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Theorem eluniima 5960
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 4061 . 2  |-  ( B  e.  U_ x  e.  A  ( F `  x )  <->  E. x  e.  A  B  e.  ( F `  x ) )
2 funiunfv 5958 . . 3  |-  ( Fun 
F  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
32eleq2d 2475 . 2  |-  ( Fun 
F  ->  ( B  e.  U_ x  e.  A  ( F `  x )  <-> 
B  e.  U. ( F " A ) ) )
41, 3syl5rbbr 252 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 177    e. wcel 1721   E.wrex 2671   U.cuni 3979   U_ciun 4057   "cima 4844   Fun wfun 5411   ` cfv 5417
This theorem is referenced by:  elunirn  5961  alephfp  7949  acsficl2d  14561  elhf  26023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-fv 5425
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