MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfima2 Unicode version

Theorem dfima2 5013
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 4701 . 2  |-  ( A
" B )  =  ran  (  A  |`  B )
2 dfrn2 4867 . 2  |-  ran  (  A  |`  B )  =  { y  |  E. x  x ( A  |`  B ) y }
3 vex 2792 . . . . . . 7  |-  y  e. 
_V
43brres 4960 . . . . . 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 2550 . . . 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 2396 . 2  |-  { y  |  E. x  x ( A  |`  B ) y }  =  {
y  |  E. x  e.  B  x A
y }
111, 2, 103eqtri 2308 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 1685   {cab 2270   E.wrex 2545   class class class wbr 4024   ran crn 4689    |` cres 4690   "cima 4691
This theorem is referenced by:  dfima3  5014  elimag  5015  imasng  5034  dfimafn  5533  isoini  5797  dffin1-5  8010  dfimafnf  23037  imgfldref2  24474  dfaimafn  27418
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 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701
  Copyright terms: Public domain W3C validator