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Theorem fvimacnv 5695
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 5352 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 5692 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
2 funfvex 5593 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
3 opelcnvg 4858 . . . . . 6  |-  ( ( ( F `  A
)  e.  _V  /\  A  e.  dom  F )  ->  ( <. ( F `  A ) ,  A >.  e.  `' F 
<-> 
<. A ,  ( F `
 A ) >.  e.  F ) )
42, 3sylancom 420 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( <. ( F `  A ) ,  A >.  e.  `' F  <->  <. A , 
( F `  A
) >.  e.  F ) )
51, 4mpbird 167 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. ( F `  A
) ,  A >.  e.  `' F )
6 elimasng 5050 . . . . 5  |-  ( ( ( F `  A
)  e.  _V  /\  A  e.  dom  F )  ->  ( A  e.  ( `' F " { ( F `  A ) } )  <->  <. ( F `  A
) ,  A >.  e.  `' F ) )
72, 6sylancom 420 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  ( `' F " { ( F `  A ) } )  <->  <. ( F `
 A ) ,  A >.  e.  `' F ) )
85, 7mpbird 167 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  A  e.  ( `' F " { ( F `
 A ) } ) )
9 snssg 3767 . . . . . . . 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 5058 . . . . . . 7  |-  ( { ( F `  A
) }  C_  B  ->  ( `' F " { ( F `  A ) } ) 
C_  ( `' F " B ) )
1210, 11biimtrdi 163 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  ->  ( `' F " { ( F `  A ) } ) 
C_  ( `' F " B ) ) )
1312imp 124 . . . . 5  |-  ( ( ( Fun  F  /\  A  e.  dom  F )  /\  ( F `  A )  e.  B
)  ->  ( `' F " { ( F `
 A ) } )  C_  ( `' F " B ) )
1413sseld 3192 . . . 4  |-  ( ( ( Fun  F  /\  A  e.  dom  F )  /\  ( F `  A )  e.  B
)  ->  ( A  e.  ( `' F " { ( F `  A ) } )  ->  A  e.  ( `' F " B ) ) )
1514ex 115 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  ->  ( A  e.  ( `' F " { ( F `  A ) } )  ->  A  e.  ( `' F " B ) ) ) )
168, 15mpid 42 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  ->  A  e.  ( `' F " B ) ) )
17 fvimacnvi 5694 . . . 4  |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  -> 
( F `  A
)  e.  B )
1817ex 115 . . 3  |-  ( Fun 
F  ->  ( A  e.  ( `' F " B )  ->  ( F `  A )  e.  B ) )
1918adantr 276 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  ( `' F " B )  ->  ( F `  A )  e.  B
) )
2016, 19impbid 129 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 104    <-> wb 105    e. wcel 2176   _Vcvv 2772    C_ wss 3166   {csn 3633   <.cop 3636   `'ccnv 4674   dom cdm 4675   "cima 4678   Fun wfun 5265   ` cfv 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279
This theorem is referenced by:  funimass3  5696  elpreima  5699  fisumss  11703  psrbaglesuppg  14434
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