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Theorem fvimacnv 5656
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 5342 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 5653 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
2 fvex 5555 . . . . . . 7  |-  ( F `
 A )  e. 
_V
3 opelcnvg 4877 . . . . . . 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 5055 . . . . . 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 3761 . . . . 5  |-  ( ( F `  A )  e.  B  <->  { ( F `  A ) }  C_  B )
12 imass2 5065 . . . . 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 3192 . . 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 5655 . . . 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 1696   _Vcvv 2801    C_ wss 3165   {csn 3653   <.cop 3656   `'ccnv 4704   dom cdm 4705   "cima 4708   Fun wfun 5265   ` cfv 5271
This theorem is referenced by:  funimass3  5657  elpreima  5661  iinpreima  5671  isr0  17444  rnelfmlem  17663  rnelfm  17664  fmfnfmlem2  17666  fmfnfmlem4  17668  fmfnfm  17669  ballotlemrv  23094  xppreima  23226  dstfrvel  23689  grpokerinj  26678  diaintclN  31870  dibintclN  31979  dihintcl  32156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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