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Theorem fvimacnv 5845
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 5527 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 5842 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
2 fvex 5742 . . . . . . 7  |-  ( F `
 A )  e. 
_V
3 opelcnvg 5052 . . . . . . 7  |-  ( ( ( F `  A
)  e.  _V  /\  A  e.  dom  F )  ->  ( <. ( F `  A ) ,  A >.  e.  `' F 
<-> 
<. A ,  ( F `
 A ) >.  e.  F ) )
42, 3mpan 652 . . . . . 6  |-  ( A  e.  dom  F  -> 
( <. ( F `  A ) ,  A >.  e.  `' F  <->  <. A , 
( F `  A
) >.  e.  F ) )
54adantl 453 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( <. ( F `  A ) ,  A >.  e.  `' F  <->  <. A , 
( F `  A
) >.  e.  F ) )
61, 5mpbird 224 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. ( F `  A
) ,  A >.  e.  `' F )
7 elimasng 5230 . . . . . 6  |-  ( ( ( F `  A
)  e.  _V  /\  A  e.  dom  F )  ->  ( A  e.  ( `' F " { ( F `  A ) } )  <->  <. ( F `  A
) ,  A >.  e.  `' F ) )
82, 7mpan 652 . . . . 5  |-  ( A  e.  dom  F  -> 
( A  e.  ( `' F " { ( F `  A ) } )  <->  <. ( F `
 A ) ,  A >.  e.  `' F ) )
98adantl 453 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  ( `' F " { ( F `  A ) } )  <->  <. ( F `
 A ) ,  A >.  e.  `' F ) )
106, 9mpbird 224 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  A  e.  ( `' F " { ( F `
 A ) } ) )
112snss 3926 . . . . 5  |-  ( ( F `  A )  e.  B  <->  { ( F `  A ) }  C_  B )
12 imass2 5240 . . . . 5  |-  ( { ( F `  A
) }  C_  B  ->  ( `' F " { ( F `  A ) } ) 
C_  ( `' F " B ) )
1311, 12sylbi 188 . . . 4  |-  ( ( F `  A )  e.  B  ->  ( `' F " { ( F `  A ) } )  C_  ( `' F " B ) )
1413sseld 3347 . . 3  |-  ( ( F `  A )  e.  B  ->  ( A  e.  ( `' F " { ( F `
 A ) } )  ->  A  e.  ( `' F " B ) ) )
1510, 14syl5com 28 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  ->  A  e.  ( `' F " B ) ) )
16 fvimacnvi 5844 . . . 4  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F `  A
)  e.  B )
1716ex 424 . . 3  |-  ( Fun 
F  ->  ( A  e.  ( `' F " B )  ->  ( F `  A )  e.  B ) )
1817adantr 452 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  ( `' F " B )  ->  ( F `  A )  e.  B
) )
1915, 18impbid 184 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 177    /\ wa 359    e. wcel 1725   _Vcvv 2956    C_ wss 3320   {csn 3814   <.cop 3817   `'ccnv 4877   dom cdm 4878   "cima 4881   Fun wfun 5448   ` cfv 5454
This theorem is referenced by:  funimass3  5846  elpreima  5850  iinpreima  5860  isr0  17769  rnelfmlem  17984  rnelfm  17985  fmfnfmlem2  17987  fmfnfmlem4  17989  fmfnfm  17990  metustidOLD  18589  metustid  18590  metustsymOLD  18591  metustsym  18592  metustexhalfOLD  18593  metustexhalf  18594  xppreima  24059  dstfrvel  24731  ballotlemrv  24777  grpokerinj  26560  diaintclN  31856  dibintclN  31965  dihintcl  32142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462
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