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Theorem fvimacnvi 5812
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 4012 . . 3  |-  ( A  e.  ( `' F " B )  ->  { A }  C_  ( `' F " B ) )
2 funimass2 5487 . . 3  |-  ( ( Fun  F  /\  { A }  C_  ( `' F " B ) )  ->  ( F " { A } ) 
C_  B )
31, 2sylan2 474 . 2  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F " { A } )  C_  B
)
4 fvex 5696 . . . 4  |-  ( F `
 A )  e. 
_V
54snss 3994 . . 3  |-  ( ( F `  A )  e.  B  <->  { ( F `  A ) }  C_  B )
6 cnvimass 5184 . . . . . 6  |-  ( `' F " B ) 
C_  dom  F
76sseli 3347 . . . . 5  |-  ( A  e.  ( `' F " B )  ->  A  e.  dom  F )
8 funfn 5442 . . . . . 6  |-  ( Fun 
F  <->  F  Fn  dom  F )
9 fnsnfv 5746 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
108, 9sylanb 472 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
117, 10sylan2 474 . . . 4  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  ->  { ( F `  A ) }  =  ( F " { A } ) )
1211sseq1d 3378 . . 3  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( { ( F `
 A ) } 
C_  B  <->  ( F " { A } ) 
C_  B ) )
135, 12syl5bb 257 . 2  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( ( F `  A )  e.  B  <->  ( F " { A } )  C_  B
) )
143, 13mpbird 232 1  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F `  A
)  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3323   {csn 3872   `'ccnv 4834   dom cdm 4835   "cima 4838   Fun wfun 5407    Fn wfn 5408   ` cfv 5413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-fv 5421
This theorem is referenced by:  fvimacnv  5813  elpreima  5818  iinpreima  5828  lmhmpreima  17106  mpfind  17597  ofco2  18307
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