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Theorem unpreima 5683
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 5284 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 elpreima 5677 . . . 4  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " ( A  u.  B ) )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  ( A  u.  B ) ) ) )
3 elun 3300 . . . . . . 7  |-  ( ( F `  x )  e.  ( A  u.  B )  <->  ( ( F `  x )  e.  A  \/  ( F `  x )  e.  B ) )
43anbi2i 457 . . . . . 6  |-  ( ( x  e.  dom  F  /\  ( F `  x
)  e.  ( A  u.  B ) )  <-> 
( x  e.  dom  F  /\  ( ( F `
 x )  e.  A  \/  ( F `
 x )  e.  B ) ) )
5 andi 819 . . . . . 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
) ) )
64, 5bitri 184 . . . . 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 ) ) )
7 elun 3300 . . . . . 6  |-  ( x  e.  ( ( `' F " A )  u.  ( `' F " B ) )  <->  ( x  e.  ( `' F " A )  \/  x  e.  ( `' F " B ) ) )
8 elpreima 5677 . . . . . . 7  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " A )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  A
) ) )
9 elpreima 5677 . . . . . . 7  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " B )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  B
) ) )
108, 9orbi12d 794 . . . . . 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 ) ) ) )
117, 10bitrid 192 . . . . 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
) ) ) )
126, 11bitr4id 199 . . . 4  |-  ( F  Fn  dom  F  -> 
( ( x  e. 
dom  F  /\  ( F `  x )  e.  ( A  u.  B
) )  <->  x  e.  ( ( `' F " A )  u.  ( `' F " B ) ) ) )
132, 12bitrd 188 . . 3  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " ( A  u.  B ) )  <-> 
x  e.  ( ( `' F " A )  u.  ( `' F " B ) ) ) )
1413eqrdv 2191 . 2  |-  ( F  Fn  dom  F  -> 
( `' F "
( A  u.  B
) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
151, 14sylbi 121 1  |-  ( Fun 
F  ->  ( `' F " ( A  u.  B ) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 709    = wceq 1364    e. wcel 2164    u. cun 3151   `'ccnv 4658   dom cdm 4659   "cima 4662   Fun wfun 5248    Fn wfn 5249   ` cfv 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262
This theorem is referenced by: (None)
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