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

Theorem unipreima 25966
Description: Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)
Assertion
Ref Expression
unipreima  |-  ( Fun 
F  ->  ( `' F " U. A )  =  U_ x  e.  A  ( `' F " x ) )
Distinct variable groups:    x, F    x, A

Proof of Theorem unipreima
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 funfn 5452 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 r19.42v 2880 . . . . . . 7  |-  ( E. x  e.  A  ( y  e.  dom  F  /\  ( F `  y
)  e.  x )  <-> 
( y  e.  dom  F  /\  E. x  e.  A  ( F `  y )  e.  x
) )
32bicomi 202 . . . . . 6  |-  ( ( y  e.  dom  F  /\  E. x  e.  A  ( F `  y )  e.  x )  <->  E. x  e.  A  ( y  e.  dom  F  /\  ( F `  y )  e.  x ) )
43a1i 11 . . . . 5  |-  ( F  Fn  dom  F  -> 
( ( y  e. 
dom  F  /\  E. x  e.  A  ( F `  y )  e.  x
)  <->  E. x  e.  A  ( y  e.  dom  F  /\  ( F `  y )  e.  x
) ) )
5 eluni2 4100 . . . . . . 7  |-  ( ( F `  y )  e.  U. A  <->  E. x  e.  A  ( F `  y )  e.  x
)
65anbi2i 694 . . . . . 6  |-  ( ( y  e.  dom  F  /\  ( F `  y
)  e.  U. A
)  <->  ( y  e. 
dom  F  /\  E. x  e.  A  ( F `  y )  e.  x
) )
76a1i 11 . . . . 5  |-  ( F  Fn  dom  F  -> 
( ( y  e. 
dom  F  /\  ( F `  y )  e.  U. A )  <->  ( y  e.  dom  F  /\  E. x  e.  A  ( F `  y )  e.  x ) ) )
8 elpreima 5828 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " x )  <-> 
( y  e.  dom  F  /\  ( F `  y )  e.  x
) ) )
98rexbidv 2741 . . . . 5  |-  ( F  Fn  dom  F  -> 
( E. x  e.  A  y  e.  ( `' F " x )  <->  E. x  e.  A  ( y  e.  dom  F  /\  ( F `  y )  e.  x
) ) )
104, 7, 93bitr4d 285 . . . 4  |-  ( F  Fn  dom  F  -> 
( ( y  e. 
dom  F  /\  ( F `  y )  e.  U. A )  <->  E. x  e.  A  y  e.  ( `' F " x ) ) )
11 elpreima 5828 . . . 4  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " U. A
)  <->  ( y  e. 
dom  F  /\  ( F `  y )  e.  U. A ) ) )
12 eliun 4180 . . . . 5  |-  ( y  e.  U_ x  e.  A  ( `' F " x )  <->  E. x  e.  A  y  e.  ( `' F " x ) )
1312a1i 11 . . . 4  |-  ( F  Fn  dom  F  -> 
( y  e.  U_ x  e.  A  ( `' F " x )  <->  E. x  e.  A  y  e.  ( `' F " x ) ) )
1410, 11, 133bitr4d 285 . . 3  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " U. A
)  <->  y  e.  U_ x  e.  A  ( `' F " x ) ) )
1514eqrdv 2441 . 2  |-  ( F  Fn  dom  F  -> 
( `' F " U. A )  =  U_ x  e.  A  ( `' F " x ) )
161, 15sylbi 195 1  |-  ( Fun 
F  ->  ( `' F " U. A )  =  U_ x  e.  A  ( `' F " x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2721   U.cuni 4096   U_ciun 4176   `'ccnv 4844   dom cdm 4845   "cima 4848   Fun wfun 5417    Fn wfn 5418   ` cfv 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-fv 5431
This theorem is referenced by:  imambfm  26682  dstrvprob  26859
  Copyright terms: Public domain W3C validator