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Theorem fnun 5685
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 5589 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 df-fn 5589 . . 3  |-  ( G  Fn  B  <->  ( Fun  G  /\  dom  G  =  B ) )
3 ineq12 3695 . . . . . . . . . . 11  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  ( dom  F  i^i  dom  G
)  =  ( A  i^i  B ) )
43eqeq1d 2469 . . . . . . . . . 10  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( dom  F  i^i  dom 
G )  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
54anbi2d 703 . . . . . . . . 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 5628 . . . . . . . . 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 5207 . . . . . . . . 9  |-  dom  ( F  u.  G )  =  ( dom  F  u.  dom  G )
9 uneq12 3653 . . . . . . . . 9  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  ( dom  F  u.  dom  G
)  =  ( A  u.  B ) )
108, 9syl5eq 2520 . . . . . . . 8  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  dom  ( F  u.  G
)  =  ( A  u.  B ) )
117, 10jctird 544 . . . . . . 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 5589 . . . . . . 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 436 . . . . 5  |-  ( ( dom  F  =  A  /\  dom  G  =  B )  ->  (
( Fun  F  /\  Fun  G )  ->  (
( A  i^i  B
)  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) ) )
1514impcom 430 . . . 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 479 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  B
) ) )
1817imp 429 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 369    = wceq 1379    u. cun 3474    i^i cin 3475   (/)c0 3785   dom cdm 4999   Fun wfun 5580    Fn wfn 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-fun 5588  df-fn 5589
This theorem is referenced by:  fnunsn  5686  fun  5746  foun  5832  f1oun  5833  undifixp  7502  brwdom2  7995  vdgrun  24577  vdgrfiun  24578  eupap1  24652  sseqfn  27969  fullfunfnv  29173  finixpnum  29615  bnj927  32906  bnj535  33027
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