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Theorem fsn2 5809
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 3756 . . . . 5  |-  A  e. 
{ A }
3 ffvelrn 5770 . . . . 5  |-  ( ( F : { A }
--> B  /\  A  e. 
{ A } )  ->  ( F `  A )  e.  B
)
42, 3mpan2 652 . . . 4  |-  ( F : { A } --> B  ->  ( F `  A )  e.  B
)
5 ffn 5495 . . . . 5  |-  ( F : { A } --> B  ->  F  Fn  { A } )
6 dffn3 5502 . . . . . . 7  |-  ( F  Fn  { A }  <->  F : { A } --> ran  F )
76biimpi 186 . . . . . 6  |-  ( F  Fn  { A }  ->  F : { A }
--> ran  F )
8 imadmrn 5127 . . . . . . . . 9  |-  ( F
" dom  F )  =  ran  F
9 fndm 5448 . . . . . . . . . 10  |-  ( F  Fn  { A }  ->  dom  F  =  { A } )
109imaeq2d 5115 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  ( F " dom  F )  =  ( F
" { A }
) )
118, 10syl5eqr 2412 . . . . . . . 8  |-  ( F  Fn  { A }  ->  ran  F  =  ( F " { A } ) )
12 fnsnfv 5689 . . . . . . . . 9  |-  ( ( F  Fn  { A }  /\  A  e.  { A } )  ->  { ( F `  A ) }  =  ( F
" { A }
) )
132, 12mpan2 652 . . . . . . . 8  |-  ( F  Fn  { A }  ->  { ( F `  A ) }  =  ( F " { A } ) )
1411, 13eqtr4d 2401 . . . . . . 7  |-  ( F  Fn  { A }  ->  ran  F  =  {
( F `  A
) } )
15 feq3 5482 . . . . . . 7  |-  ( ran 
F  =  { ( F `  A ) }  ->  ( F : { A } --> ran  F  <->  F : { A } --> { ( F `  A ) } ) )
1614, 15syl 15 . . . . . 6  |-  ( F  Fn  { A }  ->  ( F : { A } --> ran  F  <->  F : { A } --> { ( F `  A ) } ) )
177, 16mpbid 201 . . . . 5  |-  ( F  Fn  { A }  ->  F : { A }
--> { ( F `  A ) } )
185, 17syl 15 . . . 4  |-  ( F : { A } --> B  ->  F : { A } --> { ( F `
 A ) } )
194, 18jca 518 . . 3  |-  ( F : { A } --> B  ->  ( ( F `
 A )  e.  B  /\  F : { A } --> { ( F `  A ) } ) )
20 snssi 3857 . . . 4  |-  ( ( F `  A )  e.  B  ->  { ( F `  A ) }  C_  B )
21 fss 5503 . . . . 5  |-  ( ( F : { A }
--> { ( F `  A ) }  /\  { ( F `  A
) }  C_  B
)  ->  F : { A } --> B )
2221ancoms 439 . . . 4  |-  ( ( { ( F `  A ) }  C_  B  /\  F : { A } --> { ( F `
 A ) } )  ->  F : { A } --> B )
2320, 22sylan 457 . . 3  |-  ( ( ( F `  A
)  e.  B  /\  F : { A } --> { ( F `  A ) } )  ->  F : { A } --> B )
2419, 23impbii 180 . 2  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F : { A }
--> { ( F `  A ) } ) )
25 fvex 5646 . . . 4  |-  ( F `
 A )  e. 
_V
261, 25fsn 5807 . . 3  |-  ( F : { A } --> { ( F `  A ) }  <->  F  =  { <. A ,  ( F `  A )
>. } )
2726anbi2i 675 . 2  |-  ( ( ( F `  A
)  e.  B  /\  F : { A } --> { ( F `  A ) } )  <-> 
( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )
2824, 27bitri 240 1  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873    C_ wss 3238   {csn 3729   <.cop 3732   dom cdm 4792   ran crn 4793   "cima 4795    Fn wfn 5353   -->wf 5354   ` cfv 5358
This theorem is referenced by:  fnressn  5818  fressnfv  5820  mapsnconst  6956  elixpsn  6998  en1  7071  pt1hmeo  17714  0spth  27724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366
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