MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fun Unicode version

Theorem fun 5405
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 5350 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  B
)  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  B )
)
21expcom 424 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  u.  G
)  Fn  ( A  u.  B ) ) )
3 rnun 5089 . . . . . 6  |-  ran  ( F  u.  G )  =  ( ran  F  u.  ran  G )
4 unss12 3347 . . . . . 6  |-  ( ( ran  F  C_  C  /\  ran  G  C_  D
)  ->  ( ran  F  u.  ran  G ) 
C_  ( C  u.  D ) )
53, 4syl5eqss 3222 . . . . 5  |-  ( ( ran  F  C_  C  /\  ran  G  C_  D
)  ->  ran  ( F  u.  G )  C_  ( C  u.  D
) )
65a1i 10 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( ran  F  C_  C  /\  ran  G  C_  D
)  ->  ran  ( F  u.  G )  C_  ( C  u.  D
) ) )
72, 6anim12d 546 . . 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 5259 . . . . 5  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
9 df-f 5259 . . . . 5  |-  ( G : B --> D  <->  ( G  Fn  B  /\  ran  G  C_  D ) )
108, 9anbi12i 678 . . . 4  |-  ( ( F : A --> C  /\  G : B --> D )  <-> 
( ( F  Fn  A  /\  ran  F  C_  C )  /\  ( G  Fn  B  /\  ran  G  C_  D )
) )
11 an4 797 . . . 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 240 . . 3  |-  ( ( F : A --> C  /\  G : B --> D )  <-> 
( ( F  Fn  A  /\  G  Fn  B
)  /\  ( ran  F 
C_  C  /\  ran  G 
C_  D ) ) )
13 df-f 5259 . . 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 261 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( F : A --> C  /\  G : B --> D )  ->  ( F  u.  G ) : ( A  u.  B ) --> ( C  u.  D
) ) )
1514impcom 419 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 358    = wceq 1623    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ran crn 4690    Fn wfn 5250   -->wf 5251
This theorem is referenced by:  fun2  5406  fsnunf  5718  hashf  11344  cats1un  11476  axlowdimlem10  23990  pgapspf  25464  ralxpmap  26173  mapfzcons  26205  diophrw  26250  ftp  26305  diophren  26308  pwssplit1  26600  pwssplit4  26603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259
  Copyright terms: Public domain W3C validator