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Theorem fun 5738
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 5677 . . . . 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 5404 . . . . . 6  |-  ran  ( F  u.  G )  =  ( ran  F  u.  ran  G )
4 unss12 3661 . . . . . 6  |-  ( ( ran  F  C_  C  /\  ran  G  C_  D
)  ->  ( ran  F  u.  ran  G ) 
C_  ( C  u.  D ) )
53, 4syl5eqss 3533 . . . . 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 5582 . . . . 5  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
9 df-f 5582 . . . . 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 824 . . . 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 5582 . . 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 1383    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3770   ran crn 4990    Fn wfn 5573   -->wf 5574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-id 4785  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-fun 5580  df-fn 5581  df-f 5582
This theorem is referenced by:  fun2  5739  ftpg  6066  fsnunf  6094  ralxpmap  7470  hashf  12391  cats1un  12680  pwssplit1  17579  axlowdimlem10  24126  constr3trllem3  24524  eulerpartlemt  28183  sseqf  28204  mapfzcons  30623  diophrw  30667  diophren  30722  pwssplit4  31010
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