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Theorem unpreima 5549
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 5157 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 elpreima 5543 . . . 4  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " ( A  u.  B ) )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  ( A  u.  B ) ) ) )
3 elun 3218 . . . . . 6  |-  ( x  e.  ( ( `' F " A )  u.  ( `' F " B ) )  <->  ( x  e.  ( `' F " A )  \/  x  e.  ( `' F " B ) ) )
4 elpreima 5543 . . . . . . 7  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " A )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  A
) ) )
5 elpreima 5543 . . . . . . 7  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " B )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  B
) ) )
64, 5orbi12d 783 . . . . . 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 191 . . . . 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 3218 . . . . . . 7  |-  ( ( F `  x )  e.  ( A  u.  B )  <->  ( ( F `  x )  e.  A  \/  ( F `  x )  e.  B ) )
98anbi2i 453 . . . . . 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 808 . . . . . 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 183 . . . . 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 198 . . . 4  |-  ( F  Fn  dom  F  -> 
( ( x  e. 
dom  F  /\  ( F `  x )  e.  ( A  u.  B
) )  <->  x  e.  ( ( `' F " A )  u.  ( `' F " B ) ) ) )
132, 12bitrd 187 . . 3  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " ( A  u.  B ) )  <-> 
x  e.  ( ( `' F " A )  u.  ( `' F " B ) ) ) )
1413eqrdv 2138 . 2  |-  ( F  Fn  dom  F  -> 
( `' F "
( A  u.  B
) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
151, 14sylbi 120 1  |-  ( Fun 
F  ->  ( `' F " ( A  u.  B ) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 698    = wceq 1332    e. wcel 1481    u. cun 3070   `'ccnv 4542   dom cdm 4543   "cima 4546   Fun wfun 5121    Fn wfn 5122   ` cfv 5127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-sbc 2911  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-opab 3994  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-fv 5135
This theorem is referenced by: (None)
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