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Theorem unpreima 5667
Description: Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
unpreima  |-  ( Fun 
F  ->  ( `' F " ( A  u.  B ) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )

Proof of Theorem unpreima
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 funfn 5299 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 elpreima 5661 . . . 4  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " ( A  u.  B ) )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  ( A  u.  B ) ) ) )
3 elun 3329 . . . . . 6  |-  ( x  e.  ( ( `' F " A )  u.  ( `' F " B ) )  <->  ( x  e.  ( `' F " A )  \/  x  e.  ( `' F " B ) ) )
4 elpreima 5661 . . . . . . 7  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " A )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  A
) ) )
5 elpreima 5661 . . . . . . 7  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " B )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  B
) ) )
64, 5orbi12d 690 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( ( x  e.  ( `' F " A )  \/  x  e.  ( `' F " B ) )  <->  ( (
x  e.  dom  F  /\  ( F `  x
)  e.  A )  \/  ( x  e. 
dom  F  /\  ( F `  x )  e.  B ) ) ) )
73, 6syl5bb 248 . . . . 5  |-  ( F  Fn  dom  F  -> 
( x  e.  ( ( `' F " A )  u.  ( `' F " B ) )  <->  ( ( x  e.  dom  F  /\  ( F `  x )  e.  A )  \/  ( x  e.  dom  F  /\  ( F `  x )  e.  B
) ) ) )
8 elun 3329 . . . . . . 7  |-  ( ( F `  x )  e.  ( A  u.  B )  <->  ( ( F `  x )  e.  A  \/  ( F `  x )  e.  B ) )
98anbi2i 675 . . . . . 6  |-  ( ( x  e.  dom  F  /\  ( F `  x
)  e.  ( A  u.  B ) )  <-> 
( x  e.  dom  F  /\  ( ( F `
 x )  e.  A  \/  ( F `
 x )  e.  B ) ) )
10 andi 837 . . . . . 6  |-  ( ( x  e.  dom  F  /\  ( ( F `  x )  e.  A  \/  ( F `  x
)  e.  B ) )  <->  ( ( x  e.  dom  F  /\  ( F `  x )  e.  A )  \/  ( x  e.  dom  F  /\  ( F `  x )  e.  B
) ) )
119, 10bitri 240 . . . . 5  |-  ( ( x  e.  dom  F  /\  ( F `  x
)  e.  ( A  u.  B ) )  <-> 
( ( x  e. 
dom  F  /\  ( F `  x )  e.  A )  \/  (
x  e.  dom  F  /\  ( F `  x
)  e.  B ) ) )
127, 11syl6rbbr 255 . . . 4  |-  ( F  Fn  dom  F  -> 
( ( x  e. 
dom  F  /\  ( F `  x )  e.  ( A  u.  B
) )  <->  x  e.  ( ( `' F " A )  u.  ( `' F " B ) ) ) )
132, 12bitrd 244 . . 3  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " ( A  u.  B ) )  <-> 
x  e.  ( ( `' F " A )  u.  ( `' F " B ) ) ) )
1413eqrdv 2294 . 2  |-  ( F  Fn  dom  F  -> 
( `' F "
( A  u.  B
) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
151, 14sylbi 187 1  |-  ( Fun 
F  ->  ( `' F " ( A  u.  B ) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    u. cun 3163   `'ccnv 4704   dom cdm 4705   "cima 4708   Fun wfun 5265    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  unpreimaOLD  26471
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  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-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-br 4040  df-opab 4094  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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