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Theorem fun 5540
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 5484 . . . . 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 5213 . . . . . 6  |-  ran  ( F  u.  G )  =  ( ran  F  u.  ran  G )
4 unss12 3455 . . . . . 6  |-  ( ( ran  F  C_  C  /\  ran  G  C_  D
)  ->  ( ran  F  u.  ran  G ) 
C_  ( C  u.  D ) )
53, 4syl5eqss 3328 . . . . 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 5391 . . . . 5  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
9 df-f 5391 . . . . 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 5391 . . 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 1649    u. cun 3254    i^i cin 3255    C_ wss 3256   (/)c0 3564   ran crn 4812    Fn wfn 5382   -->wf 5383
This theorem is referenced by:  fun2  5541  ftpg  5848  fsnunf  5863  hashf  11545  cats1un  11710  constr3trllem3  21480  axlowdimlem10  25597  ralxpmap  26426  mapfzcons  26456  diophrw  26501  diophren  26558  pwssplit1  26850  pwssplit4  26853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-id 4432  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-fun 5389  df-fn 5390  df-f 5391
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