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Theorem abfmpunirn 27711
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 3115 . 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 5691 . . . . 5  |-  F  Fn  V
5 fnunirn 6140 . . . . 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 5939 . . . . . . 7  |-  ( ( x  e.  V  /\  { y  |  ph }  e.  _V )  ->  ( F `  x )  =  { y  |  ph } )
82, 7mpan2 669 . . . . . 6  |-  ( x  e.  V  ->  ( F `  x )  =  { y  |  ph } )
98eleq2d 2524 . . . . 5  |-  ( x  e.  V  ->  ( B  e.  ( F `  x )  <->  B  e.  { y  |  ph }
) )
109rexbiia 2955 . . . 4  |-  ( E. x  e.  V  B  e.  ( F `  x
)  <->  E. x  e.  V  B  e.  { y  |  ph } )
116, 10bitri 249 . . 3  |-  ( B  e.  U. ran  F  <->  E. x  e.  V  B  e.  { y  |  ph } )
12 abfmpunirn.3 . . . . 5  |-  ( y  =  B  ->  ( ph 
<->  ps ) )
1312elabg 3244 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  { y  |  ph }  <->  ps )
)
1413rexbidv 2965 . . 3  |-  ( B  e.  _V  ->  ( E. x  e.  V  B  e.  { y  |  ph }  <->  E. x  e.  V  ps )
)
1511, 14syl5bb 257 . 2  |-  ( B  e.  _V  ->  ( B  e.  U. ran  F  <->  E. x  e.  V  ps ) )
161, 15biadan2 640 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 184    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439   E.wrex 2805   _Vcvv 3106   U.cuni 4235    |-> cmpt 4497   ran crn 4989    Fn wfn 5565   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578
This theorem is referenced by:  rabfmpunirn  27712  isrnsigaOLD  28342  isrnsiga  28343  isrnmeas  28408
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