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Theorem unpreima 5408
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 5031 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 elpreima 5402 . . . 4  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " ( A  u.  B ) )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  ( A  u.  B ) ) ) )
3 elun 3139 . . . . . 6  |-  ( x  e.  ( ( `' F " A )  u.  ( `' F " B ) )  <->  ( x  e.  ( `' F " A )  \/  x  e.  ( `' F " B ) ) )
4 elpreima 5402 . . . . . . 7  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " A )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  A
) ) )
5 elpreima 5402 . . . . . . 7  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " B )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  B
) ) )
64, 5orbi12d 742 . . . . . 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 190 . . . . 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 3139 . . . . . . 7  |-  ( ( F `  x )  e.  ( A  u.  B )  <->  ( ( F `  x )  e.  A  \/  ( F `  x )  e.  B ) )
98anbi2i 445 . . . . . 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 767 . . . . . 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 182 . . . . 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 197 . . . 4  |-  ( F  Fn  dom  F  -> 
( ( x  e. 
dom  F  /\  ( F `  x )  e.  ( A  u.  B
) )  <->  x  e.  ( ( `' F " A )  u.  ( `' F " B ) ) ) )
132, 12bitrd 186 . . 3  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " ( A  u.  B ) )  <-> 
x  e.  ( ( `' F " A )  u.  ( `' F " B ) ) ) )
1413eqrdv 2086 . 2  |-  ( F  Fn  dom  F  -> 
( `' F "
( A  u.  B
) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
151, 14sylbi 119 1  |-  ( Fun 
F  ->  ( `' F " ( A  u.  B ) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438    u. cun 2995   `'ccnv 4427   dom cdm 4428   "cima 4431   Fun wfun 4996    Fn wfn 4997   ` cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010
This theorem is referenced by: (None)
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