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Theorem abfmpunirn 23231
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 2809 . 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 5388 . . . . 5  |-  F  Fn  V
5 fnunirn 5794 . . . . 5  |-  ( F  Fn  V  ->  ( B  e.  U. ran  F  <->  E. x  e.  V  B  e.  ( F `  x
) ) )
64, 5ax-mp 8 . . . 4  |-  ( B  e.  U. ran  F  <->  E. x  e.  V  B  e.  ( F `  x
) )
73fvmpt2 5624 . . . . . . 7  |-  ( ( x  e.  V  /\  { y  |  ph }  e.  _V )  ->  ( F `  x )  =  { y  |  ph } )
82, 7mpan2 652 . . . . . 6  |-  ( x  e.  V  ->  ( F `  x )  =  { y  |  ph } )
9 eleq2 2357 . . . . . 6  |-  ( ( F `  x )  =  { y  | 
ph }  ->  ( B  e.  ( F `  x )  <->  B  e.  { y  |  ph }
) )
108, 9syl 15 . . . . 5  |-  ( x  e.  V  ->  ( B  e.  ( F `  x )  <->  B  e.  { y  |  ph }
) )
1110rexbiia 2589 . . . 4  |-  ( E. x  e.  V  B  e.  ( F `  x
)  <->  E. x  e.  V  B  e.  { y  |  ph } )
126, 11bitri 240 . . 3  |-  ( B  e.  U. ran  F  <->  E. x  e.  V  B  e.  { y  |  ph } )
13 abfmpunirn.3 . . . . 5  |-  ( y  =  B  ->  ( ph 
<->  ps ) )
1413elabg 2928 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  { y  |  ph }  <->  ps )
)
1514rexbidv 2577 . . 3  |-  ( B  e.  _V  ->  ( E. x  e.  V  B  e.  { y  |  ph }  <->  E. x  e.  V  ps )
)
1612, 15syl5bb 248 . 2  |-  ( B  e.  _V  ->  ( B  e.  U. ran  F  <->  E. x  e.  V  ps ) )
171, 16biadan2 623 1  |-  ( B  e.  U. ran  F  <->  ( B  e.  _V  /\  E. x  e.  V  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   U.cuni 3843    e. cmpt 4093   ran crn 4706    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  rabfmpunirn  23232  isrnsigaOLD  23488  isrnsiga  23489  isrnmeas  23546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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