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Theorem dfimafn 5436
Description: Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
Assertion
Ref Expression
dfimafn  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  ( F `  x )  =  y } )
Distinct variable groups:    x, y, A   
x, F, y

Proof of Theorem dfimafn
StepHypRef Expression
1 ssel 3059 . . . . . 6  |-  ( A 
C_  dom  F  ->  ( x  e.  A  ->  x  e.  dom  F ) )
2 funbrfvb 5430 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  =  y  <-> 
x F y ) )
32ex 114 . . . . . 6  |-  ( Fun 
F  ->  ( x  e.  dom  F  ->  (
( F `  x
)  =  y  <->  x F
y ) ) )
41, 3syl9r 73 . . . . 5  |-  ( Fun 
F  ->  ( A  C_ 
dom  F  ->  ( x  e.  A  ->  (
( F `  x
)  =  y  <->  x F
y ) ) ) )
54imp31 254 . . . 4  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  x  e.  A
)  ->  ( ( F `  x )  =  y  <->  x F y ) )
65rexbidva 2409 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( E. x  e.  A  ( F `  x )  =  y  <->  E. x  e.  A  x F y ) )
76abbidv 2233 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  ->  { y  |  E. x  e.  A  ( F `  x )  =  y }  =  { y  |  E. x  e.  A  x F y } )
8 dfima2 4851 . 2  |-  ( F
" A )  =  { y  |  E. x  e.  A  x F y }
97, 8syl6reqr 2167 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  ( F `  x )  =  y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   {cab 2101   E.wrex 2392    C_ wss 3039   class class class wbr 3897   dom cdm 4507   "cima 4510   Fun wfun 5085   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by:  dfimafn2  5437  fvelimab  5443
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