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Theorem dfimafn 5353
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 3019 . . . . . 6  |-  ( A 
C_  dom  F  ->  ( x  e.  A  ->  x  e.  dom  F ) )
2 funbrfvb 5347 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  =  y  <-> 
x F y ) )
32ex 113 . . . . . 6  |-  ( Fun 
F  ->  ( x  e.  dom  F  ->  (
( F `  x
)  =  y  <->  x F
y ) ) )
41, 3syl9r 72 . . . . 5  |-  ( Fun 
F  ->  ( A  C_ 
dom  F  ->  ( x  e.  A  ->  (
( F `  x
)  =  y  <->  x F
y ) ) ) )
54imp31 252 . . . 4  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  x  e.  A
)  ->  ( ( F `  x )  =  y  <->  x F y ) )
65rexbidva 2377 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( E. x  e.  A  ( F `  x )  =  y  <->  E. x  e.  A  x F y ) )
76abbidv 2205 . 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 4776 . 2  |-  ( F
" A )  =  { y  |  E. x  e.  A  x F y }
97, 8syl6reqr 2139 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074   E.wrex 2360    C_ wss 2999   class class class wbr 3845   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:  dfimafn2  5354  fvelimab  5360
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