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Theorem fnun 5518
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 5424 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 df-fn 5424 . . 3  |-  ( G  Fn  B  <->  ( Fun  G  /\  dom  G  =  B ) )
3 ineq12 3505 . . . . . . . . . . 11  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  ( dom  F  i^i  dom  G
)  =  ( A  i^i  B ) )
43eqeq1d 2420 . . . . . . . . . 10  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( dom  F  i^i  dom 
G )  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
54anbi2d 685 . . . . . . . . 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 5462 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
75, 6syl6bir 221 . . . . . . . 8  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( ( Fun  F  /\  Fun  G )  /\  ( A  i^i  B )  =  (/) )  ->  Fun  ( F  u.  G
) ) )
8 dmun 5043 . . . . . . . . 9  |-  dom  ( F  u.  G )  =  ( dom  F  u.  dom  G )
9 uneq12 3464 . . . . . . . . 9  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  ( dom  F  u.  dom  G
)  =  ( A  u.  B ) )
108, 9syl5eq 2456 . . . . . . . 8  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  dom  ( F  u.  G
)  =  ( A  u.  B ) )
117, 10jctird 529 . . . . . . 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 5424 . . . . . . 7  |-  ( ( F  u.  G )  Fn  ( A  u.  B )  <->  ( Fun  ( F  u.  G
)  /\  dom  ( F  u.  G )  =  ( A  u.  B
) ) )
1311, 12syl6ibr 219 . . . . . 6  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( ( Fun  F  /\  Fun  G )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) )
1413exp3a 426 . . . . 5  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( Fun  F  /\  Fun  G )  ->  (
( A  i^i  B
)  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) ) )
1514impcom 420 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  =  A  /\  dom  G  =  B ) )  ->  ( ( A  i^i  B )  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B )
) )
1615an4s 800 . . 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 466 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) )
1817imp 419 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 359    = wceq 1649    u. cun 3286    i^i cin 3287   (/)c0 3596   dom cdm 4845   Fun wfun 5415    Fn wfn 5416
This theorem is referenced by:  fnunsn  5519  fun  5574  foun  5660  f1oun  5661  undifixp  7065  brwdom2  7505  vdgrun  21633  vdgrfiun  21634  eupap1  21659  fullfunfnv  25707  bnj927  28857  bnj535  28979
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-id 4466  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-fun 5423  df-fn 5424
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