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Theorem fvimacnvi 5643
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 3748 . . 3  |-  ( A  e.  ( `' F " B )  ->  { A }  C_  ( `' F " B ) )
2 funimass2 5306 . . 3  |-  ( ( Fun  F  /\  { A }  C_  ( `' F " B ) )  ->  ( F " { A } ) 
C_  B )
31, 2sylan2 286 . 2  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F " { A } )  C_  B
)
4 cnvimass 5003 . . . . 5  |-  ( `' F " B ) 
C_  dom  F
54sseli 3163 . . . 4  |-  ( A  e.  ( `' F " B )  ->  A  e.  dom  F )
6 funfvex 5544 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
7 snssg 3738 . . . . 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 286 . . 3  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( ( F `  A )  e.  B  <->  { ( F `  A
) }  C_  B
) )
10 funfn 5258 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
11 fnsnfv 5588 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
1210, 11sylanb 284 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
135, 12sylan2 286 . . . 4  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  ->  { ( F `  A ) }  =  ( F " { A } ) )
1413sseq1d 3196 . . 3  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( { ( F `
 A ) } 
C_  B  <->  ( F " { A } ) 
C_  B ) )
159, 14bitrd 188 . 2  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( ( F `  A )  e.  B  <->  ( F " { A } )  C_  B
) )
163, 15mpbird 167 1  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F `  A
)  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1363    e. wcel 2158   _Vcvv 2749    C_ wss 3141   {csn 3604   `'ccnv 4637   dom cdm 4638   "cima 4641   Fun wfun 5222    Fn wfn 5223   ` cfv 5228
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236
This theorem is referenced by:  fvimacnv  5644  elpreima  5648
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