MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fun2 Structured version   Unicode version

Theorem fun2 5564
Description: The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
Assertion
Ref Expression
fun2  |-  ( ( ( F : A --> C  /\  G : B --> C )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B
) --> C )

Proof of Theorem fun2
StepHypRef Expression
1 fun 5563 . 2  |-  ( ( ( F : A --> C  /\  G : B --> C )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B
) --> ( C  u.  C ) )
2 unidm 3487 . . 3  |-  ( C  u.  C )  =  C
3 feq3 5532 . . 3  |-  ( ( C  u.  C )  =  C  ->  (
( F  u.  G
) : ( A  u.  B ) --> ( C  u.  C )  <-> 
( F  u.  G
) : ( A  u.  B ) --> C ) )
42, 3ax-mp 5 . 2  |-  ( ( F  u.  G ) : ( A  u.  B ) --> ( C  u.  C )  <->  ( F  u.  G ) : ( A  u.  B ) --> C )
51, 4sylib 196 1  |-  ( ( ( F : A --> C  /\  G : B --> C )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B
) --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    u. cun 3314    i^i cin 3315   (/)c0 3625   -->wf 5402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-opab 4339  df-id 4623  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-fun 5408  df-fn 5409  df-f 5410
This theorem is referenced by:  fresaun  5570  mapunen  7468  ac6sfi  7544  axdc3lem4  8610  fseq1p1m1  11518  axlowdimlem5  23015  axlowdimlem7  23017  uhgraun  23068  umgraun  23085  eupap1  23420  resf1o  25855
  Copyright terms: Public domain W3C validator