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

Theorem eluniima 5897
Description: Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.)
Assertion
Ref Expression
eluniima  |-  ( Fun 
F  ->  ( B  e.  U. ( F " A )  <->  E. x  e.  A  B  e.  ( F `  x ) ) )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem eluniima
StepHypRef Expression
1 eliun 4011 . 2  |-  ( B  e.  U_ x  e.  A  ( F `  x )  <->  E. x  e.  A  B  e.  ( F `  x ) )
2 funiunfv 5895 . . 3  |-  ( Fun 
F  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
32eleq2d 2433 . 2  |-  ( Fun 
F  ->  ( B  e.  U_ x  e.  A  ( F `  x )  <-> 
B  e.  U. ( F " A ) ) )
41, 3syl5rbbr 251 1  |-  ( Fun 
F  ->  ( B  e.  U. ( F " A )  <->  E. x  e.  A  B  e.  ( F `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1715   E.wrex 2629   U.cuni 3929   U_ciun 4007   "cima 4795   Fun wfun 5352   ` cfv 5358
This theorem is referenced by:  elunirn  5898  alephfp  7882  acsficl2d  14489  elhf  25546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-fv 5366
  Copyright terms: Public domain W3C validator