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Theorem fsn2 6047
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 4044 . . . . 5  |-  A  e. 
{ A }
3 ffvelrn 6005 . . . . 5  |-  ( ( F : { A }
--> B  /\  A  e. 
{ A } )  ->  ( F `  A )  e.  B
)
42, 3mpan2 669 . . . 4  |-  ( F : { A } --> B  ->  ( F `  A )  e.  B
)
5 ffn 5713 . . . . 5  |-  ( F : { A } --> B  ->  F  Fn  { A } )
6 dffn3 5720 . . . . . . 7  |-  ( F  Fn  { A }  <->  F : { A } --> ran  F )
76biimpi 194 . . . . . 6  |-  ( F  Fn  { A }  ->  F : { A }
--> ran  F )
8 imadmrn 5335 . . . . . . . . 9  |-  ( F
" dom  F )  =  ran  F
9 fndm 5662 . . . . . . . . . 10  |-  ( F  Fn  { A }  ->  dom  F  =  { A } )
109imaeq2d 5325 . . . . . . . . 9  |-  ( F  Fn  { A }  ->  ( F " dom  F )  =  ( F
" { A }
) )
118, 10syl5eqr 2509 . . . . . . . 8  |-  ( F  Fn  { A }  ->  ran  F  =  ( F " { A } ) )
12 fnsnfv 5908 . . . . . . . . 9  |-  ( ( F  Fn  { A }  /\  A  e.  { A } )  ->  { ( F `  A ) }  =  ( F
" { A }
) )
132, 12mpan2 669 . . . . . . . 8  |-  ( F  Fn  { A }  ->  { ( F `  A ) }  =  ( F " { A } ) )
1411, 13eqtr4d 2498 . . . . . . 7  |-  ( F  Fn  { A }  ->  ran  F  =  {
( F `  A
) } )
1514feq3d 5701 . . . . . 6  |-  ( F  Fn  { A }  ->  ( F : { A } --> ran  F  <->  F : { A } --> { ( F `  A ) } ) )
167, 15mpbid 210 . . . . 5  |-  ( F  Fn  { A }  ->  F : { A }
--> { ( F `  A ) } )
175, 16syl 16 . . . 4  |-  ( F : { A } --> B  ->  F : { A } --> { ( F `
 A ) } )
184, 17jca 530 . . 3  |-  ( F : { A } --> B  ->  ( ( F `
 A )  e.  B  /\  F : { A } --> { ( F `  A ) } ) )
19 snssi 4160 . . . 4  |-  ( ( F `  A )  e.  B  ->  { ( F `  A ) }  C_  B )
20 fss 5721 . . . . 5  |-  ( ( F : { A }
--> { ( F `  A ) }  /\  { ( F `  A
) }  C_  B
)  ->  F : { A } --> B )
2120ancoms 451 . . . 4  |-  ( ( { ( F `  A ) }  C_  B  /\  F : { A } --> { ( F `
 A ) } )  ->  F : { A } --> B )
2219, 21sylan 469 . . 3  |-  ( ( ( F `  A
)  e.  B  /\  F : { A } --> { ( F `  A ) } )  ->  F : { A } --> B )
2318, 22impbii 188 . 2  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F : { A }
--> { ( F `  A ) } ) )
24 fvex 5858 . . . 4  |-  ( F `
 A )  e. 
_V
251, 24fsn 6045 . . 3  |-  ( F : { A } --> { ( F `  A ) }  <->  F  =  { <. A ,  ( F `  A )
>. } )
2625anbi2i 692 . 2  |-  ( ( ( F `  A
)  e.  B  /\  F : { A } --> { ( F `  A ) } )  <-> 
( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )
2723, 26bitri 249 1  |-  ( F : { A } --> B 
<->  ( ( F `  A )  e.  B  /\  F  =  { <. A ,  ( F `
 A ) >. } ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   {csn 4016   <.cop 4022   dom cdm 4988   ran crn 4989   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578
This theorem is referenced by:  fnressn  6059  fressnfv  6061  mapsnconst  7457  elixpsn  7501  en1  7575  mat1dimelbas  19143  pt1hmeo  20476  0spth  24778  ldepsnlinclem1  33379  ldepsnlinclem2  33380
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