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Theorem fun 5599
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 5543 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  B )
)
21expcom 425 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  u.  G
)  Fn  ( A  u.  B ) ) )
3 rnun 5272 . . . . . 6  |-  ran  ( F  u.  G )  =  ( ran  F  u.  ran  G )
4 unss12 3511 . . . . . 6  |-  ( ( ran  F  C_  C  /\  ran  G  C_  D
)  ->  ( ran  F  u.  ran  G ) 
C_  ( C  u.  D ) )
53, 4syl5eqss 3384 . . . . 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 547 . . 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 5450 . . . . 5  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
9 df-f 5450 . . . . 5  |-  ( G : B --> D  <->  ( G  Fn  B  /\  ran  G  C_  D ) )
108, 9anbi12i 679 . . . 4  |-  ( ( F : A --> C  /\  G : B --> D )  <-> 
( ( F  Fn  A  /\  ran  F  C_  C )  /\  ( G  Fn  B  /\  ran  G  C_  D )
) )
11 an4 798 . . . 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 241 . . 3  |-  ( ( F : A --> C  /\  G : B --> D )  <-> 
( ( F  Fn  A  /\  G  Fn  B
)  /\  ( ran  F 
C_  C  /\  ran  G 
C_  D ) ) )
13 df-f 5450 . . 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 262 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( F : A --> C  /\  G : B --> D )  ->  ( F  u.  G ) : ( A  u.  B ) --> ( C  u.  D
) ) )
1514impcom 420 1  |-  ( ( ( F : A --> C  /\  G : B --> D )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B
) --> ( C  u.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   ran crn 4871    Fn wfn 5441   -->wf 5442
This theorem is referenced by:  fun2  5600  ftpg  5908  fsnunf  5923  hashf  11617  cats1un  11782  constr3trllem3  21631  axlowdimlem10  25882  ralxpmap  26733  mapfzcons  26763  diophrw  26808  diophren  26865  pwssplit1  27156  pwssplit4  27159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450
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