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Theorem fvimacnvi 5427
Description: A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)
Assertion
Ref Expression
fvimacnvi  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F `  A
)  e.  B )

Proof of Theorem fvimacnvi
StepHypRef Expression
1 snssi 3587 . . 3  |-  ( A  e.  ( `' F " B )  ->  { A }  C_  ( `' F " B ) )
2 funimass2 5105 . . 3  |-  ( ( Fun  F  /\  { A }  C_  ( `' F " B ) )  ->  ( F " { A } ) 
C_  B )
31, 2sylan2 281 . 2  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F " { A } )  C_  B
)
4 cnvimass 4808 . . . . 5  |-  ( `' F " B ) 
C_  dom  F
54sseli 3022 . . . 4  |-  ( A  e.  ( `' F " B )  ->  A  e.  dom  F )
6 funfvex 5335 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
7 snssg 3579 . . . . 5  |-  ( ( F `  A )  e.  _V  ->  (
( F `  A
)  e.  B  <->  { ( F `  A ) }  C_  B ) )
86, 7syl 14 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  <->  { ( F `  A
) }  C_  B
) )
95, 8sylan2 281 . . 3  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( ( F `  A )  e.  B  <->  { ( F `  A
) }  C_  B
) )
10 funfn 5058 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
11 fnsnfv 5376 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
1210, 11sylanb 279 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
135, 12sylan2 281 . . . 4  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  ->  { ( F `  A ) }  =  ( F " { A } ) )
1413sseq1d 3054 . . 3  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( { ( F `
 A ) } 
C_  B  <->  ( F " { A } ) 
C_  B ) )
159, 14bitrd 187 . 2  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( ( F `  A )  e.  B  <->  ( F " { A } )  C_  B
) )
163, 15mpbird 166 1  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F `  A
)  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   _Vcvv 2620    C_ wss 3000   {csn 3450   `'ccnv 4450   dom cdm 4451   "cima 4454   Fun wfun 5022    Fn wfn 5023   ` cfv 5028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-fv 5036
This theorem is referenced by:  fvimacnv  5428  elpreima  5432
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