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

Theorem elima 5335
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 5334 . 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 1762   E.wrex 2810   _Vcvv 3108   class class class wbr 4442   "cima 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-xp 5000  df-cnv 5002  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007
This theorem is referenced by:  elima2  5336  rninxp  5439  imaco  5505  isarep1  5660  eliman0  5888  funimass4  5911  isomin  6214  dfsup2  7893  dfsup2OLD  7894  dfac10b  8510  hausmapdom  19762  pi1blem  21269  adjbd1o  26668  brimage  29141  dfrdg4  29165  tfrqfree  29166  dfint3  29167  imagesset  29168
  Copyright terms: Public domain W3C validator