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Theorem fun 5686
Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
Assertion
Ref Expression
fun  |-  ( ( ( F : A --> C  /\  G : B --> D )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B
) --> ( C  u.  D ) )

Proof of Theorem fun
StepHypRef Expression
1 fnun 5628 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  B )
)
21expcom 435 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  u.  G
)  Fn  ( A  u.  B ) ) )
3 rnun 5356 . . . . . 6  |-  ran  ( F  u.  G )  =  ( ran  F  u.  ran  G )
4 unss12 3639 . . . . . 6  |-  ( ( ran  F  C_  C  /\  ran  G  C_  D
)  ->  ( ran  F  u.  ran  G ) 
C_  ( C  u.  D ) )
53, 4syl5eqss 3511 . . . . 5  |-  ( ( ran  F  C_  C  /\  ran  G  C_  D
)  ->  ran  ( F  u.  G )  C_  ( C  u.  D
) )
65a1i 11 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( ran  F  C_  C  /\  ran  G  C_  D
)  ->  ran  ( F  u.  G )  C_  ( C  u.  D
) ) )
72, 6anim12d 563 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( ran  F 
C_  C  /\  ran  G 
C_  D ) )  ->  ( ( F  u.  G )  Fn  ( A  u.  B
)  /\  ran  ( F  u.  G )  C_  ( C  u.  D
) ) ) )
8 df-f 5533 . . . . 5  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
9 df-f 5533 . . . . 5  |-  ( G : B --> D  <->  ( G  Fn  B  /\  ran  G  C_  D ) )
108, 9anbi12i 697 . . . 4  |-  ( ( F : A --> C  /\  G : B --> D )  <-> 
( ( F  Fn  A  /\  ran  F  C_  C )  /\  ( G  Fn  B  /\  ran  G  C_  D )
) )
11 an4 820 . . . 4  |-  ( ( ( F  Fn  A  /\  ran  F  C_  C
)  /\  ( G  Fn  B  /\  ran  G  C_  D ) )  <->  ( ( F  Fn  A  /\  G  Fn  B )  /\  ( ran  F  C_  C  /\  ran  G  C_  D ) ) )
1210, 11bitri 249 . . 3  |-  ( ( F : A --> C  /\  G : B --> D )  <-> 
( ( F  Fn  A  /\  G  Fn  B
)  /\  ( ran  F 
C_  C  /\  ran  G 
C_  D ) ) )
13 df-f 5533 . . 3  |-  ( ( F  u.  G ) : ( A  u.  B ) --> ( C  u.  D )  <->  ( ( F  u.  G )  Fn  ( A  u.  B
)  /\  ran  ( F  u.  G )  C_  ( C  u.  D
) ) )
147, 12, 133imtr4g 270 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( F : A --> C  /\  G : B --> D )  ->  ( F  u.  G ) : ( A  u.  B ) --> ( C  u.  D
) ) )
1514impcom 430 1  |-  ( ( ( F : A --> C  /\  G : B --> D )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B
) --> ( C  u.  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    u. cun 3437    i^i cin 3438    C_ wss 3439   (/)c0 3748   ran crn 4952    Fn wfn 5524   -->wf 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-id 4747  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-fun 5531  df-fn 5532  df-f 5533
This theorem is referenced by:  fun2  5687  ftpg  6004  fsnunf  6028  ralxpmap  7375  hashf  12231  cats1un  12492  pwssplit1  17273  axlowdimlem10  23376  constr3trllem3  23717  eulerpartlemt  26921  sseqf  26942  mapfzcons  29223  diophrw  29268  diophren  29323  pwssplit4  29613
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