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Theorem fvimacnv 5640
Description: The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 5326 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
Assertion
Ref Expression
fvimacnv  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  <->  A  e.  ( `' F " B ) ) )

Proof of Theorem fvimacnv
StepHypRef Expression
1 funfvop 5637 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
2 fvex 5539 . . . . . . 7  |-  ( F `
 A )  e. 
_V
3 opelcnvg 4861 . . . . . . 7  |-  ( ( ( F `  A
)  e.  _V  /\  A  e.  dom  F )  ->  ( <. ( F `  A ) ,  A >.  e.  `' F 
<-> 
<. A ,  ( F `
 A ) >.  e.  F ) )
42, 3mpan 651 . . . . . 6  |-  ( A  e.  dom  F  -> 
( <. ( F `  A ) ,  A >.  e.  `' F  <->  <. A , 
( F `  A
) >.  e.  F ) )
54adantl 452 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( <. ( F `  A ) ,  A >.  e.  `' F  <->  <. A , 
( F `  A
) >.  e.  F ) )
61, 5mpbird 223 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. ( F `  A
) ,  A >.  e.  `' F )
7 elimasng 5039 . . . . . 6  |-  ( ( ( F `  A
)  e.  _V  /\  A  e.  dom  F )  ->  ( A  e.  ( `' F " { ( F `  A ) } )  <->  <. ( F `  A
) ,  A >.  e.  `' F ) )
82, 7mpan 651 . . . . 5  |-  ( A  e.  dom  F  -> 
( A  e.  ( `' F " { ( F `  A ) } )  <->  <. ( F `
 A ) ,  A >.  e.  `' F ) )
98adantl 452 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  ( `' F " { ( F `  A ) } )  <->  <. ( F `
 A ) ,  A >.  e.  `' F ) )
106, 9mpbird 223 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  A  e.  ( `' F " { ( F `
 A ) } ) )
112snss 3748 . . . . 5  |-  ( ( F `  A )  e.  B  <->  { ( F `  A ) }  C_  B )
12 imass2 5049 . . . . 5  |-  ( { ( F `  A
) }  C_  B  ->  ( `' F " { ( F `  A ) } ) 
C_  ( `' F " B ) )
1311, 12sylbi 187 . . . 4  |-  ( ( F `  A )  e.  B  ->  ( `' F " { ( F `  A ) } )  C_  ( `' F " B ) )
1413sseld 3179 . . 3  |-  ( ( F `  A )  e.  B  ->  ( A  e.  ( `' F " { ( F `
 A ) } )  ->  A  e.  ( `' F " B ) ) )
1510, 14syl5com 26 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  ->  A  e.  ( `' F " B ) ) )
16 fvimacnvi 5639 . . . 4  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F `  A
)  e.  B )
1716ex 423 . . 3  |-  ( Fun 
F  ->  ( A  e.  ( `' F " B )  ->  ( F `  A )  e.  B ) )
1817adantr 451 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  ( `' F " B )  ->  ( F `  A )  e.  B
) )
1915, 18impbid 183 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  <->  A  e.  ( `' F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {csn 3640   <.cop 3643   `'ccnv 4688   dom cdm 4689   "cima 4692   Fun wfun 5249   ` cfv 5255
This theorem is referenced by:  funimass3  5641  elpreima  5645  iinpreima  5655  isr0  17428  rnelfmlem  17647  rnelfm  17648  fmfnfmlem2  17650  fmfnfmlem4  17652  fmfnfm  17653  ballotlemrv  23078  xppreima  23211  dstfrvel  23674  grpokerinj  26575  diaintclN  31248  dibintclN  31357  dihintcl  31534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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