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Theorem fvimacnvi 5333
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 3549 . . 3  |-  ( A  e.  ( `' F " B )  ->  { A }  C_  ( `' F " B ) )
2 funimass2 5028 . . 3  |-  ( ( Fun  F  /\  { A }  C_  ( `' F " B ) )  ->  ( F " { A } ) 
C_  B )
31, 2sylan2 280 . 2  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F " { A } )  C_  B
)
4 cnvimass 4738 . . . . 5  |-  ( `' F " B ) 
C_  dom  F
54sseli 3004 . . . 4  |-  ( A  e.  ( `' F " B )  ->  A  e.  dom  F )
6 funfvex 5243 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
7 snssg 3541 . . . . 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 280 . . 3  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( ( F `  A )  e.  B  <->  { ( F `  A
) }  C_  B
) )
10 funfn 4981 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
11 fnsnfv 5284 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
1210, 11sylanb 278 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
135, 12sylan2 280 . . . 4  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  ->  { ( F `  A ) }  =  ( F " { A } ) )
1413sseq1d 3035 . . 3  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( { ( F `
 A ) } 
C_  B  <->  ( F " { A } ) 
C_  B ) )
159, 14bitrd 186 . 2  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( ( F `  A )  e.  B  <->  ( F " { A } )  C_  B
) )
163, 15mpbird 165 1  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F `  A
)  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   _Vcvv 2610    C_ wss 2982   {csn 3416   `'ccnv 4390   dom cdm 4391   "cima 4394   Fun wfun 4946    Fn wfn 4947   ` cfv 4952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960
This theorem is referenced by:  fvimacnv  5334  elpreima  5338
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