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Theorem fsn2 5455
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 5147 . . 3  |-  ( F : { A } --> B  ->  F  Fn  { A } )
2 fsn2.1 . . . . 5  |-  A  e. 
_V
32snid 3470 . . . 4  |-  A  e. 
{ A }
4 funfvex 5306 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
54funfni 5100 . . . 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 2630 . . 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 5416 . . . . . 6  |-  ( ( F : { A }
--> B  /\  A  e. 
{ A } )  ->  ( F `  A )  e.  B
)
113, 10mpan2 416 . . . . 5  |-  ( F : { A } --> B  ->  ( F `  A )  e.  B
)
12 dffn3 5156 . . . . . . . 8  |-  ( F  Fn  { A }  <->  F : { A } --> ran  F )
1312biimpi 118 . . . . . . 7  |-  ( F  Fn  { A }  ->  F : { A }
--> ran  F )
14 imadmrn 4771 . . . . . . . . . 10  |-  ( F
" dom  F )  =  ran  F
15 fndm 5099 . . . . . . . . . . 11  |-  ( F  Fn  { A }  ->  dom  F  =  { A } )
1615imaeq2d 4761 . . . . . . . . . 10  |-  ( F  Fn  { A }  ->  ( F " dom  F )  =  ( F
" { A }
) )
1714, 16syl5eqr 2134 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  ran  F  =  ( F " { A } ) )
18 fnsnfv 5347 . . . . . . . . . 10  |-  ( ( F  Fn  { A }  /\  A  e.  { A } )  ->  { ( F `  A ) }  =  ( F
" { A }
) )
193, 18mpan2 416 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  { ( F `  A ) }  =  ( F " { A } ) )
2017, 19eqtr4d 2123 . . . . . . . 8  |-  ( F  Fn  { A }  ->  ran  F  =  {
( F `  A
) } )
21 feq3 5133 . . . . . . . 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 3576 . . . . 5  |-  ( ( F `  A )  e.  B  ->  { ( F `  A ) }  C_  B )
27 fss 5157 . . . . . 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 5454 . . . . 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 655 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 1289    e. wcel 1438   _Vcvv 2619    C_ wss 2997   {csn 3441   <.cop 3444   dom cdm 4428   ran crn 4429   "cima 4431    Fn wfn 4997   -->wf 4998   ` cfv 5002
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010
This theorem is referenced by:  fnressn  5467  fressnfv  5468  mapsnconst  6431  en1  6496
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