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Theorem fvun 5785
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 5487 . . 3  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
2 funfv 5782 . . 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 5277 . . . 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 4018 . 2  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  U. ( ( F  u.  G ) " { A } )  =  U. ( ( F " { A } )  u.  ( G " { A } ) ) )
7 uniun 4026 . . 3  |-  U. (
( F " { A } )  u.  ( G " { A }
) )  =  ( U. ( F " { A } )  u. 
U. ( G " { A } ) )
8 funfv 5782 . . . . . . 7  |-  ( Fun 
F  ->  ( F `  A )  =  U. ( F " { A } ) )
98eqcomd 2440 . . . . . 6  |-  ( Fun 
F  ->  U. ( F " { A }
)  =  ( F `
 A ) )
10 funfv 5782 . . . . . . 7  |-  ( Fun 
G  ->  ( G `  A )  =  U. ( G " { A } ) )
1110eqcomd 2440 . . . . . 6  |-  ( Fun 
G  ->  U. ( G " { A }
)  =  ( G `
 A ) )
129, 11anim12i 550 . . . . 5  |-  ( ( Fun  F  /\  Fun  G )  ->  ( U. ( F " { A } )  =  ( F `  A )  /\  U. ( G
" { A }
)  =  ( G `
 A ) ) )
1312adantr 452 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( U. ( F
" { A }
)  =  ( F `
 A )  /\  U. ( G " { A } )  =  ( G `  A ) ) )
14 uneq12 3488 . . . 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 2479 . 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 2471 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 359    = wceq 1652    u. cun 3310    i^i cin 3311   (/)c0 3620   {csn 3806   U.cuni 4007   dom cdm 4870   "cima 4873   Fun wfun 5440   ` cfv 5446
This theorem is referenced by:  fvun1  5786  undifixp  7090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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