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Theorem fnun 5429
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 5337 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 df-fn 5337 . . 3  |-  ( G  Fn  B  <->  ( Fun  G  /\  dom  G  =  B ) )
3 ineq12 3441 . . . . . . . . . . 11  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  ( dom  F  i^i  dom  G
)  =  ( A  i^i  B ) )
43eqeq1d 2366 . . . . . . . . . 10  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( dom  F  i^i  dom 
G )  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
54anbi2d 684 . . . . . . . . 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 5375 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
75, 6syl6bir 220 . . . . . . . 8  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( ( Fun  F  /\  Fun  G )  /\  ( A  i^i  B )  =  (/) )  ->  Fun  ( F  u.  G
) ) )
8 dmun 4964 . . . . . . . . 9  |-  dom  ( F  u.  G )  =  ( dom  F  u.  dom  G )
9 uneq12 3400 . . . . . . . . 9  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  ( dom  F  u.  dom  G
)  =  ( A  u.  B ) )
108, 9syl5eq 2402 . . . . . . . 8  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  dom  ( F  u.  G
)  =  ( A  u.  B ) )
117, 10jctird 528 . . . . . . 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 5337 . . . . . . 7  |-  ( ( F  u.  G )  Fn  ( A  u.  B )  <->  ( Fun  ( F  u.  G
)  /\  dom  ( F  u.  G )  =  ( A  u.  B
) ) )
1311, 12syl6ibr 218 . . . . . 6  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( ( Fun  F  /\  Fun  G )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) )
1413exp3a 425 . . . . 5  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( Fun  F  /\  Fun  G )  ->  (
( A  i^i  B
)  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) ) )
1514impcom 419 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  =  A  /\  dom  G  =  B ) )  ->  ( ( A  i^i  B )  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B )
) )
1615an4s 799 . . 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 465 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) )
1817imp 418 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 358    = wceq 1642    u. cun 3226    i^i cin 3227   (/)c0 3531   dom cdm 4768   Fun wfun 5328    Fn wfn 5329
This theorem is referenced by:  fnunsn  5430  fun  5485  foun  5571  f1oun  5572  undifixp  6937  brwdom2  7374  vdgrun  24297  eupap1  24304  fullfunfnv  25043  vdgreun  27792  bnj927  28545  bnj535  28667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-id 4388  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-fun 5336  df-fn 5337
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