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Theorem fvun 5938
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 5630 . . 3  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
2 funfv 5935 . . 3  |-  ( Fun  ( F  u.  G
)  ->  ( ( F  u.  G ) `  A )  =  U. ( ( F  u.  G ) " { A } ) )
31, 2syl 16 . 2  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( ( F  u.  G ) `  A
)  =  U. (
( F  u.  G
) " { A } ) )
4 imaundir 5419 . . . 4  |-  ( ( F  u.  G )
" { A }
)  =  ( ( F " { A } )  u.  ( G " { A }
) )
54a1i 11 . . 3  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( ( F  u.  G ) " { A } )  =  ( ( F " { A } )  u.  ( G " { A }
) ) )
65unieqd 4255 . 2  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  U. ( ( F  u.  G ) " { A } )  =  U. ( ( F " { A } )  u.  ( G " { A } ) ) )
7 uniun 4264 . . 3  |-  U. (
( F " { A } )  u.  ( G " { A }
) )  =  ( U. ( F " { A } )  u. 
U. ( G " { A } ) )
8 funfv 5935 . . . . . . 7  |-  ( Fun 
F  ->  ( F `  A )  =  U. ( F " { A } ) )
98eqcomd 2475 . . . . . 6  |-  ( Fun 
F  ->  U. ( F " { A }
)  =  ( F `
 A ) )
10 funfv 5935 . . . . . . 7  |-  ( Fun 
G  ->  ( G `  A )  =  U. ( G " { A } ) )
1110eqcomd 2475 . . . . . 6  |-  ( Fun 
G  ->  U. ( G " { A }
)  =  ( G `
 A ) )
129, 11anim12i 566 . . . . 5  |-  ( ( Fun  F  /\  Fun  G )  ->  ( U. ( F " { A } )  =  ( F `  A )  /\  U. ( G
" { A }
)  =  ( G `
 A ) ) )
1312adantr 465 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( U. ( F
" { A }
)  =  ( F `
 A )  /\  U. ( G " { A } )  =  ( G `  A ) ) )
14 uneq12 3653 . . . 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 16 . . 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 2520 . 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 2512 1  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( ( F  u.  G ) `  A
)  =  ( ( F `  A )  u.  ( G `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    u. cun 3474    i^i cin 3475   (/)c0 3785   {csn 4027   U.cuni 4245   dom cdm 4999   "cima 5002   Fun wfun 5582   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596
This theorem is referenced by:  fvun1  5939  undifixp  7506
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