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Theorem dfima2 4967
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 4647 . 2  |-  ( A
" B )  =  ran  (  A  |`  B )
2 dfrn2 4821 . 2  |-  ran  (  A  |`  B )  =  { y  |  E. x  x ( A  |`  B ) y }
3 vex 2743 . . . . . . 7  |-  y  e. 
_V
43brres 4914 . . . . . 6  |-  ( x ( A  |`  B ) y  <->  ( x A y  /\  x  e.  B ) )
5 ancom 439 . . . . . 6  |-  ( ( x A y  /\  x  e.  B )  <->  ( x  e.  B  /\  x A y ) )
64, 5bitri 242 . . . . 5  |-  ( x ( A  |`  B ) y  <->  ( x  e.  B  /\  x A y ) )
76exbii 1580 . . . 4  |-  ( E. x  x ( A  |`  B ) y  <->  E. x
( x  e.  B  /\  x A y ) )
8 df-rex 2521 . . . 4  |-  ( E. x  e.  B  x A y  <->  E. x
( x  e.  B  /\  x A y ) )
97, 8bitr4i 245 . . 3  |-  ( E. x  x ( A  |`  B ) y  <->  E. x  e.  B  x A
y )
109abbii 2368 . 2  |-  { y  |  E. x  x ( A  |`  B ) y }  =  {
y  |  E. x  e.  B  x A
y }
111, 2, 103eqtri 2280 1  |-  ( A
" B )  =  { y  |  E. x  e.  B  x A y }
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   {cab 2242   E.wrex 2517   class class class wbr 3963   ran crn 4627    |` cres 4628   "cima 4629
This theorem is referenced by:  dfima3  4968  elimag  4969  imasng  4988  dfimafn  5470  isoini  5734  dffin1-5  7947  dfimafnf  22967  imgfldref2  24395
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-opab 4018  df-xp 4640  df-cnv 4642  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647
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