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Theorem fsn2 5686
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 5361 . . 3  |-  ( F : { A } --> B  ->  F  Fn  { A } )
2 fsn2.1 . . . . 5  |-  A  e. 
_V
32snid 3622 . . . 4  |-  A  e. 
{ A }
4 funfvex 5528 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
54funfni 5312 . . . 4  |-  ( ( F  Fn  { A }  /\  A  e.  { A } )  ->  ( F `  A )  e.  _V )
63, 5mpan2 425 . . 3  |-  ( F  Fn  { A }  ->  ( F `  A
)  e.  _V )
71, 6syl 14 . 2  |-  ( F : { A } --> B  ->  ( F `  A )  e.  _V )
8 elex 2748 . . 3  |-  ( ( F `  A )  e.  B  ->  ( F `  A )  e.  _V )
98adantr 276 . 2  |-  ( ( ( F `  A
)  e.  B  /\  F  =  { <. A , 
( F `  A
) >. } )  -> 
( F `  A
)  e.  _V )
10 ffvelcdm 5645 . . . . . 6  |-  ( ( F : { A }
--> B  /\  A  e. 
{ A } )  ->  ( F `  A )  e.  B
)
113, 10mpan2 425 . . . . 5  |-  ( F : { A } --> B  ->  ( F `  A )  e.  B
)
12 dffn3 5372 . . . . . . . 8  |-  ( F  Fn  { A }  <->  F : { A } --> ran  F )
1312biimpi 120 . . . . . . 7  |-  ( F  Fn  { A }  ->  F : { A }
--> ran  F )
14 imadmrn 4976 . . . . . . . . . 10  |-  ( F
" dom  F )  =  ran  F
15 fndm 5311 . . . . . . . . . . 11  |-  ( F  Fn  { A }  ->  dom  F  =  { A } )
1615imaeq2d 4966 . . . . . . . . . 10  |-  ( F  Fn  { A }  ->  ( F " dom  F )  =  ( F
" { A }
) )
1714, 16eqtr3id 2224 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  ran  F  =  ( F " { A } ) )
18 fnsnfv 5571 . . . . . . . . . 10  |-  ( ( F  Fn  { A }  /\  A  e.  { A } )  ->  { ( F `  A ) }  =  ( F
" { A }
) )
193, 18mpan2 425 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  { ( F `  A ) }  =  ( F " { A } ) )
2017, 19eqtr4d 2213 . . . . . . . 8  |-  ( F  Fn  { A }  ->  ran  F  =  {
( F `  A
) } )
21 feq3 5346 . . . . . . . 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 147 . . . . . 6  |-  ( F  Fn  { A }  ->  F : { A }
--> { ( F `  A ) } )
241, 23syl 14 . . . . 5  |-  ( F : { A } --> B  ->  F : { A } --> { ( F `
 A ) } )
2511, 24jca 306 . . . 4  |-  ( F : { A } --> B  ->  ( ( F `
 A )  e.  B  /\  F : { A } --> { ( F `  A ) } ) )
26 snssi 3735 . . . . 5  |-  ( ( F `  A )  e.  B  ->  { ( F `  A ) }  C_  B )
27 fss 5373 . . . . . 6  |-  ( ( F : { A }
--> { ( F `  A ) }  /\  { ( F `  A
) }  C_  B
)  ->  F : { A } --> B )
2827ancoms 268 . . . . 5  |-  ( ( { ( F `  A ) }  C_  B  /\  F : { A } --> { ( F `
 A ) } )  ->  F : { A } --> B )
2926, 28sylan 283 . . . 4  |-  ( ( ( F `  A
)  e.  B  /\  F : { A } --> { ( F `  A ) } )  ->  F : { A } --> B )
3025, 29impbii 126 . . 3  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F : { A }
--> { ( F `  A ) } ) )
31 fsng 5685 . . . . 5  |-  ( ( A  e.  _V  /\  ( F `  A )  e.  _V )  -> 
( F : { A } --> { ( F `
 A ) }  <-> 
F  =  { <. A ,  ( F `  A ) >. } ) )
322, 31mpan 424 . . . 4  |-  ( ( F `  A )  e.  _V  ->  ( F : { A } --> { ( F `  A ) }  <->  F  =  { <. A ,  ( F `  A )
>. } ) )
3332anbi2d 464 . . 3  |-  ( ( F `  A )  e.  _V  ->  (
( ( F `  A )  e.  B  /\  F : { A }
--> { ( F `  A ) } )  <-> 
( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) ) )
3430, 33bitrid 192 . 2  |-  ( ( F `  A )  e.  _V  ->  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) ) )
357, 9, 34pm5.21nii 704 1  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   _Vcvv 2737    C_ wss 3129   {csn 3591   <.cop 3594   dom cdm 4623   ran crn 4624   "cima 4626    Fn wfn 5207   -->wf 5208   ` cfv 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220
This theorem is referenced by:  fnressn  5698  fressnfv  5699  mapsnconst  6688  elixpsn  6729  en1  6793
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