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Theorem abfmpunirn 28197
Description: Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.)
Hypotheses
Ref Expression
abfmpunirn.1  |-  F  =  ( x  e.  V  |->  { y  |  ph } )
abfmpunirn.2  |-  { y  |  ph }  e.  _V
abfmpunirn.3  |-  ( y  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
abfmpunirn  |-  ( B  e.  U. ran  F  <->  ( B  e.  _V  /\  E. x  e.  V  ps ) )
Distinct variable groups:    x, y, B    x, F, y    x, V, y    ps, y
Allowed substitution hints:    ph( x, y)    ps( x)

Proof of Theorem abfmpunirn
StepHypRef Expression
1 elex 3031 . 2  |-  ( B  e.  U. ran  F  ->  B  e.  _V )
2 abfmpunirn.2 . . . . . 6  |-  { y  |  ph }  e.  _V
3 abfmpunirn.1 . . . . . 6  |-  F  =  ( x  e.  V  |->  { y  |  ph } )
42, 3fnmpti 5667 . . . . 5  |-  F  Fn  V
5 fnunirn 6117 . . . . 5  |-  ( F  Fn  V  ->  ( B  e.  U. ran  F  <->  E. x  e.  V  B  e.  ( F `  x
) ) )
64, 5ax-mp 5 . . . 4  |-  ( B  e.  U. ran  F  <->  E. x  e.  V  B  e.  ( F `  x
) )
73fvmpt2 5917 . . . . . . 7  |-  ( ( x  e.  V  /\  { y  |  ph }  e.  _V )  ->  ( F `  x )  =  { y  |  ph } )
82, 7mpan2 675 . . . . . 6  |-  ( x  e.  V  ->  ( F `  x )  =  { y  |  ph } )
98eleq2d 2491 . . . . 5  |-  ( x  e.  V  ->  ( B  e.  ( F `  x )  <->  B  e.  { y  |  ph }
) )
109rexbiia 2865 . . . 4  |-  ( E. x  e.  V  B  e.  ( F `  x
)  <->  E. x  e.  V  B  e.  { y  |  ph } )
116, 10bitri 252 . . 3  |-  ( B  e.  U. ran  F  <->  E. x  e.  V  B  e.  { y  |  ph } )
12 abfmpunirn.3 . . . . 5  |-  ( y  =  B  ->  ( ph 
<->  ps ) )
1312elabg 3161 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  { y  |  ph }  <->  ps )
)
1413rexbidv 2878 . . 3  |-  ( B  e.  _V  ->  ( E. x  e.  V  B  e.  { y  |  ph }  <->  E. x  e.  V  ps )
)
1511, 14syl5bb 260 . 2  |-  ( B  e.  _V  ->  ( B  e.  U. ran  F  <->  E. x  e.  V  ps ) )
161, 15biadan2 646 1  |-  ( B  e.  U. ran  F  <->  ( B  e.  _V  /\  E. x  e.  V  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2414   E.wrex 2715   _Vcvv 3022   U.cuni 4162    |-> cmpt 4425   ran crn 4797    Fn wfn 5539   ` cfv 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-fv 5552
This theorem is referenced by:  rabfmpunirn  28198  isrnsigaOLD  28886  isrnsiga  28887  isrnmeas  28974
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