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Theorem fvimacnv 5414
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 5092 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 5411 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
2 funfvex 5322 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
3 opelcnvg 4616 . . . . . 6  |-  ( ( ( F `  A
)  e.  _V  /\  A  e.  dom  F )  ->  ( <. ( F `  A ) ,  A >.  e.  `' F 
<-> 
<. A ,  ( F `
 A ) >.  e.  F ) )
42, 3sylancom 411 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( <. ( F `  A ) ,  A >.  e.  `' F  <->  <. A , 
( F `  A
) >.  e.  F ) )
51, 4mpbird 165 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. ( F `  A
) ,  A >.  e.  `' F )
6 elimasng 4800 . . . . 5  |-  ( ( ( F `  A
)  e.  _V  /\  A  e.  dom  F )  ->  ( A  e.  ( `' F " { ( F `  A ) } )  <->  <. ( F `  A
) ,  A >.  e.  `' F ) )
72, 6sylancom 411 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  ( `' F " { ( F `  A ) } )  <->  <. ( F `
 A ) ,  A >.  e.  `' F ) )
85, 7mpbird 165 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  A  e.  ( `' F " { ( F `
 A ) } ) )
9 snssg 3573 . . . . . . . 8  |-  ( ( F `  A )  e.  _V  ->  (
( F `  A
)  e.  B  <->  { ( F `  A ) }  C_  B ) )
102, 9syl 14 . . . . . . 7  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  <->  { ( F `  A
) }  C_  B
) )
11 imass2 4808 . . . . . . 7  |-  ( { ( F `  A
) }  C_  B  ->  ( `' F " { ( F `  A ) } ) 
C_  ( `' F " B ) )
1210, 11syl6bi 161 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  ->  ( `' F " { ( F `  A ) } ) 
C_  ( `' F " B ) ) )
1312imp 122 . . . . 5  |-  ( ( ( Fun  F  /\  A  e.  dom  F )  /\  ( F `  A )  e.  B
)  ->  ( `' F " { ( F `
 A ) } )  C_  ( `' F " B ) )
1413sseld 3024 . . . 4  |-  ( ( ( Fun  F  /\  A  e.  dom  F )  /\  ( F `  A )  e.  B
)  ->  ( A  e.  ( `' F " { ( F `  A ) } )  ->  A  e.  ( `' F " B ) ) )
1514ex 113 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  ->  ( A  e.  ( `' F " { ( F `  A ) } )  ->  A  e.  ( `' F " B ) ) ) )
168, 15mpid 41 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  ->  A  e.  ( `' F " B ) ) )
17 fvimacnvi 5413 . . . 4  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F `  A
)  e.  B )
1817ex 113 . . 3  |-  ( Fun 
F  ->  ( A  e.  ( `' F " B )  ->  ( F `  A )  e.  B ) )
1918adantr 270 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  ( `' F " B )  ->  ( F `  A )  e.  B
) )
2016, 19impbid 127 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    /\ wa 102    <-> wb 103    e. wcel 1438   _Vcvv 2619    C_ wss 2999   {csn 3446   <.cop 3449   `'ccnv 4437   dom cdm 4438   "cima 4441   Fun wfun 5009   ` cfv 5015
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by:  funimass3  5415  elpreima  5418  fisumss  10780
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