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Theorem fvimacnvi 5583
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 3702 . . 3  |-  ( A  e.  ( `' F " B )  ->  { A }  C_  ( `' F " B ) )
2 funimass2 5250 . . 3  |-  ( ( Fun  F  /\  { A }  C_  ( `' F " B ) )  ->  ( F " { A } ) 
C_  B )
31, 2sylan2 284 . 2  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F " { A } )  C_  B
)
4 cnvimass 4951 . . . . 5  |-  ( `' F " B ) 
C_  dom  F
54sseli 3124 . . . 4  |-  ( A  e.  ( `' F " B )  ->  A  e.  dom  F )
6 funfvex 5487 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
7 snssg 3694 . . . . 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 284 . . 3  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( ( F `  A )  e.  B  <->  { ( F `  A
) }  C_  B
) )
10 funfn 5202 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
11 fnsnfv 5529 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
1210, 11sylanb 282 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
135, 12sylan2 284 . . . 4  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  ->  { ( F `  A ) }  =  ( F " { A } ) )
1413sseq1d 3157 . . 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 1335    e. wcel 2128   _Vcvv 2712    C_ wss 3102   {csn 3561   `'ccnv 4587   dom cdm 4588   "cima 4591   Fun wfun 5166    Fn wfn 5167   ` cfv 5172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-fv 5180
This theorem is referenced by:  fvimacnv  5584  elpreima  5588
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