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Theorem fvun 5733
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 5436 . . 3  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
2 funfv 5730 . . 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 5226 . . . 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 3969 . 2  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  U. ( ( F  u.  G ) " { A } )  =  U. ( ( F " { A } )  u.  ( G " { A } ) ) )
7 uniun 3977 . . 3  |-  U. (
( F " { A } )  u.  ( G " { A }
) )  =  ( U. ( F " { A } )  u. 
U. ( G " { A } ) )
8 funfv 5730 . . . . . . 7  |-  ( Fun 
F  ->  ( F `  A )  =  U. ( F " { A } ) )
98eqcomd 2393 . . . . . 6  |-  ( Fun 
F  ->  U. ( F " { A }
)  =  ( F `
 A ) )
10 funfv 5730 . . . . . . 7  |-  ( Fun 
G  ->  ( G `  A )  =  U. ( G " { A } ) )
1110eqcomd 2393 . . . . . 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 3440 . . . 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 2432 . 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 2424 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 1649    u. cun 3262    i^i cin 3263   (/)c0 3572   {csn 3758   U.cuni 3958   dom cdm 4819   "cima 4822   Fun wfun 5389   ` cfv 5395
This theorem is referenced by:  fvun1  5734  undifixp  7035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-fv 5403
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