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Theorem fsn2 5363
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 5071 . . 3  |-  ( F : { A } --> B  ->  F  Fn  { A } )
2 fsn2.1 . . . . 5  |-  A  e. 
_V
32snid 3427 . . . 4  |-  A  e. 
{ A }
4 funfvex 5217 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
54funfni 5024 . . . 4  |-  ( ( F  Fn  { A }  /\  A  e.  { A } )  ->  ( F `  A )  e.  _V )
63, 5mpan2 416 . . 3  |-  ( F  Fn  { A }  ->  ( F `  A
)  e.  _V )
71, 6syl 14 . 2  |-  ( F : { A } --> B  ->  ( F `  A )  e.  _V )
8 elex 2611 . . 3  |-  ( ( F `  A )  e.  B  ->  ( F `  A )  e.  _V )
98adantr 270 . 2  |-  ( ( ( F `  A
)  e.  B  /\  F  =  { <. A , 
( F `  A
) >. } )  -> 
( F `  A
)  e.  _V )
10 ffvelrn 5326 . . . . . 6  |-  ( ( F : { A }
--> B  /\  A  e. 
{ A } )  ->  ( F `  A )  e.  B
)
113, 10mpan2 416 . . . . 5  |-  ( F : { A } --> B  ->  ( F `  A )  e.  B
)
12 dffn3 5078 . . . . . . . 8  |-  ( F  Fn  { A }  <->  F : { A } --> ran  F )
1312biimpi 118 . . . . . . 7  |-  ( F  Fn  { A }  ->  F : { A }
--> ran  F )
14 imadmrn 4702 . . . . . . . . . 10  |-  ( F
" dom  F )  =  ran  F
15 fndm 5023 . . . . . . . . . . 11  |-  ( F  Fn  { A }  ->  dom  F  =  { A } )
1615imaeq2d 4692 . . . . . . . . . 10  |-  ( F  Fn  { A }  ->  ( F " dom  F )  =  ( F
" { A }
) )
1714, 16syl5eqr 2128 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  ran  F  =  ( F " { A } ) )
18 fnsnfv 5258 . . . . . . . . . 10  |-  ( ( F  Fn  { A }  /\  A  e.  { A } )  ->  { ( F `  A ) }  =  ( F
" { A }
) )
193, 18mpan2 416 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  { ( F `  A ) }  =  ( F " { A } ) )
2017, 19eqtr4d 2117 . . . . . . . 8  |-  ( F  Fn  { A }  ->  ran  F  =  {
( F `  A
) } )
21 feq3 5057 . . . . . . . 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 145 . . . . . 6  |-  ( F  Fn  { A }  ->  F : { A }
--> { ( F `  A ) } )
241, 23syl 14 . . . . 5  |-  ( F : { A } --> B  ->  F : { A } --> { ( F `
 A ) } )
2511, 24jca 300 . . . 4  |-  ( F : { A } --> B  ->  ( ( F `
 A )  e.  B  /\  F : { A } --> { ( F `  A ) } ) )
26 snssi 3531 . . . . 5  |-  ( ( F `  A )  e.  B  ->  { ( F `  A ) }  C_  B )
27 fss 5079 . . . . . 6  |-  ( ( F : { A }
--> { ( F `  A ) }  /\  { ( F `  A
) }  C_  B
)  ->  F : { A } --> B )
2827ancoms 264 . . . . 5  |-  ( ( { ( F `  A ) }  C_  B  /\  F : { A } --> { ( F `
 A ) } )  ->  F : { A } --> B )
2926, 28sylan 277 . . . 4  |-  ( ( ( F `  A
)  e.  B  /\  F : { A } --> { ( F `  A ) } )  ->  F : { A } --> B )
3025, 29impbii 124 . . 3  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F : { A }
--> { ( F `  A ) } ) )
31 fsng 5362 . . . . 5  |-  ( ( A  e.  _V  /\  ( F `  A )  e.  _V )  -> 
( F : { A } --> { ( F `
 A ) }  <-> 
F  =  { <. A ,  ( F `  A ) >. } ) )
322, 31mpan 415 . . . 4  |-  ( ( F `  A )  e.  _V  ->  ( F : { A } --> { ( F `  A ) }  <->  F  =  { <. A ,  ( F `  A )
>. } ) )
3332anbi2d 452 . . 3  |-  ( ( F `  A )  e.  _V  ->  (
( ( F `  A )  e.  B  /\  F : { A }
--> { ( F `  A ) } )  <-> 
( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) ) )
3430, 33syl5bb 190 . 2  |-  ( ( F `  A )  e.  _V  ->  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) ) )
357, 9, 34pm5.21nii 653 1  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   _Vcvv 2602    C_ wss 2974   {csn 3400   <.cop 3403   dom cdm 4365   ran crn 4366   "cima 4368    Fn wfn 4921   -->wf 4922   ` cfv 4926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934
This theorem is referenced by:  fnressn  5375  fressnfv  5376  en1  6338
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