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Theorem unpreima 5857
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 5483 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 elpreima 5851 . . . 4  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " ( A  u.  B ) )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  ( A  u.  B ) ) ) )
3 elun 3489 . . . . . 6  |-  ( x  e.  ( ( `' F " A )  u.  ( `' F " B ) )  <->  ( x  e.  ( `' F " A )  \/  x  e.  ( `' F " B ) ) )
4 elpreima 5851 . . . . . . 7  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " A )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  A
) ) )
5 elpreima 5851 . . . . . . 7  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " B )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  B
) ) )
64, 5orbi12d 692 . . . . . 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 250 . . . . 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 3489 . . . . . . 7  |-  ( ( F `  x )  e.  ( A  u.  B )  <->  ( ( F `  x )  e.  A  \/  ( F `  x )  e.  B ) )
98anbi2i 677 . . . . . 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 839 . . . . . 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 242 . . . . 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 257 . . . 4  |-  ( F  Fn  dom  F  -> 
( ( x  e. 
dom  F  /\  ( F `  x )  e.  ( A  u.  B
) )  <->  x  e.  ( ( `' F " A )  u.  ( `' F " B ) ) ) )
132, 12bitrd 246 . . 3  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " ( A  u.  B ) )  <-> 
x  e.  ( ( `' F " A )  u.  ( `' F " B ) ) ) )
1413eqrdv 2435 . 2  |-  ( F  Fn  dom  F  -> 
( `' F "
( A  u.  B
) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
151, 14sylbi 189 1  |-  ( Fun 
F  ->  ( `' F " ( A  u.  B ) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    u. cun 3319   `'ccnv 4878   dom cdm 4879   "cima 4882   Fun wfun 5449    Fn wfn 5450   ` cfv 5455
This theorem is referenced by:  sibfof  24655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463
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