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Theorem dfima2 4809
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 4480 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
2 dfrn2 4655 . 2  |-  ran  ( A  |`  B )  =  { y  |  E. x  x ( A  |`  B ) y }
3 vex 2636 . . . . . . 7  |-  y  e. 
_V
43brres 4751 . . . . . 6  |-  ( x ( A  |`  B ) y  <->  ( x A y  /\  x  e.  B ) )
5 ancom 263 . . . . . 6  |-  ( ( x A y  /\  x  e.  B )  <->  ( x  e.  B  /\  x A y ) )
64, 5bitri 183 . . . . 5  |-  ( x ( A  |`  B ) y  <->  ( x  e.  B  /\  x A y ) )
76exbii 1548 . . . 4  |-  ( E. x  x ( A  |`  B ) y  <->  E. x
( x  e.  B  /\  x A y ) )
8 df-rex 2376 . . . 4  |-  ( E. x  e.  B  x A y  <->  E. x
( x  e.  B  /\  x A y ) )
97, 8bitr4i 186 . . 3  |-  ( E. x  x ( A  |`  B ) y  <->  E. x  e.  B  x A
y )
109abbii 2210 . 2  |-  { y  |  E. x  x ( A  |`  B ) y }  =  {
y  |  E. x  e.  B  x A
y }
111, 2, 103eqtri 2119 1  |-  ( A
" B )  =  { y  |  E. x  e.  B  x A y }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1296   E.wex 1433    e. wcel 1445   {cab 2081   E.wrex 2371   class class class wbr 3867   ran crn 4468    |` cres 4469   "cima 4470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-xp 4473  df-cnv 4475  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480
This theorem is referenced by:  dfima3  4810  elimag  4811  imasng  4830  imadiflem  5127  imadif  5128  imainlem  5129  imain  5130  funimaexglem  5131  dfimafn  5388  isoini  5635
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