { ( F `
 A ) }  <-> 
F  =  { <. A ,  ( F `  A ) >. } ) )
322, 31mpan 420 . . . 4  |-  ( ( F `  A )  e.  _V  ->  ( F : { A } --> { ( F `  A ) }  <->  F  =  { <. A ,  ( F `  A )
>. } ) )
3332anbi2d 459 . . 3  |-  ( ( F `  A )  e.  _V  ->  (
( ( F `  A )  e.  B  /\  F : { A }
--> { ( F `  A ) } )  <-> 
( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) ) )
3430, 33syl5bb 191 . 2  |-  ( ( F `  A )  e.  _V  ->  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) ) )
357, 9, 34pm5.21nii 678 1  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   _Vcvv 2660    C_ wss 3041   {csn 3497   <.cop 3500   dom cdm 4509   ran crn 4510   "cima 4512    Fn wfn 5088   -->wf 5089   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101
This theorem is referenced by:  fnressn  5574  fressnfv  5575  mapsnconst  6556  elixpsn  6597  en1  6661
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