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Theorem dfimafn 5897
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 3435 . . . . . 6  |-  ( A 
C_  dom  F  ->  ( x  e.  A  ->  x  e.  dom  F ) )
2 funbrfvb 5890 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  =  y  <-> 
x F y ) )
32ex 432 . . . . . 6  |-  ( Fun 
F  ->  ( x  e.  dom  F  ->  (
( F `  x
)  =  y  <->  x F
y ) ) )
41, 3syl9r 71 . . . . 5  |-  ( Fun 
F  ->  ( A  C_ 
dom  F  ->  ( x  e.  A  ->  (
( F `  x
)  =  y  <->  x F
y ) ) ) )
54imp31 430 . . . 4  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  x  e.  A
)  ->  ( ( F `  x )  =  y  <->  x F y ) )
65rexbidva 2914 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( E. x  e.  A  ( F `  x )  =  y  <->  E. x  e.  A  x F y ) )
76abbidv 2538 . 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 5158 . 2  |-  ( F
" A )  =  { y  |  E. x  e.  A  x F y }
97, 8syl6reqr 2462 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 367    = wceq 1405    e. wcel 1842   {cab 2387   E.wrex 2754    C_ wss 3413   class class class wbr 4394   dom cdm 4822   "cima 4825   Fun wfun 5562   ` cfv 5568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-fv 5576
This theorem is referenced by:  dfimafn2  5898  fvelimab  5904  curry2ima  27957
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