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Theorem fnun 5554
Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
Assertion
Ref Expression
fnun  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  B )
)

Proof of Theorem fnun
StepHypRef Expression
1 df-fn 5460 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 df-fn 5460 . . 3  |-  ( G  Fn  B  <->  ( Fun  G  /\  dom  G  =  B ) )
3 ineq12 3539 . . . . . . . . . . 11  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  ( dom  F  i^i  dom  G
)  =  ( A  i^i  B ) )
43eqeq1d 2446 . . . . . . . . . 10  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( dom  F  i^i  dom 
G )  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
54anbi2d 686 . . . . . . . . 9  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G )  =  (/) )  <->  ( ( Fun  F  /\  Fun  G
)  /\  ( A  i^i  B )  =  (/) ) ) )
6 funun 5498 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
75, 6syl6bir 222 . . . . . . . 8  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( ( Fun  F  /\  Fun  G )  /\  ( A  i^i  B )  =  (/) )  ->  Fun  ( F  u.  G
) ) )
8 dmun 5079 . . . . . . . . 9  |-  dom  ( F  u.  G )  =  ( dom  F  u.  dom  G )
9 uneq12 3498 . . . . . . . . 9  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  ( dom  F  u.  dom  G
)  =  ( A  u.  B ) )
108, 9syl5eq 2482 . . . . . . . 8  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  dom  ( F  u.  G
)  =  ( A  u.  B ) )
117, 10jctird 530 . . . . . . 7  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( ( Fun  F  /\  Fun  G )  /\  ( A  i^i  B )  =  (/) )  ->  ( Fun  ( F  u.  G
)  /\  dom  ( F  u.  G )  =  ( A  u.  B
) ) ) )
12 df-fn 5460 . . . . . . 7  |-  ( ( F  u.  G )  Fn  ( A  u.  B )  <->  ( Fun  ( F  u.  G
)  /\  dom  ( F  u.  G )  =  ( A  u.  B
) ) )
1311, 12syl6ibr 220 . . . . . 6  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( ( Fun  F  /\  Fun  G )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) )
1413exp3a 427 . . . . 5  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( Fun  F  /\  Fun  G )  ->  (
( A  i^i  B
)  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) ) )
1514impcom 421 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  =  A  /\  dom  G  =  B ) )  ->  ( ( A  i^i  B )  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B )
) )
1615an4s 801 . . 3  |-  ( ( ( Fun  F  /\  dom  F  =  A )  /\  ( Fun  G  /\  dom  G  =  B ) )  ->  (
( A  i^i  B
)  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) )
171, 2, 16syl2anb 467 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) )
1817imp 420 1  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    u. cun 3320    i^i cin 3321   (/)c0 3630   dom cdm 4881   Fun wfun 5451    Fn wfn 5452
This theorem is referenced by:  fnunsn  5555  fun  5610  foun  5696  f1oun  5697  undifixp  7101  brwdom2  7544  vdgrun  21677  vdgrfiun  21678  eupap1  21703  fullfunfnv  25796  bnj927  29213  bnj535  29335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-id 4501  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-fun 5459  df-fn 5460
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