Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  abfmpunirn Structured version   Unicode version

Theorem abfmpunirn 26117
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 3085 . 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 5646 . . . . 5  |-  F  Fn  V
5 fnunirn 6078 . . . . 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 5889 . . . . . . 7  |-  ( ( x  e.  V  /\  { y  |  ph }  e.  _V )  ->  ( F `  x )  =  { y  |  ph } )
82, 7mpan2 671 . . . . . 6  |-  ( x  e.  V  ->  ( F `  x )  =  { y  |  ph } )
98eleq2d 2524 . . . . 5  |-  ( x  e.  V  ->  ( B  e.  ( F `  x )  <->  B  e.  { y  |  ph }
) )
109rexbiia 2869 . . . 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 3212 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  { y  |  ph }  <->  ps )
)
1413rexbidv 2864 . . 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 642 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 369    = wceq 1370    e. wcel 1758   {cab 2439   E.wrex 2799   _Vcvv 3076   U.cuni 4198    |-> cmpt 4457   ran crn 4948    Fn wfn 5520   ` cfv 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-fv 5533
This theorem is referenced by:  rabfmpunirn  26118  isrnsigaOLD  26699  isrnsiga  26700  isrnmeas  26758
  Copyright terms: Public domain W3C validator