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Theorem dfima2 5014
Description: Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dfima2  |-  ( A
" B )  =  { y  |  E. x  e.  B  x A y }
Distinct variable groups:    x, y, A    x, B, y

Proof of Theorem dfima2
StepHypRef Expression
1 df-ima 4702 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
2 dfrn2 4868 . 2  |-  ran  ( A  |`  B )  =  { y  |  E. x  x ( A  |`  B ) y }
3 vex 2791 . . . . . . 7  |-  y  e. 
_V
43brres 4961 . . . . . 6  |-  ( x ( A  |`  B ) y  <->  ( x A y  /\  x  e.  B ) )
5 ancom 437 . . . . . 6  |-  ( ( x A y  /\  x  e.  B )  <->  ( x  e.  B  /\  x A y ) )
64, 5bitri 240 . . . . 5  |-  ( x ( A  |`  B ) y  <->  ( x  e.  B  /\  x A y ) )
76exbii 1569 . . . 4  |-  ( E. x  x ( A  |`  B ) y  <->  E. x
( x  e.  B  /\  x A y ) )
8 df-rex 2549 . . . 4  |-  ( E. x  e.  B  x A y  <->  E. x
( x  e.  B  /\  x A y ) )
97, 8bitr4i 243 . . 3  |-  ( E. x  x ( A  |`  B ) y  <->  E. x  e.  B  x A
y )
109abbii 2395 . 2  |-  { y  |  E. x  x ( A  |`  B ) y }  =  {
y  |  E. x  e.  B  x A
y }
111, 2, 103eqtri 2307 1  |-  ( A
" B )  =  { y  |  E. x  e.  B  x A y }
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   class class class wbr 4023   ran crn 4690    |` cres 4691   "cima 4692
This theorem is referenced by:  dfima3  5015  elimag  5016  imasng  5035  dfimafn  5571  isoini  5835  dffin1-5  8014  dfimafnf  23041  imgfldref2  25064  dfaimafn  28027
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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