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Theorem fvun 5942
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 5634 . . 3  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
2 funfv 5939 . . 3  |-  ( Fun  ( F  u.  G
)  ->  ( ( F  u.  G ) `  A )  =  U. ( ( F  u.  G ) " { A } ) )
31, 2syl 17 . 2  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( ( F  u.  G ) `  A
)  =  U. (
( F  u.  G
) " { A } ) )
4 imaundir 5260 . . . 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 4223 . 2  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  U. ( ( F  u.  G ) " { A } )  =  U. ( ( F " { A } )  u.  ( G " { A } ) ) )
7 uniun 4232 . . 3  |-  U. (
( F " { A } )  u.  ( G " { A }
) )  =  ( U. ( F " { A } )  u. 
U. ( G " { A } ) )
8 funfv 5939 . . . . . . 7  |-  ( Fun 
F  ->  ( F `  A )  =  U. ( F " { A } ) )
98eqcomd 2428 . . . . . 6  |-  ( Fun 
F  ->  U. ( F " { A }
)  =  ( F `
 A ) )
10 funfv 5939 . . . . . . 7  |-  ( Fun 
G  ->  ( G `  A )  =  U. ( G " { A } ) )
1110eqcomd 2428 . . . . . 6  |-  ( Fun 
G  ->  U. ( G " { A }
)  =  ( G `
 A ) )
129, 11anim12i 568 . . . . 5  |-  ( ( Fun  F  /\  Fun  G )  ->  ( U. ( F " { A } )  =  ( F `  A )  /\  U. ( G
" { A }
)  =  ( G `
 A ) ) )
1312adantr 466 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( U. ( F
" { A }
)  =  ( F `
 A )  /\  U. ( G " { A } )  =  ( G `  A ) ) )
14 uneq12 3612 . . . 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 17 . . 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 2473 . 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 2465 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 370    = wceq 1437    u. cun 3431    i^i cin 3432   (/)c0 3758   {csn 3993   U.cuni 4213   dom cdm 4845   "cima 4848   Fun wfun 5586   ` cfv 5592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-fv 5600
This theorem is referenced by:  fvun1  5943  undifixp  7557
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