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Theorem fnun 5608
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 5512 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 df-fn 5512 . . 3  |-  ( G  Fn  B  <->  ( Fun  G  /\  dom  G  =  B ) )
3 ineq12 3622 . . . . . . . . . . 11  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  ( dom  F  i^i  dom  G
)  =  ( A  i^i  B ) )
43eqeq1d 2394 . . . . . . . . . 10  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( dom  F  i^i  dom 
G )  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
54anbi2d 701 . . . . . . . . 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 5551 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
75, 6syl6bir 229 . . . . . . . 8  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( ( Fun  F  /\  Fun  G )  /\  ( A  i^i  B )  =  (/) )  ->  Fun  ( F  u.  G
) ) )
8 dmun 5135 . . . . . . . . 9  |-  dom  ( F  u.  G )  =  ( dom  F  u.  dom  G )
9 uneq12 3580 . . . . . . . . 9  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  ( dom  F  u.  dom  G
)  =  ( A  u.  B ) )
108, 9syl5eq 2445 . . . . . . . 8  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  dom  ( F  u.  G
)  =  ( A  u.  B ) )
117, 10jctird 542 . . . . . . 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 5512 . . . . . . 7  |-  ( ( F  u.  G )  Fn  ( A  u.  B )  <->  ( Fun  ( F  u.  G
)  /\  dom  ( F  u.  G )  =  ( A  u.  B
) ) )
1311, 12syl6ibr 227 . . . . . 6  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( ( Fun  F  /\  Fun  G )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) )
1413expd 434 . . . . 5  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( Fun  F  /\  Fun  G )  ->  (
( A  i^i  B
)  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) ) )
1514impcom 428 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  =  A  /\  dom  G  =  B ) )  ->  ( ( A  i^i  B )  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B )
) )
1615an4s 824 . . 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 477 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) )
1817imp 427 1  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    u. cun 3400    i^i cin 3401   (/)c0 3724   dom cdm 4926   Fun wfun 5503    Fn wfn 5504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pr 4614
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rab 2751  df-v 3049  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-sn 3958  df-pr 3960  df-op 3964  df-br 4381  df-opab 4439  df-id 4722  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-fun 5511  df-fn 5512
This theorem is referenced by:  fnunsn  5609  fun  5669  foun  5755  f1oun  5756  undifixp  7442  brwdom2  7932  vdgrun  25043  vdgrfiun  25044  eupap1  25118  sseqfn  28548  fullfunfnv  29785  finixpnum  30239  bnj927  34209  bnj535  34330
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