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Theorem dfima2 5016
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 4704 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
2 dfrn2 4870 . 2  |-  ran  ( A  |`  B )  =  { y  |  E. x  x ( A  |`  B ) y }
3 vex 2793 . . . . . . 7  |-  y  e. 
_V
43brres 4963 . . . . . 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 1571 . . . 4  |-  ( E. x  x ( A  |`  B ) y  <->  E. x
( x  e.  B  /\  x A y ) )
8 df-rex 2551 . . . 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 2397 . 2  |-  { y  |  E. x  x ( A  |`  B ) y }  =  {
y  |  E. x  e.  B  x A
y }
111, 2, 103eqtri 2309 1  |-  ( A
" B )  =  { y  |  E. x  e.  B  x A y }
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1530    = wceq 1625    e. wcel 1686   {cab 2271   E.wrex 2546   class class class wbr 4025   ran crn 4692    |` cres 4693   "cima 4694
This theorem is referenced by:  dfima3  5017  elimag  5018  imasng  5037  dfimafn  5573  isoini  5837  dffin1-5  8016  dfimafnf  23043  imgfldref2  25075  dfaimafn  28038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-xp 4697  df-cnv 4699  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704
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