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Theorem dfimafn 5735
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 3345 . . . . . 6  |-  ( A 
C_  dom  F  ->  ( x  e.  A  ->  x  e.  dom  F ) )
2 funbrfvb 5729 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  =  y  <-> 
x F y ) )
32ex 434 . . . . . 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 432 . . . 4  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  x  e.  A
)  ->  ( ( F `  x )  =  y  <->  x F y ) )
65rexbidva 2727 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( E. x  e.  A  ( F `  x )  =  y  <->  E. x  e.  A  x F y ) )
76abbidv 2552 . 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 5166 . 2  |-  ( F
" A )  =  { y  |  E. x  e.  A  x F y }
97, 8syl6reqr 2489 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  ( F `  x )  =  y } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2424   E.wrex 2711    C_ wss 3323   class class class wbr 4287   dom cdm 4835   "cima 4838   Fun wfun 5407   ` cfv 5413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-fv 5421
This theorem is referenced by:  dfimafn2  5736  fvelimab  5742  curry2ima  25954
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