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Theorem fun2 5685
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 5684 . 2  |-  ( ( ( F : A --> C  /\  G : B --> C )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B
) --> ( C  u.  C ) )
2 unidm 3608 . . 3  |-  ( C  u.  C )  =  C
3 feq3 5653 . . 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 1370    u. cun 3435    i^i cin 3436   (/)c0 3746   -->wf 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-id 4745  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-fun 5529  df-fn 5530  df-f 5531
This theorem is referenced by:  fresaun  5691  mapunen  7591  ac6sfi  7668  axdc3lem4  8734  fseq1p1m1  11652  axlowdimlem5  23345  axlowdimlem7  23347  uhgraun  23398  umgraun  23415  eupap1  23750  resf1o  26182
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