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

Theorem elima 5162
Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
Hypothesis
Ref Expression
elima.1  |-  A  e. 
_V
Assertion
Ref Expression
elima  |-  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem elima
StepHypRef Expression
1 elima.1 . 2  |-  A  e. 
_V
2 elimag 5161 . 2  |-  ( A  e.  _V  ->  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A ) )
31, 2ax-mp 5 1  |-  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1842   E.wrex 2755   _Vcvv 3059   class class class wbr 4395   "cima 4826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-cnv 4831  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836
This theorem is referenced by:  elima2  5163  rninxp  5264  imaco  5328  isarep1  5648  eliman0  5878  funimass4  5900  isomin  6216  dfsup2  7936  dfsup2OLD  7937  dfac10b  8551  hausmapdom  20293  pi1blem  21831  adjbd1o  27417  brimage  30264  dfrecs2  30288  dfrdg4  30289  dfint3  30290  imagesset  30291  elimaint  35627  elintima  35632
  Copyright terms: Public domain W3C validator