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Theorem fun2 5731
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 5730 . 2  |-  ( ( ( F : A --> C  /\  G : B --> C )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B
) --> ( C  u.  C ) )
2 unidm 3633 . . 3  |-  ( C  u.  C )  =  C
3 feq3 5697 . . 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 367    = wceq 1398    u. cun 3459    i^i cin 3460   (/)c0 3783   -->wf 5566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-fun 5572  df-fn 5573  df-f 5574
This theorem is referenced by:  fresaun  5738  mapunen  7679  ac6sfi  7756  axdc3lem4  8824  fseq1p1m1  11756  axlowdimlem5  24451  axlowdimlem7  24453  uhgraun  24513  umgraun  24530  eupap1  25178  resf1o  27784  locfinref  28079  uhgun  32752
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