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Theorem fvun 5605
Description: Value of the union of two functions when the domains are separate. (Contributed by FL, 7-Nov-2011.)
Assertion
Ref Expression
fvun  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( ( F  u.  G ) `  A
)  =  ( ( F `  A )  u.  ( G `  A ) ) )

Proof of Theorem fvun
StepHypRef Expression
1 funun 5312 . . 3  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
2 funfv 5602 . . 3  |-  ( Fun  ( F  u.  G
)  ->  ( ( F  u.  G ) `  A )  =  U. ( ( F  u.  G ) " { A } ) )
31, 2syl 15 . 2  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( ( F  u.  G ) `  A
)  =  U. (
( F  u.  G
) " { A } ) )
4 imaundir 5110 . . . 4  |-  ( ( F  u.  G )
" { A }
)  =  ( ( F " { A } )  u.  ( G " { A }
) )
54a1i 10 . . 3  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( ( F  u.  G ) " { A } )  =  ( ( F " { A } )  u.  ( G " { A }
) ) )
65unieqd 3854 . 2  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  U. ( ( F  u.  G ) " { A } )  =  U. ( ( F " { A } )  u.  ( G " { A } ) ) )
7 uniun 3862 . . 3  |-  U. (
( F " { A } )  u.  ( G " { A }
) )  =  ( U. ( F " { A } )  u. 
U. ( G " { A } ) )
8 funfv 5602 . . . . . . 7  |-  ( Fun 
F  ->  ( F `  A )  =  U. ( F " { A } ) )
98eqcomd 2301 . . . . . 6  |-  ( Fun 
F  ->  U. ( F " { A }
)  =  ( F `
 A ) )
10 funfv 5602 . . . . . . 7  |-  ( Fun 
G  ->  ( G `  A )  =  U. ( G " { A } ) )
1110eqcomd 2301 . . . . . 6  |-  ( Fun 
G  ->  U. ( G " { A }
)  =  ( G `
 A ) )
129, 11anim12i 549 . . . . 5  |-  ( ( Fun  F  /\  Fun  G )  ->  ( U. ( F " { A } )  =  ( F `  A )  /\  U. ( G
" { A }
)  =  ( G `
 A ) ) )
1312adantr 451 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( U. ( F
" { A }
)  =  ( F `
 A )  /\  U. ( G " { A } )  =  ( G `  A ) ) )
14 uneq12 3337 . . . 4  |-  ( ( U. ( F " { A } )  =  ( F `  A
)  /\  U. ( G " { A }
)  =  ( G `
 A ) )  ->  ( U. ( F " { A }
)  u.  U. ( G " { A }
) )  =  ( ( F `  A
)  u.  ( G `
 A ) ) )
1513, 14syl 15 . . 3  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( U. ( F
" { A }
)  u.  U. ( G " { A }
) )  =  ( ( F `  A
)  u.  ( G `
 A ) ) )
167, 15syl5eq 2340 . 2  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  U. ( ( F " { A } )  u.  ( G " { A } ) )  =  ( ( F `  A )  u.  ( G `  A )
) )
173, 6, 163eqtrd 2332 1  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( ( F  u.  G ) `  A
)  =  ( ( F `  A )  u.  ( G `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   U.cuni 3843   dom cdm 4705   "cima 4708   Fun wfun 5265   ` cfv 5271
This theorem is referenced by:  fvun1  5606  undifixp  6868
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-13 1698  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-pow 4204  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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